estimation of deformation induced martensite in austenitic stainless steels
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Materials Science and Engineering A 529 (2011) 9– 20
Contents lists available at SciVerse ScienceDirect
Materials Science and Engineering A
jo ur n al hom epage: www.elsev ier .com/ locate /msea
stimation of deformation induced martensite in austenitic stainless steels
rpan Dasa,b,∗, Soumitra Tarafdera, Pravash Chandra Chakrabortib
Fatigue & Fracture Group, National Metallurgical Laboratory (Council of Scientific & Industrial Research), Jamshedpur 831007, IndiaDepartment of Metallurgical & Material Engineering, Jadavpur University, Kolkata 700 032, India
r t i c l e i n f o
rticle history:eceived 4 February 2011eceived in revised form 23 June 2011ccepted 19 August 2011vailable online 27 August 2011
a b s t r a c t
The extent of deformation induced martensite (DIM) is controlled by steel chemistry, strain rate, stress,strain, grain size, stress state, initial texture and temperature of deformation. In this research, a neuralnetwork model within a Bayesian framework has been created using extensive published data correlatingthe extent of DIM with its influencing parameters in a variety of austenitic grade stainless steels. TheBayesian method puts error bars on the predicted value of the rate and allows the significance of each
eywords:eformation induced martensiteustenitic stainless steelsartensitic transformation
ignificanceayesian neural network
individual parameter to be estimated. In addition, it is possible to estimate the isolated influence ofparticular variable such as grain size, which cannot in practice be varied independently. This demonstratesthe ability of the method to investigate the new phenomena in cases where the information cannot beaccessed experimentally. The model has been applied to confirm that the predictions are reasonable inthe context of metallurgical principles, present experimental data and other recent data published in theliteratures.
© 2011 Elsevier B.V. All rights reserved.
. Introduction
Austenitic stainless steels are extensively used in engineer-ng applications, nuclear power plant components, automobilend pharmaceutical industries due to their excellent corrosionesistance, weldability and mechanical properties. Metastableustenitic stainless steels undergo DIM transformation, where the
(fcc) austenite is transformed to thermodynamically more stable′ (bcc) martensite due to plastic deformation. This phase trans-
ormation enhances the work hardening of these steels, and affectsheir ductility [1].
Furthermore, the microstructural evolution and the mechani-al behaviour are sensitive to chemical composition, temperature,tress, strain, strain path, strain rate, stress state, grain size, andnitial crystallographic microtexture. Understanding the influencef these factors, resulting microstructures and the correspondingechanical behaviour are the most important part not only in terms
f the selection of the best material, but also in the optimal develop-ent of material models, which are nowadays extensively applied
n the automobile and nuclear power plant industries to under-tand their forming and crash related performances. Considerablettention was given in the past to the microstructure of austenitic
∗ Corresponding author at: Fatigue & Fracture Group, National Metallurgical Lab-ratory (Council of Scientific & Industrial Research), Jamshedpur 831007, India.el.: +91 9934328051; fax: +91 657 2345213.
E-mail address: [email protected] (A. Das).
921-5093/$ – see front matter © 2011 Elsevier B.V. All rights reserved.oi:10.1016/j.msea.2011.08.039
stainless steels, the stability of the phases present in these steelsand the effects of amount and distribution of the phases presenton the mechanical behaviour of the material under service. Themechanical properties of metastable austenitic stainless steels arestrongly influenced by the morphologies and the extent of defor-mation induced phase transformation.
In the past, there has been a constantly increasing interest forneural network modelling in different fields of materials science [2].Several models have been developed for prediction of mechanicalproperties, phase transformations, optimizing alloy composition,processing parameters, heat treatment conditions, on line corro-sion monitoring, improving weldability, etc. [2]. This empiricalapproach becomes more attractive as it is fairly the robust tech-nique and in most cases, it rapidly converges to a target solution.This provides a range of powerful new techniques for solving prob-lems in pattern recognition, data analysis and control.
The purpose of the work presented here is to develop a model,which makes possible the estimation of DIM content as func-tion of its influencing variables using neural network techniquewithin a Bayesian framework [3]. This model would tremendouslyhelp to the nuclear power plant, automobiles and pharmaceuti-cal industries to design their components under service. In thepresent context, the optimization process needs access to a quanti-tative relationship between the chemical composition of austenitic
stainless steels, grain size, stress, strain, temperature, strain rateand the ultimate the extent of DIM. A neural network methodhas been developed to correlate those and applied extensively forapplications.
10 A. Das et al. / Materials Science and Engineering A 529 (2011) 9– 20
Table 1Statistics of database used for neural network analysis. SR: strain rate, T: temperature, GS: grain size, TSS: true stress, TSN: true strain, DIM: deformation induced martensiteand SD: standard deviation. The column marked ‘Example’ is a specific case used to generate Fig. 14.
Inputs Units Maximum Minimum Mean SD Example
C wt% 0.10 0.007 0.05 0.03 0.028Mn wt% 8.92 0.42 1.761 1.61 1.32Cr wt% 18.58 15.40 17.78 0.68 18.13Ni wt% 13.53 2.75 8.03 1.59 8.32Mo wt% 2.53 0 0.31 0.53 0.15N wt% 0.24 0 0.05 0.05 0.044Cu wt% 0.70 0 0.16 0.15 0.26Nb wt% 0.11 0 0.006 0.016 0.015Co wt% 0.20 0 0.042 0.07 0.10Ti wt% 0.67 0 0.01 0.08 0.01SR s−1 200 0.0001 6.44 34.74 0.000125T ◦C 200 −196 −1.49 66.79 24GS �m 200 5.90 29.19 28.16 23.8TSS MPa 1951.17 14.22 848.33 282.78 1078.33TSN – 0.65 0 0.25 0.14 0.37
Output Units Maximum Minimum Mean SD
DIM – 1 0 0.22 0.24
2
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. Results and Discussion
.1. Analytical procedure
We have extensively carried out literature study to understandhe martensitic transformation micro-mechanisms and their inter-retation while explaining the mechanical behaviour of austenitictainless steels under various operating conditions. For the presentodel, inputs are chosen according to the knowledge gained
rom the published literatures and from the industrial experi-nces. The inputs of the model are chosen to be: chemistry ofustenitic stainless steels, strain rate, initial austenite grain size,emperature of testing, true stress and true strain. The target (i.e.utput) is the extent of DIM. The other influencing parametersor martensitic transformations are stress state, initial microtex-ure of austenite and strain path, which were not, included asnput parameters because there is lack of published data avail-ble. In most of those literatures, we considered those studies,hich deal with different grades of austenitic stainless steels withifferent grain sizes under uniaxial loading at various testing con-itions. In most of those literatures, DIM is generally representeds strain induced martensite. Various techniques have been usedor quantifying the DIM formation in those literatures, such as:RD, magnetic methods etc. The most common and availableraphs are found to be: (a) stress–strain and (b) DIM as a func-ion of strain. We have extensively extracted data (i.e. strain, DIMraction and corresponding stress value) from those two graphsith their corresponding testing conditions and material history
eported in corresponding literatures. We have tabulated all theata in a single spreadsheet and the size of the database, whichre used for neural network analysis to be 1600 rows. The litera-ures (1954–2010), from where those data digitized are mentionedo be [1,4–27]. The statistics of the whole database are given inable 1. It is emphasised that unlike linear regression analysis,he ranges stated in Table 1 cannot be used to define the rangef applicability of the neural network analysis. This is becausehe inputs are in general expected to interact each other. Wehall see later that it is the Bayesian framework of our neuraletwork analysis, which makes possible the calculation of error
ars whose magnitude varies with the position in the input space,hich define the range of useful applicability of the trained net-ork. A visual impression of the spread of the data is shown inig. 1(a–o).
2.2. Empirical modelling
A neural network is a general method of regression analy-sis in which a very flexible non-linear function is fitted to theexperimental data. It can capture the enormous complexity in thedatabase, which avoid over fitting [28]. It is nevertheless useful todiscuss some salient features, to place the technique in the con-text. The Bayesian framework of neural network has been used inthis present study. A neural network is generally trained on a setof examples of input and output data with repetitive representa-tions. The outcome of training is a set of co-efficient (i.e. weights)and a specification of the functions, which in combination withthe weights correlating the inputs to the output. The training pro-cess itself involves a search for the optimum non-linear correlationbetween the inputs to the output and is computer intensive. Oncethe neural network is trained, the estimation of the output for anygiven inputs is very easy. The details of this method used here haverecently been comprehensively reviewed by MacKay [29] and theoriginal method is described thoroughly elsewhere [29–34]. Oneof the difficulties with the blind data modelling is that of over fit-ting, in which spurious details and the noise in the training dataare over fitted by the model. This gives rise to solutions that gen-eralise poorly. MacKay [29–34] and Neal [35] have developed aBayesian framework for neural networks in which the appropriatemodel complexity is inferred from the database. This Bayesian neu-ral network modelling has two important advantages. Firstly, thesignificance of all the input variables is quantified automatically,which is extremely important to understand the response of eachvariable. Consequently, the model perceived significance of eachinput variable can be compared against the existing metallurgicaltheory. Secondly, the neural network’s predictions are accompa-nied by error bars, which depend on the specific position in inputspace. This quantifies the model’s certainty about its predictions. Inthis present study, both the inputs and output variables were firstnormalised within the range ±0.5 as follows:
xN = x − xmin
xmax − xmin− 0.5 (1)
where xN is the normalised value of x; xmin and xmax are respec-
tively the minimum and maximum values of x in the entire dataset(Table 1). The normalisation is straightforward for all the quantita-tive variables. The normalisation is not necessary for this analysisbut facilitates the subsequent comparison of the significance of
A. Das et al. / Materials Science and Engineering A 529 (2011) 9– 20 11
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Fig. 1. The database values of each variables: (a) C, (b) Mn, (c) Cr, (d) Ni, (e) Mo, (f) N, (g) Cu, (h) Nb, (i) Co, (j) Ti, (k) strain rate, (l) temperature, (m) grain size, (n) true stressand (o) true strain versus the volume fraction of DIM in a variety of austenitic grade stainless steels.
12 A. Das et al. / Materials Science and E
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Fig. 2. A typical network used in the analysis.
ach of the variables. The neural network consisted of fifteen inputodes, one hidden unit and an output node representing the extentf DIM (schematically shown in Fig. 2). Several models were cre-ted with various combinations of hidden units with varying theirorresponding nodes. The database was randomised properly andhen partitioned equally into testing and training data sets. Theater was used to create a large variety of neural networks models
hereas the testing data set was used to see how the trained mod-ls generalised on unseen data. The inputs and output as shownre connected through hidden units where the inputs xj are oper-ted by a hyperbolic tangent transfer function to obtain the hiddennits, hi defined as:
i = tanh
⎛⎝∑
j
w(1)ij
xj + �(1)i
⎞⎠ (2)
The bias is designated �i and is analogous to the constant thatppears in linear regression analysis. The strength of the transferunction is in each case determined by the weight wij. The transfero the output y is linear:
=∑
i
w(2)i
hi + �(2) (3)
here w(2)i
and �(2) are a new set of weights and a bias, respec-ively. Eqs. (2) and (3) define the neural network structure thatonnects the inputs to the output. The weights and biases, however,re unknown to be determined through training process using theayesian back propagation scheme, which involves a minimizationf the energy function [30]; the minimization was implementedsing a variable metric optimizer [36]. The gradient of M(w) wasomputed using back propagation algorithm [37]. The energy func-ion consists of the error function, ED and regularisation Ew.
(w) = ˇED +∑
c
acEw(c) (4)
The error function, ED(w) is the sum squared error as follows:
D(w) = 12
∑m
∑i
(yi(xm, w) − tm)2 (5)
ngineering A 529 (2011) 9– 20
where the data set {xm, tm} consists of xm inputs related to a par-ticular target tm (m is a level of pairs). The objective is to determinea set of weights in a neural network that minimizes ED but with-out over fitting to noise. Thus the regularisers, Ew are included sothat smooth solutions of y(xm,w) are favoured and the possibility offitting to noise in the experimental data can be reduced. The coeffi-cients w and biases �, which are shown in Eqs. (2) and (3) make upthe parameter vector w. A number of regularisers Ew(c) are added tothe data error. The regularisers favour functions y (x; w), which aresmooth functions of x. The simplest regularisations method uses asingle regulariser:
Ew = 12
∑w2
i (6)
Here, however, we have used a slightly more complicated regu-larisation method known as the automatic relevance determinationmodel which has been described elsewhere [3,33]. Each weight isassigned to a class c depending on which neurons it connects. Foreach of the input, all the weights connecting that input to the hid-den units are in a single class. The hidden units’ biases are in anotherclass, and all the weights from the hidden units to the outputs are ina final class. Ew(c) is defined to the sum of the squares of the weightsin class c [31]:
Ew(c)(w) = 12
∑i=c
w2i (7)
The additional terms favour small values of w and decreasethe tendency of a model to overfit noise in the data set. The con-trol parameters ˛c and ̌ together with the number of hiddenunits determine the complexity of the model. These hyper param-eters define the assumed Gaussian noise level of: �2
v = 1/ ̌ andthe assumed weight variances: �2
w(c) = 1/˛c . �v is the noise levelinferred by the model. The parameter ̨ has the effect of encourag-ing the weights to decay. Therefore, a high value of �w implies thatthe inputs parameter concerned explains a relatively large amountof the variation in the output. Thus �w is regarded as a good expres-sion of the significance of each input thought not of the sensitivityof the output to that input. This has been described thoroughlyelsewhere [3,31].
The error, ED is expected to increase if important input vari-ables have been excluded from the analysis. Whereas ED gives anoverall perceived level of noise in the output parameter, it is, on itsown, an unsatisfying description of the uncertainties of prediction.MacKay [29–33] has developed a particularly useful treatment ofneural network analysis in a Bayesian framework, which allows thecalculation of error bars representing the uncertainty in the fittingparameters. The method recognises that there are many functionsthat can be fitted or extrapolated into certain regions of the inputspace, without unduly compromising the fit in adjacent regions,which are rich in accurate data. Instead of calculating a uniqueset of weights, a probability distribution of set of weights is usedto define the fitting uncertainty. The error bars therefore, becomelarge when data are sparse or locally noisy. The error bars presentedthroughout the whole work, therefore, represent a combination ofthe perceived level of noise in the output (i.e. DIM) and the fittinguncertainty as described above. This specification of the networkstructure, together with the set of weights is a complete descrip-tion of the formula relating to the inputs to the output. The weightsare determined by training the network; the details are describedthoroughly elsewhere [29–34]. The number of hidden units useddetermines the complexity of the neural network analysis and more
accurate predictions occur with increased number of hidden units(Fig. 3). The training for each network is started with a variety ofrandom seeds. The term, �v used here is the framework estimat-ing the overall noise level of the database. The complexity of the
A. Das et al. / Materials Science and Engineering A 529 (2011) 9– 20 13
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ig. 3. Variation in �v as a function of the number of hidden units. Several valuesre presented for each set of hidden units because the training for each network wastarted with a variety of random seeds.
odel is controlled by the number of hidden units and the val-es of the regularisation constants (�w), one associated with eachf the inputs, one for biases, and one for all weights connected tohe output. Fig. 3 shows that the inferred noise level, �v decreases
onotonically as the number of hidden units increase. However,he complexity of the model also increases with the number of hid-en units. A high degree of complexity may not be justified, and inhe extreme case, the model may in a meaningless way attempt tot the noise in the experimental data.
More complex relations can be modelled with a large numberf hidden units. However, the function may then be over fitted, ashown in Fig. 4, because experimental data always contain errors.his has been discussed in detail elsewhere [3]. In order to reducever fitting, the test error (the value of the error function for non-rained data set) was measured, using 800 randomly chosen rowsf data, which were not included in the training set. Fig. 5 showshe change in test error as function of hidden units.
MacKay [29–33] has made a detailed study of this problem andas defined a quantity, evidence, which comments on the proba-ility of a model. In circumstances where two models give similarind of results over the known dataset, the more probable modelould be predicted to be that which is simpler; this simple modelould have a higher value of evidence. The evidence framework wassed to control the regularisation constants and �v. The numberf hidden units was set by examining performance on test dataFig. 3). A combination of Bayesian and pragmatic statistical tech-iques were, therefore, used to control the model complexity. A
urther procedure used to avoid the over fitting problem was toivide the experimental data randomly into two equal sets namelyhe training and test data sets. The models are created using the
Real function
Overfitted
Fig. 4. Overestimation of function [3].
Hidden units
Fig. 5. The test error as a function of the number of hidden units.
training data only. The unseen test data are then used to assess howwell the model generalises. A good model would produce similarlevels of error in both the test and training data whereas an overfitted model might accurately predict the training data but badlyestimate the unseen test data. Once the correct complexity of themodel has been determined using this procedure, it can be retainedusing all the data with a small but significant reduction in the error.The test error, Te is a measure of the deviation of the predicted valuefrom the experimental one in the test data:
Te = 12
∑n
(yn − tn)2 (8)
where yn is the predicted amount of DIM fraction and tn is its mea-sured value.
The test error defined as the value of the error function forunseen data is shown in Fig. 5. The best model may be definedas that with the smallest test error. This would be appropriate,for situation where only scalar prediction (i.e. no error bars) arerequired. MacKay has shown when making predictions with errorbars, the best model should be decided according to a quantity thelog predicted error (LPE). Using the LPE, unlike the test error, wildpredictions are penalised less if they have large error bars whenusing noisy data, common in many experimental situation, somewild predictions must be expected. Assuming that for each exam-ple n the model gives a prediction with error (yn, �2
v ), the LPE is(Fig. 6):
LPE =∑
n
[0.5(tn − yn)2
�2n
+ log(√
2��v)
](9)
The behaviour of the training and testing data are shown inFigs. 7 and 8 respectively which show a similar degree of scatterin both the graphs, indicating that the complexity of the respectivemodel is optimum. It should be noted that the test data cover awide range of DIM fraction value and, for a very few cases at thehighest amount of DIM, the model under predicts the measuredvalues. Over fitting would lead to an apparently better accuracy inthe prediction of training data when compared with the test dataset. The error bars in Figs. 7 and 8 include the error bars on theunderlying function and the inferred noise level in the dataset �v.In all other subsequent predictions discussed below, the error bars
include the former component only. It is often the case with noisydata that models with different complexity make different predic-tions. In these circumstances, the prediction made by a committeeof models may be more reliable than using a single model. Fig. 9
14 A. Das et al. / Materials Science and Engineering A 529 (2011) 9– 20
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Fig. 6. LPE as a function of the number of hidden units.
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Fig. 7. Plot of the estimated versus measured DIM fraction – training dataset.
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Fig. 8. Plot of the estimated versus measured DIM fraction – test dataset.
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Fig. 9. Combined test error as a function of the number of models in committee.
describes a population of models that can be ranked according tothe magnitude of the test error. We start a committee by using Nmodels ranked by LPE. The committee is formed through combin-ing the best N models (where N = 1, 2, 3. . .) such that the meanprediction of the committee is:
yn = 1N
N∑i=1
yi (10)
with associated error in yn expressed as:
�2 = 1N
N∑i=1
�2i + 1
N
N∑i=1
(yi − yn)2 (11)
Fig. 9 shows the changes in the test error with the number ofmodels used to form a committee. The figure shows that fifteenmodels committee is favourable for the DIM fraction. Committeepredictions are compared against experimental data in Fig. 10. Thebehaviour of the committee model consisting of individual mod-
els retrained on the entire data set is illustrated in this figure. Theinputs to output mapping becomes more accurate after retraining.The purpose of the division into training and test data was to iden-tify models with the optimal level of complexity. Once that is done,1.00.80.60.40.20.00.0
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Fig. 10. Training data for best committee model (training was done on wholedataset); error bars includes errors in underlying functions only.
A. Das et al. / Materials Science and Engineering A 529 (2011) 9– 20 15
1.00.80.60.40.20.00.0
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Cal
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Measured martensite fractionig. 11. Application of the best model for the blind literature data [38] source.
t is quite reasonable to use the entire data set for retraining, butithout changing the complexity of the model. Since the committee
omplexity is not changed after retraining, its ability to generalises not significantly affected.
.3. Application of the model
We now examine the metallurgical significance of the results.he optimized committee model was used to study the effect ofndividual variables on the formation of DIM to find out whetherhe results are compatible with known metallurgical principles andther published data. Figs. 11–13 show the application of our com-ittee model from the very recent published data by Lindgren et al.
38], Hauild et al. [39] and the present authors [40,41] respectivelyhich were not included in the training as well as in the testing
f neural network. From these three graphs, it is noted that ourodel is robust to predict the total blind data. As it has been shown
n Figs. 11–13, the correlation coefficient (i.e. adjusted R2) for best
1.00.80.60.40.20.00.0
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Measured martensite fraction
ig. 12. Application of the best model for the blind literature data [39] source.
Fig. 13. Application of the best model for the present experimental data [40,41]source.
fit for the data set is 0.9421, 0.95579 and 0.88271 respectively. Thismeans that a good correlation (i.e. reasonable accurate) betweenthe measured and calculated data have been obtained for all theapplications. These figures show that the used network could becapable for prediction with a minimum error.
Note that the error bars length is different for each data(Figs. 11 and 12) point such that they represent true confidenceof prediction that is one of the strong points of Bayesian neuralnetwork. Since reserved data are equivalent to new experiments,it may be possible to predict the extent of DIM of a new austeniticstainless steel type with a similar precision as long as the inputs arein the range of Table 1. Though the model tends to slightly underpredicts and over predicts at lower and higher DIM fractions respec-tively, the predictions are quite close to the experimental results.At small strain, defects introduced into the austenite enhance thenucleation rate; therefore the actual fraction of martensite will begreater than when only stress is accounted for. At large strains,the defects oppose transformation by mechanical stabilisation. Soa calculation based on stress alone will overestimate the amount ofmartensite (Figs. 11 and 12).
2.4. Prediction of the model
The optimized committee model has been used to predict theinfluence of individual input variables on the formation of DIM inthe following subsections. The prediction (Fig. 14) has been madefor a specific 304 stainless steel. It is important to note that pre-dictions are for the case where just one input variable is altered,keeping all other fixed. This may not be possible when conductingexperiments. Fig. 16 also shows the significance of input variableson DIM formation. Now we shall investigate the isolated influenceof all the individual variables on martensitic transformation withthe support of extensive literature study.
2.4.1. Effect of alloying elements on martensitic transformationAlloying elements of austenitic stainless steels significantly
influence the formation of DIM. So in the following paragraphs,the predicted effect of altering the chemical composition on the
martensitic transformation is described.Carbon is the most important alloying element, which deter-mines the austenite stability. The present model predicts thevariation of DIM formation as a function of carbon content
16 A. Das et al. / Materials Science and Engineering A 529 (2011) 9– 20
0 2
0.4
0.6
0.8
1.0ic
ted
mar
tens
ite fr
actio
n
0 2
0.4
0.6
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icte
d m
arte
nsite
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tion
0.4
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cted
mar
tens
ite fr
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0.02 0.0 4 0.0 6 0.08 0.100.0
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i
C - Content / wt%1 2 3 4 5 6 7 8 9
0.0
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Pred
i
Mn - Conte nt / wt%16.0 16 .5 17 .0 17.5 18.0 18.5
0.0
0.2
Cr - Con tent / wt%
Pred
ic
0 4
0.6
0.8
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mar
tens
ite fr
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n
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n
0 4
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(b)(a) (c)
0 6
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nsite
frac
tion
3 4 5 6 7 8 9 10 11 120.0
0.2
0.4
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icte
d m
Ni - Con tent / wt%0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8
0.0
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0.4
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icte
d m
Mo - Con tent / wt%0.00 0.05 0.10 0.15 0.20
0.0
0.2
0.4
N - Conte nt / wt%
Pred
icte
d m
0 6
0.8
1.0
site
frac
tion
(d) (e) (f)
(i)0.8
1.0
site
frac
tion
0.00 0.02 0.04 0.0 6 0.0 8 0.1 00.0
0.2
0.4
0.6
Pred
icte
d m
arte
n
Nb - Conte nt / wt%0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
0.0
0.2
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0.6
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icte
d m
arte
ns
Cu - Con tent / wt%
0.8
1.0
actio
n
0 8
1.0
actio
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actio
n(g) (h)
0.00 0.05 0.1 0 0.1 5 0.20
0.2
0.4
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Pred
icte
d m
arte
ns
Co - Con tent / wt%
0.0 0.1 0.2 0.3 0.4 0.5 0.60.0
0.2
0.4
0.6
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Pred
icte
d m
arte
nsite
fra
Ti - Conte nt / wt%0 20 40 60 80 10 0 12 0 140 160 18 0 200
0.0
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Strain rate / s-1
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nsite
fra
-50 -25 0 25 50 75 100 125 150 175 2000.0
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fra
Temperat ure / 0C
1.0 1.0
(j) (k) (l)
0 25 50 75 100 125 15 0 17 5 2000.2
0.4
0.6
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icte
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arte
nsite
frac
tion
Grain size / μm600 80 0 100 0 1200 140 0 1600
0.0
0.2
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arte
nsite
frac
tion
True stress / MPa0.0 0.1 0.2 0.3 0.4 0.5 0.6
0.0
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0.4
0.6
0.8
True strain
Pred
icte
d m
arte
nsite
frac
tion
(m)
(n) (o)
μ True stress / MPa
Fig. 14. Influence of (a) C, (b) Mn, (c) Cr, (d) Ni, (e) Mo, (f) N, (g) Cu, (h) Nb, (i) Co, (j) Ti, (k) strain rate, (l) temperature, (m) grain size, (n) true stress and (o) true strain on theformation of DIM in austenitic grade stainless steels, predicted by the model. Note: the small error bars indicate that the scatter in the database is very small and the largeerror bars suggest lack of sufficient data in the range examined.
A. Das et al. / Materials Science and E
43210
0
20
40
60
80
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Cold rolled Cold drawn
Mar
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ite fr
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%
Equivalent strain
F[
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ig. 15. Changes in the volume fraction of DIM during cold rolling and cold drawing62].
Fig. 14(a)). With the increase in carbon content, DIM formationecreases drastically and getting saturated beyond 0.06%. Kruppt al. [42] investigated that the effect of carbon content and/or theemperature on the formation of DIM is very strong in such a wayhat high carbon content and elevated temperatures stabilises theustenite phase.
The effect of manganese on martensitic transformation is shownn Fig. 14(b). It is predicted that with the increase in manganeseontent, DIM formation decreases. Manganese by itself tends toncrease the stability of austenite by decreasing its Ms temperature.his is a good example of the safety of the predictions made by theodel, in that the error bars are large when the model is uncertain.Chromium is the most essential element added in stainless
teels mainly responsible for corrosion resistance. The estimatedffect of chromium on DIM transformation is shown in Fig. 14(c).t is predicted that with the increase in chromium, DIM forma-ion decreases. It is worthy to mention that although chromiums a weak ferrite stabiliser, its influence on Ms temperature is verytrong.
Fig. 14(d) shows the prediction of DIM formation as a functionf nickel content. Small increment in nickel concentration drasti-ally reduces the rate of isothermal transformation of DIM. Nickels responsible for high toughness and high strength at both high
0.0
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3.0
Strain
Stress
Grain s
ize
Temperat
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rate
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Niobium
Coppe
r
Nitroge
n
Molybd
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Manga
nese
Sign
ifica
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ig. 16. Bar chart showing a measure of the model perceived significance of each ofhe input variables in influencing martensitic transformation.
ngineering A 529 (2011) 9– 20 17
and low temperatures without sacrificing the ductility in stainlesssteels.
The effect of adding molybdenum on the stability of austenite aspredicted by the model is shown in Fig. 14(e). It increases the resis-tance towards crevice and pitting corrosion. From Fig. 14(e), it ispredicted that as molybdenum content increases, DIM transforma-tion suppresses. This result is in accordance with those reportedin the literature demonstrating the role of molybdenum, whichslightly increases Ms temperature [43].
Nitrogen is added in austenitic stainless steel to improve theyield strength by refining its grains. It is noted from Fig. 14(f)that with the increase in nitrogen content up to 0.10% (approxi-mately), DIM transformation decreases. According to Lee et al. [44],increasing the nitrogen content causes the transient strain for DIMformation to shift at higher strain and, finally, DIM will not formwhen nitrogen content is beyond 0.50 wt%. Lee et al. [45] evalu-ated the effect of nitrogen on DIM formation in 304 stainless steelsbased on the proposed kinetics relation between DIM and inelas-tic strain and reported that nitrogen addition reduces the austenitestability parameter, which is inversely proportional to the austenitestability, leading to decrease in Md30 temperature. Recently, Brackeet al. [46] reported that chromium and nitrogen suppressed the DIMtransformation in Fe–Mn–Cr–N steels and the differences in trans-formation behaviour are attributed to the change in the intrinsicstacking fault energy.
Copper is normally present in austenitic stainless steels as resid-ual alloying element and it increases the tensile strength. However,it is added to a few alloys to produce precipitations hardeningproperties or to enhance the corrosion resistance. Present modelpredicts that copper does not have any significant effect on DIMformation in austenitic stainless steels (Fig. 14(g)). The effect ofcopper on Ms temperature is not as clear as the austenite stabilis-ing elements (i.e. Mn, Ni etc.). Capdevilla et al. [43] has suggestedthat for copper concentration up to 1.0 wt%, this element does notinfluence Ms temperature. On the other hand, Hong and Koo [47]suggested that the addition of copper can suppress the formationof DIM during tensile testing to prevent strain hardening.
The effect of cobalt content on DIM formation is shown inFig. 14(i). It is predicted that with the increase in cobalt concen-tration, the tendency of DIM formation increases. Present result isfully consistent with those reported in the literature demonstrat-ing the role of cobalt increasing Ms temperature [43]. Capdevillaet al. [43] also suggested that the additions of cobalt change thetendency of Ms temperature depending on the chromium concen-tration. Co–Cr alloys are well suited to high temperature creep andfatigue resistance applications.
The effect of adding micro alloying elements on the stabilityof the austenite (i.e. DIM fraction) as predicted by the model isshown in Fig. 14(h and j). Since the error bars in Fig. 14(h) are largebecause of the insufficient experimental data, it cannot be con-cluded that niobium really affects DIM transformation. In austeniticstainless steels, it is generally added to improve the resistance tointergranular corrosion but it also enhances the mechanical prop-erties at high temperatures. From Fig. 14(j), it is predicted thattitanium enhances the martensitic transformation. The addition oftitanium can influence the stability of austenite phase especiallyunder strained condition. Titanium is generally added for carbidestabilisation especially when austenitic stainless steels are to bewelded.
2.4.2. Effect of strain rate on martensitic transformationMost investigations carried out in order to clarify the effect of
strain rate on DIM formation, which has indicated that the trans-formation is suppressed with increasing strain rate. This has beenmostly explained in terms of the adiabatic heating, which decreasesthe chemical driving force of the transformation. However,
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8 A. Das et al. / Materials Science
taudhammer et al. [48] suggested that high strain rate may pro-ote more irregular shear band arrays compared to the low strain
ate. This may lead to a reduced probability of the formation of ˛′
bcc) martensite embryos of the critical size, and thus, suppresseshe formation of ˛′ (bcc) martensite. On the other hand, it has beenound that high strain rate, (103 s−1) promoted shear band forma-ion in 304 stainless steel compared to low strain rate, (10−3 s−1).his led to an increased number of shear band intersections andigher volume fraction of ˛′ (bcc) martensite at the early stages ofensile deformation, as illustrated by Hecker et al. [6] and Murr et al.49]. They have found that at strains higher than 0.25, the ˛′ (bcc)
artensite transformation was suppressed at high strain rate. Thisas attributed to the rise in temperature (i.e. adiabatic heating) atigh strain rates which would suppress the martensitic transforma-ion. Present authors [40,41] have already investigated the role oftrain rate on the formation of DIM in 304LN stainless steels exper-mentally. In the present model, we have predicted the influence oftrain rate on DIM formation (Fig. 14(k)). It has been found that withhe increase in strain rate, martensitic transformation suppressesrastically.
.4.3. Effect of testing temperature on martensitic transformationMartensitic transformation is affected by two composition
ependent parameters (i.e. stacking fault energy and �GCHEM).ince both of these parameters are temperature dependent, theendency to the DIM formation is sensitive to the temperature, asell. It is well known that the DIM formation is suppressed with
ncreasing temperature [4]. The behaviour is normally attributedo the decrease in the �GCHEM with increasing temperature. In theresent model, we have demonstrated the influence of tempera-ure on DIM formation (Fig. 14(l)). It is noted that with the increasen temperature, DIM transformation decreases. As it is well knownhat stability of austenite increases with increase in temperaturend on increasing the temperature beyond a limit Md, no trans-ormation takes place [50]. The predicted DIM fraction, which ishown in Fig. 14(l) decreases with increase in testing tempera-ure from −75 to 100 ◦C (approximately) and beyond that theres uncertainty in prediction. Sugimoto et al. [51] found that onncreasing the temperature further to 300 ◦C, the predicted stabil-ty of austenite decreases for TRIP aided dual phase steel, which haseen reported elsewhere [51], where the authors have found thateyond a certain temperature, austenite undergoes strain inducedainitic transformation, which results in a decrease in stability. It
s noted from (Fig. 14(l)) that the error bars associated with theseigh temperature predictions (i.e. beyond 100 ◦C) are very large.his is a reflection of the limited number of high temperature datan the training data set, which result in rather uncertain predic-ions. Nevertheless, it is noteworthy that in spite of the limitedata the network is able to capture quite accurately the effect ofemperature.
.4.4. Effect of austenite grain size on martensitic transformationGonzales et al. [52] studied the effect of austenite grain size on
artensitic transformation in 304 stainless steel. The transforma-ion was found to be enhanced by large grain size. Varma et al. [7]lso found that the large grain size promoted DIM formation duringensile and cold rolling of 304 and 316 stainless steels both. In con-rast, Srinivas et al. [53] found that the formation of DIM during coldolling increases with decreasing grain size in 304 stainless steelnd is grain size independent in 316 stainless steel. The predictedffect of grain size on the martensitic transformation is shown inig. 14(m). From this figure, it is noted that with the increase in
ustenite grain size, DIM formation enhances drastically. Accord-ng to Yang and Bhadeshia [54], the extent of DIM in the early stagesf transformation is proportional to the cube of the austenite grainize. They have clearly explained that there is a large dependencengineering A 529 (2011) 9– 20
of Ms temperature as a function of austenite grain size. Accord-ing to Guimarães et al. [55], a fine austenitic grain size should beexpected to shift MS
� and �MS� to a lower temperature and a higher
stress value, respectively, as observed experimentally. Hence, theobservation strongly conforms to the published theory.
2.4.5. Effect of stress on martensitic transformationPatel and Cohen [56] have described the criterion for the appli-
cation of external stress on martensitic transformation in theirelegant study. According to them, when the external force is act-ing, the resulting effect on the Ms temperature is calculated fromthe mechanical work done (i.e. �GMECH) on or by the transformingregion as the resolved shear stress and normal components of theapplied stress are carried through the corresponding transforma-tion strains. This work done (U) on or by the transformation due tothe action of applied stress is comprised of: (��0), the shear stressresolved along a potential habit plane times the transformationshear strain, and (�∈0), the normal stress resolved perpendicular tothe habit plane times the normal component of the transformationstrain. Thus
U = ��0 + �ε0 (12)
� is numerically positive when the normal stress is tensile, andnegative when this component is compressive. � is always takento be positive because the many habit permutations (±{259} inthese alloys) virtually permit shearing in either sense [56]. Hencein effect, shear stresses will stimulate the phase transformation,but normal stresses may aid or oppose it depending upon whether� is tensile or compressive [56]. Under uniaxial tensile test, thetransformation is aided by both the shear and (positive) normalcomponents of stress, and therefore the Ms is raised even more thanin the case of uniaxial compression. At temperatures just aboveMs, transformation can be induced via stress assisted nucleation onthe same heterogeneous sites responsible for the transformationon cooling according to Stringfellow et al. [57]. Hong-zhuang et al.[58] also concluded in their study that as the applied stress, � isincreased, the maximum magnitude of the transformation inducedplasticity effect along the longitudinal direction decreases. Recentlywe have demonstrated that a large amount of published data relat-ing the fraction of DIM to plastic strain can in fact be described interms of the pure thermodynamic effect of applied stress [59]. In thepresent model, we have demonstrated the effect of true stress onDIM formation (Fig. 14(n)). It shows that with the increase in truestress, the formation of DIM increases drastically. Hence, the roleof stress on martensitic transformation is convincingly revealed.
2.4.6. Effect of stress-state and strain on martensitictransformation
In the temperature range above MS� and below Md, the trans-
formation is dominated by strain induced nucleation on potentnucleation sites created by plastic strain [60]. Iwamoto et al. [61]found that in compression the transformation rate was initiallyhigher than in tension, but at higher strains the relation wasreversed. Hecker et al. [6] also found that more ˛′ (bcc) marten-site was formed in biaxial tension than in uniaxial tension. Fig. 15demonstrates the influence of stress-state on the formation of DIMin cold rolled and cold drawing 316 stainless steel [62]. Due tolack of data in the published domain, we could not incorporatestress-state as of input parameter, which plays a significant roleon martensitic transformation. We have also predicted the influ-ence of true strain on DIM formation (Fig. 14(o)). It has been foundthat strain alone is having very little influence on martensitic trans-
formation. As strain is increasing, DIM volume fraction increasesslightly. The independent effect of strain is seen to be minor andmore uncertain. These results confirm the earlier conclusion that inexperiments where martensite is stimulated during a tensile test, it
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A. Das et al. / Materials Science
s the mechanical driving force, which plays a dominant role ratherhan strain induced transformation. This empirical analysis of aide range of data supports the predominant role of stress overlastic strain.
.4.7. Effect of microtexture of parent austenite on martensiticransformation
Initial microtexture of austenite is an important parameter toontrol the amount of DIM while application of external stress.ccording to Wang et al. [63], cold rolling generates residual stress,hich is dependent not only on the specimen sections but also
n the grain orientations. Nakamura and Wakasa [64] have alsonvestigated the variation of DIM in a two phase steel having tex-ures. In the present model, it was not possible to include the
icrotexture data of the initial austenite as one of the input param-ters because of lack of data availability in the published domain.ecently Kundu and Bhadeshia [65] have clearly demonstrated inheir elegant transformation texture model that the initial micro-exture of austenite strongly influence the variant selection of DIMransformation in stainless steels.
.4.8. Significance of input variables on martensiticransformation
In the present model, it has been possible to show the isolatednfluence of input parameters. The Bayesian neural network mod-lling has an excellent advantage to calculate the significance of thenput variables, which has been clearly demonstrated by MacKay29–33] in his pioneer studies. Fig. 16 shows the perceived signifi-ance of the inputs for the best model. The parameter, �w is ratherike a partial correlation coefficient in linear regression analysis inhat it represents the amount of variation in the output that can bettributed to any particular input parameter and does not neces-arily represent the sensitivity of the output to each of the inputs.t should be noted that it does not indicate the sensitivity of theutput to the inputs. The sign of the effect is indicated in Fig. 16 forach variable that has been perceived to have a high significance. Its clearly understood that the effect of temperature and stress are
ore predominant than the rest. From these graph, we can drawhe importance of each input variables on DIM formation.
. Conclusions
Estimation of DIM has been investigated with its influencingarameters in a variety of austenitic grade stainless steels throughayesian neural network modelling. The obtained results are sum-arised as follows:
a) A neural network model has been developed to predict thenature of variation of DIM with its influencing parameterswhich can be applied for designing the nuclear power plantcomponents, automobile industries etc. The model has beenextensively applied for recent published data, which matchesalmost accurately.
) The model predictions confirm the important effect of C, Mn, Ni,Mo, N, Co, Ti on the variation of DIM formation. It also interest-ingly brings out the insignificant effect of few alloying elements(i.e. Cu, Nb, etc.).
c) The increase in austenite grain size enhances martensitic trans-formation. DIM formation suppresses while increasing strainrate, which is mainly attributed to the increase in adiabatic heat.With the increase in temperature, DIM formation triggers dras-tically.
) The remarkable result is that the role of stress is convincinglyrevealed, whereas the independent effect of strain is seen tobe minor and more uncertain. These results confirm the earlierconclusion that in experiments where martensite is stimulated
[[[
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ngineering A 529 (2011) 9– 20 19
during a tensile test, it is the mechanical driving force, whichplays a dominant role rather than strain induced transformation.
Acknowledgements
Arpan Das is grateful to the Ministry of Science & Technology,Department of Science & Technology, Government of India for theBOYSCAST Fellowship. The authors are extremely grateful to Pro-fessor H. K. D. H. Bhadeshia, University of Cambridge, UK, for theprovision of Neuromat Neural Network software for the presentanalyses. The authors would like to thank the respected reviewersfor their positive and constructive comments for this manuscript.
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