Estimation of deformation induced martensite in austenitic stainless steels

Materials Science and Engineering A 529 (2011) 9–20

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Estimation of deformation induced martensite in austenitic stainless steels Arpan Das a,b,∗ , Soumitra Tarafder a , Pravash Chandra Chakraborti b a b

Fatigue & Fracture Group, National Metallurgical Laboratory (Council of Scientific & Industrial Research), Jamshedpur 831007, India Department of Metallurgical & Material Engineering, Jadavpur University, Kolkata 700 032, India

a r t i c l e

i n f o

Article history: Received 4 February 2011 Received in revised form 23 June 2011 Accepted 19 August 2011 Available online 27 August 2011 Keywords: Deformation induced martensite Austenitic stainless steels Martensitic transformation Significance Bayesian neural network

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 neural network model within a Bayesian framework has been created using extensive published data correlating the extent of DIM with its influencing parameters in a variety of austenitic grade stainless steels. The Bayesian method puts error bars on the predicted value of the rate and allows the significance of each individual parameter to be estimated. In addition, it is possible to estimate the isolated influence of particular variable such as grain size, which cannot in practice be varied independently. This demonstrates the ability of the method to investigate the new phenomena in cases where the information cannot be accessed experimentally. The model has been applied to confirm that the predictions are reasonable in the context of metallurgical principles, present experimental data and other recent data published in the literatures. © 2011 Elsevier B.V. All rights reserved.

1. Introduction Austenitic stainless steels are extensively used in engineering applications, nuclear power plant components, automobile and pharmaceutical industries due to their excellent corrosion resistance, weldability and mechanical properties. Metastable austenitic stainless steels undergo DIM transformation, where the  (fcc) austenite is transformed to thermodynamically more stable ˛ (bcc) martensite due to plastic deformation. This phase transformation enhances the work hardening of these steels, and affects their ductility [1]. Furthermore, the microstructural evolution and the mechanical behaviour are sensitive to chemical composition, temperature, stress, strain, strain path, strain rate, stress state, grain size, and initial crystallographic microtexture. Understanding the influence of these factors, resulting microstructures and the corresponding mechanical behaviour are the most important part not only in terms of the selection of the best material, but also in the optimal development of material models, which are nowadays extensively applied in the automobile and nuclear power plant industries to understand their forming and crash related performances. Considerable attention was given in the past to the microstructure of austenitic

∗ Corresponding author at: Fatigue & Fracture Group, National Metallurgical Laboratory (Council of Scientific & Industrial Research), Jamshedpur 831007, India. Tel.: +91 9934328051; fax: +91 657 2345213. E-mail address: [email protected] (A. Das). 0921-5093/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.msea.2011.08.039

stainless steels, the stability of the phases present in these steels and the effects of amount and distribution of the phases present on the mechanical behaviour of the material under service. The mechanical properties of metastable austenitic stainless steels are strongly influenced by the morphologies and the extent of deformation induced phase transformation. In the past, there has been a constantly increasing interest for neural network modelling in different fields of materials science [2]. Several models have been developed for prediction of mechanical properties, phase transformations, optimizing alloy composition, processing parameters, heat treatment conditions, on line corrosion monitoring, improving weldability, etc. [2]. This empirical approach becomes more attractive as it is fairly the robust technique and in most cases, it rapidly converges to a target solution. This provides a range of powerful new techniques for solving problems 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 function of its influencing variables using neural network technique within a Bayesian framework [3]. This model would tremendously help to the nuclear power plant, automobiles and pharmaceutical industries to design their components under service. In the present context, the optimization process needs access to a quantitative relationship between the chemical composition of austenitic stainless steels, grain size, stress, strain, temperature, strain rate and the ultimate the extent of DIM. A neural network method has been developed to correlate those and applied extensively for applications.

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A. Das et al. / Materials Science and Engineering A 529 (2011) 9–20

Table 1 Statistics of database used for neural network analysis. SR: strain rate, T: temperature, GS: grain size, TSS: true stress, TSN: true strain, DIM: deformation induced martensite and SD: standard deviation. The column marked ‘Example’ is a specific case used to generate Fig. 14. Inputs

Units

Maximum

Minimum

Mean

SD

Example

C Mn Cr Ni Mo N Cu Nb Co Ti SR T GS TSS TSN

wt% wt% wt% wt% wt% wt% wt% wt% wt% wt% s−1 ◦ C ␮m MPa –

0.10 8.92 18.58 13.53 2.53 0.24 0.70 0.11 0.20 0.67 200 200 200 1951.17 0.65

0.007 0.42 15.40 2.75 0 0 0 0 0 0 0.0001 −196 5.90 14.22 0

0.05 1.761 17.78 8.03 0.31 0.05 0.16 0.006 0.042 0.01 6.44 −1.49 29.19 848.33 0.25

0.03 1.61 0.68 1.59 0.53 0.05 0.15 0.016 0.07 0.08 34.74 66.79 28.16 282.78 0.14

0.028 1.32 18.13 8.32 0.15 0.044 0.26 0.015 0.10 0.01 0.000125 24 23.8 1078.33 0.37

Output

Units

Maximum

Minimum

Mean

SD

DIM

–

1

0

0.22

0.24

2. Results and Discussion

2.2. Empirical modelling

2.1. Analytical procedure

A neural network is a general method of regression analysis in which a very flexible non-linear function is fitted to the experimental data. It can capture the enormous complexity in the database, which avoid over fitting [28]. It is nevertheless useful to discuss some salient features, to place the technique in the context. The Bayesian framework of neural network has been used in this present study. A neural network is generally trained on a set of examples of input and output data with repetitive representations. The outcome of training is a set of co-efficient (i.e. weights) and a specification of the functions, which in combination with the weights correlating the inputs to the output. The training process itself involves a search for the optimum non-linear correlation between the inputs to the output and is computer intensive. Once the neural network is trained, the estimation of the output for any given inputs is very easy. The details of this method used here have recently been comprehensively reviewed by MacKay [29] and the original method is described thoroughly elsewhere [29–34]. One of the difficulties with the blind data modelling is that of over fitting, in which spurious details and the noise in the training data are over fitted by the model. This gives rise to solutions that generalise poorly. MacKay [29–34] and Neal [35] have developed a Bayesian framework for neural networks in which the appropriate model complexity is inferred from the database. This Bayesian neural network modelling has two important advantages. Firstly, the significance of all the input variables is quantified automatically, which is extremely important to understand the response of each variable. Consequently, the model perceived significance of each input variable can be compared against the existing metallurgical theory. Secondly, the neural network’s predictions are accompanied by error bars, which depend on the specific position in input space. This quantifies the model’s certainty about its predictions. In this present study, both the inputs and output variables were first normalised within the range ±0.5 as follows:

We have extensively carried out literature study to understand the martensitic transformation micro-mechanisms and their interpretation while explaining the mechanical behaviour of austenitic stainless steels under various operating conditions. For the present model, inputs are chosen according to the knowledge gained from the published literatures and from the industrial experiences. The inputs of the model are chosen to be: chemistry of austenitic stainless steels, strain rate, initial austenite grain size, temperature of testing, true stress and true strain. The target (i.e. output) is the extent of DIM. The other influencing parameters for martensitic transformations are stress state, initial microtexture of austenite and strain path, which were not, included as input parameters because there is lack of published data available. In most of those literatures, we considered those studies, which deal with different grades of austenitic stainless steels with different grain sizes under uniaxial loading at various testing conditions. In most of those literatures, DIM is generally represented as strain induced martensite. Various techniques have been used for quantifying the DIM formation in those literatures, such as: XRD, magnetic methods etc. The most common and available graphs are found to be: (a) stress–strain and (b) DIM as a function of strain. We have extensively extracted data (i.e. strain, DIM fraction and corresponding stress value) from those two graphs with their corresponding testing conditions and material history reported in corresponding literatures. We have tabulated all the data in a single spreadsheet and the size of the database, which are used for neural network analysis to be 1600 rows. The literatures (1954–2010), from where those data digitized are mentioned to be [1,4–27]. The statistics of the whole database are given in Table 1. It is emphasised that unlike linear regression analysis, the ranges stated in Table 1 cannot be used to define the range of applicability of the neural network analysis. This is because the inputs are in general expected to interact each other. We shall see later that it is the Bayesian framework of our neural network analysis, which makes possible the calculation of error bars whose magnitude varies with the position in the input space, which define the range of useful applicability of the trained network. A visual impression of the spread of the data is shown in Fig. 1(a–o).

xN =

x − xmin − 0.5 xmax − xmin

(1)

where xN is the normalised value of x; xmin and xmax are respectively the minimum and maximum values of x in the entire dataset (Table 1). The normalisation is straightforward for all the quantitative variables. The normalisation is not necessary for this analysis but facilitates the subsequent comparison of the significance of

1.0

(a)

0.6 0.4 0.2 0 2 0.0 0.00

0.02

0.04

0.06

0.08

0.10

0.8 0.6 0.4 0.2 0 2 0.0

0

1

2

C - Content / wt% Measured martensite m fraction

0.4 0 4 0.2 0

3

6

9

12

15

9 10

(e)

0.8 0.6 0 4 0.4 0.2 0.0 0.0

0.5

1.0

1.5

2.0

2.5

3.0

0.6 0 6 0.4 0.2 0.0 0.0

0.2

0.4

0.6

0.8

(h)

0.8 0.6 0 6 0.4 0.2 0.0 0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.4 0.2 0.15

0.30

0.45

0.60

0.75

(k)

0 8 0.8 0.6 0.4 0.2 0.0

0

50

Ti - Content / wt%

100

150

Strain rate / s

200

250

0.8 0.6 0.4 0.2 0

25

50

75 100 125 150 175 200

Grain size / μm

(n)

0.8 0.6 0.4 0.2 0.0

18.0

18.5

(f)

0.8 0.6 0 4 0.4 0.2 0.0 0.00

0.05

0.10

0.15

0.20

0.25

1.0

(i)

0.8 0.6 0 6 0.4 0.2 0.0 0.00

0.05

0.10

0.15

0.20

0

300

600

900

1200 1500 1800

True stress / MPa

(l)

0 8 0.8 0.6 0.4 0.2 0.0

-1

1.0

(m)

Measured martensite fraction

1.0

17.5

1.0 Measured martensite fraction

0.6

Measured martensite fraction

(j)

17.0

Co - Content / wt%

1.0

0.0 0.00

16.5

Nb - Content / wt%

1.0 0 8 0.8

0.0 16.0

N - Content / wt% nsite fraction Measured marten

(g)

0.8

0 2 0.2

Cr - Content / wt%

1.0 Measured martensiite fraction

Measured martensiite fraction

8

0.4

1.0

Cu - Content / wt% Measured martensite fraction

7

1.0

Ni - Content / wt%

Measured martensite fraction

6

0.6

Mo - Content / wt%

1.0

0.0

5

0

50

100

150

200

250

Temperature / ºC Measured martensite fraction

Measured martensite fraction

(d)

0.6

0.0

4

(c)

0.8

Mn - Content / wt%

1.0 0.8

3

Measured martensite fraction

0.8

11

1.0

(b)

Measurred martensite fraction

1.0 Measurred martensite fraction

Measu ured martensite fraction

A. Das et al. / Materials Science and Engineering A 529 (2011) 9–20

1.0

(o)

0.8 0.6 0.4 0.2 0.0 0.0

0.2

0.4

0.6

0.8

True strain

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 stress and (o) true strain versus the volume fraction of DIM in a variety of austenitic grade stainless steels.

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A. Das et al. / Materials Science and Engineering A 529 (2011) 9–20

where the data set {xm , tm } consists of xm inputs related to a particular target tm (m is a level of pairs). The objective is to determine a set of weights in a neural network that minimizes ED but without over fitting to noise. Thus the regularisers, Ew are included so that smooth solutions of y(xm ,w) are favoured and the possibility of fitting to noise in the experimental data can be reduced. The coefficients w and biases , which are shown in Eqs. (2) and (3) make up the parameter vector w. A number of regularisers Ew(c) are added to the data error. The regularisers favour functions y (x; w), which are smooth functions of x. The simplest regularisations method uses a single regulariser:

θ1 h1

w11 x1

θ2

w12

h2

w13

x2

θ θ3

w14

h3

w1 w2 w3

θ

y

Ew =

θi

wij



hi

Fig. 2. A typical network used in the analysis.

each of the variables. The neural network consisted of fifteen input nodes, one hidden unit and an output node representing the extent of DIM (schematically shown in Fig. 2). Several models were created with various combinations of hidden units with varying their corresponding nodes. The database was randomised properly and then partitioned equally into testing and training data sets. The later was used to create a large variety of neural networks models whereas the testing data set was used to see how the trained models generalised on unseen data. The inputs and output as shown are connected through hidden units where the inputs xj are operated by a hyperbolic tangent transfer function to obtain the hidden units, hi defined as:

⎛



hi = tanh ⎝

⎞

(1)

(1)

wij xj + i

⎠

(2)

j

The bias is designated  i and is analogous to the constant that appears in linear regression analysis. The strength of the transfer function is in each case determined by the weight wij . The transfer to the output y is linear: y=



(2)

wi hi +  (2)

(3)

i (2)

where wi and  (2) are a new set of weights and a bias, respectively. Eqs. (2) and (3) define the neural network structure that connects the inputs to the output. The weights and biases, however, are unknown to be determined through training process using the Bayesian back propagation scheme, which involves a minimization of the energy function [30]; the minimization was implemented using a variable metric optimizer [36]. The gradient of M(w) was computed using back propagation algorithm [37]. The energy function consists of the error function, ED and regularisation Ew . M(w) = ˇED +



ac Ew(c)

(4)

c

The error function, ED (w) is the sum squared error as follows: ED (w) =

1  2 (yi (xm , w) − t m ) 2 m

i

wi2

(6)

Here, however, we have used a slightly more complicated regularisation method known as the automatic relevance determination model which has been described elsewhere [3,33]. Each weight is assigned to a class c depending on which neurons it connects. For each of the input, all the weights connecting that input to the hidden units are in a single class. The hidden units’ biases are in another class, and all the weights from the hidden units to the outputs are in a final class. Ew(c) is defined to the sum of the squares of the weights in class c [31]:

wi xj

1 2

(5)

Ew(c) (w) =

1 2 wi 2

(7)

i=c

The additional terms favour small values of w and decrease the tendency of a model to overfit noise in the data set. The control parameters ˛c and ˇ together with the number of hidden units determine the complexity of the model. These hyper parameters define the assumed Gaussian noise level of: v2 = 1/ˇ and 2 the assumed weight variances: w(c) = 1/˛c . v is the noise level inferred by the model. The parameter ˛ has the effect of encouraging the weights to decay. Therefore, a high value of  w implies that the inputs parameter concerned explains a relatively large amount of the variation in the output. Thus  w is regarded as a good expression of the significance of each input thought not of the sensitivity of the output to that input. This has been described thoroughly elsewhere [3,31]. The error, ED is expected to increase if important input variables have been excluded from the analysis. Whereas ED 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 [29–33] has developed a particularly useful treatment of neural network analysis in a Bayesian framework, which allows the calculation of error bars representing the uncertainty in the fitting parameters. The method recognises that there are many functions that can be fitted or extrapolated into certain 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 set of weights is used to define the fitting uncertainty. The error bars therefore, become large when data are sparse or locally noisy. The error bars presented throughout the whole work, therefore, represent a combination of the perceived level of noise in the output (i.e. DIM) and the fitting uncertainty as described above. This specification of the network structure, together with the set of weights is a complete description of the formula relating to the inputs to the output. The weights are determined by training the network; the details are described thoroughly elsewhere [29–34]. The number of hidden units used determines 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 of random seeds. The term,  v used here is the framework estimating the overall noise level of the database. The complexity of the

A. Das et al. / Materials Science and Engineering A 529 (2011) 9–20

0.090

13

3.5

3.0

0.075

Test error

σν

2.5

0.060

0.045

2.0

1.5

0.030 1.0

0

3

6

9

12

15

18

0.5

Hidden units

0

Real function

Overfitted Fig. 4. Overestimation of function [3].

10

15

20

Hidden units

Fig. 3. Variation in  v as a function of the number of hidden units. Several values are presented for each set of hidden units because the training for each network was started with a variety of random seeds.

model is controlled by the number of hidden units and the values of the regularisation constants ( w ), one associated with each of the inputs, one for biases, and one for all weights connected to the output. Fig. 3 shows that the inferred noise level,  v decreases monotonically as the number of hidden units increase. However, the complexity of the model also increases with the number of hidden units. A high degree of complexity may not be justified, and in the extreme case, the model may in a meaningless way attempt to fit the noise in the experimental data. More complex relations can be modelled with a large number of hidden units. However, the function may then be over fitted, as shown in Fig. 4, because experimental data always contain errors. This has been discussed in detail elsewhere [3]. In order to reduce over fitting, the test error (the value of the error function for nontrained data set) was measured, using 800 randomly chosen rows of data, which were not included in the training set. Fig. 5 shows the change in test error as function of hidden units. MacKay [29–33] has made a detailed study of this problem and has defined a quantity, evidence, which comments on the probability of a model. In circumstances where two models give similar kind of results over the known dataset, the more probable model would be predicted to be that which is simpler; this simple model would have a higher value of evidence. The evidence framework was used to control the regularisation constants and  v . The number of hidden units was set by examining performance on test data (Fig. 3). A combination of Bayesian and pragmatic statistical techniques were, therefore, used to control the model complexity. A further procedure used to avoid the over fitting problem was to divide the experimental data randomly into two equal sets namely the training and test data sets. The models are created using the

5

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 how well the model generalises. A good model would produce similar levels of error in both the test and training data whereas an over fitted model might accurately predict the training data but badly estimate the unseen test data. Once the correct complexity of the model has been determined using this procedure, it can be retained using 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 value from the experimental one in the test data: Te =

1 (yn − tn )2 2

(8)

n

where yn is the predicted amount of DIM fraction and tn is its measured value. The test error defined as the value of the error function for unseen data is shown in Fig. 5. The best model may be defined as that with the smallest test error. This would be appropriate, for situation where only scalar prediction (i.e. no error bars) are required. MacKay has shown when making predictions with error bars, the best model should be decided according to a quantity the log predicted error (LPE). Using the LPE, unlike the test error, wild predictions are penalised less if they have large error bars when using noisy data, common in many experimental situation, some wild predictions must be expected. Assuming that for each example n the model gives a prediction with error (yn , v2 ), the LPE is (Fig. 6): LPE =

  0.5(tn − yn )2 n

n2



+ log(



2v )

(9)

The behaviour of the training and testing data are shown in Figs. 7 and 8 respectively which show a similar degree of scatter in both the graphs, indicating that the complexity of the respective model is optimum. It should be noted that the test data cover a wide range of DIM fraction value and, for a very few cases at the highest amount of DIM, the model under predicts the measured values. Over fitting would lead to an apparently better accuracy in the prediction of training data when compared with the test data set. The error bars in Figs. 7 and 8 include the error bars on the underlying 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 noisy data that models with different complexity make different predictions. In these circumstances, the prediction made by a committee of models may be more reliable than using a single model. Fig. 9

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A. Das et al. / Materials Science and Engineering A 529 (2011) 9–20

1.00

1000

0.95

Combined test error

Log predictive error

950

900

850

800

0.90 0.85 0.80 0.75 0.70 0.65

750 0

5

10

15

0

20

5

Fig. 6. LPE as a function of the number of hidden units.

15

20

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 to the magnitude of the test error. We start a committee by using N models ranked by LPE. The committee is formed through combining the best N models (where N = 1, 2, 3. . .) such that the mean prediction of the committee is:

0.6

Training data

Predicted martensite fraction

10

Number of models

Hidden units

0.3

1 yi N N

yn =

(10)

i=1

with associated error in yn expressed as:

0.0

2 = -0.3

-0.6 -0.6

-0.3

0.0

0.3

0.6

Measured martensite fraction Fig. 7. Plot of the estimated versus measured DIM fraction – training dataset.

1 2 1 i + (yi − yn )2 N N N

N

i=1

i=1

(11)

Fig. 9 shows the changes in the test error with the number of models used to form a committee. The figure shows that fifteen models committee is favourable for the DIM fraction. Committee predictions are compared against experimental data in Fig. 10. The behaviour of the committee model consisting of individual models retrained on the entire data set is illustrated in this figure. The inputs to output mapping becomes more accurate after retraining. The purpose of the division into training and test data was to identify models with the optimal level of complexity. Once that is done, 1.0

0.6

Testing data Calculated martensite fraction

Predicted martensite fraction

0.4

0.2

0.0

-0.2

0.8

0.6

0.4

0.2

-0.4 0.0

-0.6 -0.6

0.0

-0.3

0.0

0.3

0.6

0.2

0.4

0.6

0.8

1.0

Measured martensite fraction

Measured martensite fraction Fig. 8. Plot of the estimated versus measured DIM fraction – test dataset.

Fig. 10. Training data for best committee model (training was done on whole dataset); error bars includes errors in underlying functions only.

A. Das et al. / Materials Science and Engineering A 529 (2011) 9–20

0.20

0.8

Calculated martensite fraction

Calculated martensite fraction

1.0

15

0.6

0.4

0.15

0.10

0.05

0.2 2

R = 0.88271

2

R = 0.9421

0.00

0.0

0.00 0.0

0.2

0.4

0.6

0.8

1.0

0.05

0.10

0.15

0.20

Measured martensite fraction

Measured martensite fraction Fig. 11. Application of the best model for the blind literature data [38] source.

it is quite reasonable to use the entire data set for retraining, but without changing the complexity of the model. Since the committee complexity is not changed after retraining, its ability to generalise is not significantly affected. 2.3. Application of the model We now examine the metallurgical significance of the results. The optimized committee model was used to study the effect of individual variables on the formation of DIM to find out whether the results are compatible with known metallurgical principles and other published data. Figs. 11–13 show the application of our committee model from the very recent published data by Lindgren et al. [38], Hauild et al. [39] and the present authors [40,41] respectively which were not included in the training as well as in the testing of neural network. From these three graphs, it is noted that our model is robust to predict the total blind data. As it has been shown in Figs. 11–13, the correlation coefficient (i.e. adjusted R2 ) for best

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. This means that a good correlation (i.e. reasonable accurate) between the measured and calculated data have been obtained for all the applications. These figures show that the used network could be capable 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 confidence of prediction that is one of the strong points of Bayesian neural network. Since reserved data are equivalent to new experiments, it may be possible to predict the extent of DIM of a new austenitic stainless steel type with a similar precision as long as the inputs are in the range of Table 1. Though the model tends to slightly under predicts and over predicts at lower and higher DIM fractions respectively, the predictions are quite close to the experimental results. At small strain, defects introduced into the austenite enhance the nucleation rate; therefore the actual fraction of martensite will be greater than when only stress is accounted for. At large strains, the defects oppose transformation by mechanical stabilisation. So a calculation based on stress alone will overestimate the amount of martensite (Figs. 11 and 12).

1.0

Calculated martensite fraction

2.4. Prediction of the model The optimized committee model has been used to predict the influence of individual input variables on the formation of DIM in the following subsections. The prediction (Fig. 14) has been made for a specific 304 stainless steel. It is important to note that predictions are for the case where just one input variable is altered, keeping all other fixed. This may not be possible when conducting experiments. Fig. 16 also shows the significance of input variables on DIM formation. Now we shall investigate the isolated influence of all the individual variables on martensitic transformation with the support of extensive literature study.

0.8

0.6

0.4

0.2 2

R = 0.95579 0.0 0.0

0.2

0.4

0.6

0.8

1.0

Measured martensite fraction Fig. 12. Application of the best model for the blind literature data [39] source.

2.4.1. Effect of alloying elements on martensitic transformation Alloying 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 determines the austenite stability. The present model predicts the variation of DIM formation as a function of carbon content

A. Das et al. / Materials Science and Engineering A 529 (2011) 9–20

1.0 0.8 0.6 0.4 02 0.2

(a) 0.02

0.04 0.06 0.08 C - Content / wt%

0.8 0.6 0.4 02 0.2

(b) 0.0

0.10

1

2

1.0

6

7

8

9

0.6 04 0.4

(d) 4

5

6

7

8

9

10

0.8 0.6 04 0.4 0.2

(e)

11

0.0 0.0

12

06 0.6 0.4

(g) 0.1

0.2

0.3

0.4

0.5

0.6

0.7

Cu - Content / wt%

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

0.8 06 0.6 0.4 0.2

(h)

0.0 0.00

0.4

(j)

0.0 0.0

0.1

0.2

0.3

0.4

0.5

0.02 0.04 0.06 0.08 Nb - Content / wt%

Predicted martensite fraction

0.6

0.4

0.2 50

75

100

125

Grain size / μm

150

0.2

(f) 0.05

175

200

0.10

0.15

0.20

(i)

0.2 0.00

0.05 0.10 0.15 Co - Content / wt%

0.20

0.6 0.4 0.2

(k)

0.8 0.6 0.4 0.2

(l)

0.0 20

40

60

-50 -25

80 100 120 140 160 180 200 -1

Strain rate / s

0.8

25

04 0.4

0.4

0.10

08 0.8

0

(m)

0

0.6

1.0

Ti - Content / wt%

1.0

18.5

0.6

0.0

0.6

18.0

0.8

0.8

Predicted martensite fraaction

Predicted martensite fraaction

0.6

17.5

1.0

1.0

0.8

17.0

N - Content / wt%

1.0

1.0

0.2

16.5

0.0 0.00

Predicted martenssite fraction

Predicted marten nsite fraction

0.8

0.0 0.0

(c)

Mo - Content / wt%

1.0

0.2

0.2

Cr - Content / wt%

Predicted m martensite fraction

0.8

0.2

0.4

1.0

Ni - Content / wt%

Predicted martenssite fraction

5

1.0

3

Predicted martensite fraaction

4

0.6

Mn - Content / wt%

0.0

Predicted martensite fraction

3

0.8

0.0 16.0

25

50

75 100 125 150 175 200 0

C

1.0

1.0 0.8 0.6 0.4 0.2 0.0 600

0

Temperature /

(n) 800

1000

1200

1400

True stress / MPa

1600

Predicted martensite fraction

martensite fraction Predicted m

0.0

1.0

Prediccted martensite fraction

Prediicted martensite fraction

1.0

Predicted m martensite fraction

Prediicted martensite fraction

16

0.8 0.6 0.4 0.2

(o) 0.0 0.0

0.1

0.2

0.3

0.4

0.5

0.6

True strain

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 the formation 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 large error bars suggest lack of sufficient data in the range examined.

A. Das et al. / Materials Science and Engineering A 529 (2011) 9–20

100

Martensite fraction / %

80

60

40

20

Cold rolled Cold drawn

0 0

1

2

3

4

Equivalent strain Fig. 15. Changes in the volume fraction of DIM during cold rolling and cold drawing [62].

(Fig. 14(a)). With the increase in carbon content, DIM formation decreases drastically and getting saturated beyond 0.06%. Krupp et al. [42] investigated that the effect of carbon content and/or the temperature on the formation of DIM is very strong in such a way that high carbon content and elevated temperatures stabilises the austenite phase. The effect of manganese on martensitic transformation is shown in Fig. 14(b). It is predicted that with the increase in manganese content, DIM formation decreases. Manganese by itself tends to increase the stability of austenite by decreasing its Ms temperature. This is a good example of the safety of the predictions made by the model, in that the error bars are large when the model is uncertain. Chromium is the most essential element added in stainless steels mainly responsible for corrosion resistance. The estimated effect of chromium on DIM transformation is shown in Fig. 14(c). It is predicted that with the increase in chromium, DIM formation decreases. It is worthy to mention that although chromium is a weak ferrite stabiliser, its influence on Ms temperature is very strong. Fig. 14(d) shows the prediction of DIM formation as a function of nickel content. Small increment in nickel concentration drastically reduces the rate of isothermal transformation of DIM. Nickel is responsible for high toughness and high strength at both high 3.0

Significance

2.4

1.8

1.2

0.6

C M arb an o ga n Ch ne ro se m iu m M Nic ol ke yb l dn N um itr og Co en p N per io bi um Co Ti bal ta t St nium Te rain m ra pe te ra G tur ra e in siz e St re ss St ra in

0.0

Fig. 16. Bar chart showing a measure of the model perceived significance of each of the input variables in influencing martensitic transformation.

17

and low temperatures without sacrificing the ductility in stainless steels. The effect of adding molybdenum on the stability of austenite as predicted by the model is shown in Fig. 14(e). It increases the resistance towards crevice and pitting corrosion. From Fig. 14(e), it is predicted that as molybdenum content increases, DIM transformation suppresses. This result is in accordance with those reported in the literature demonstrating the role of molybdenum, which slightly increases Ms temperature [43]. Nitrogen is added in austenitic stainless steel to improve the yield strength by refining its grains. It is noted from Fig. 14(f) that with the increase in nitrogen content up to 0.10% (approximately), DIM transformation decreases. According to Lee et al. [44], increasing the nitrogen content causes the transient strain for DIM formation to shift at higher strain and, finally, DIM will not form when nitrogen content is beyond 0.50 wt%. Lee et al. [45] evaluated the effect of nitrogen on DIM formation in 304 stainless steels based on the proposed kinetics relation between DIM and inelastic strain and reported that nitrogen addition reduces the austenite stability parameter, which is inversely proportional to the austenite stability, leading to decrease in Md30 temperature. Recently, Bracke et al. [46] reported that chromium and nitrogen suppressed the DIM transformation in Fe–Mn–Cr–N steels and the differences in transformation behaviour are attributed to the change in the intrinsic stacking fault energy. Copper is normally present in austenitic stainless steels as residual alloying element and it increases the tensile strength. However, it is added to a few alloys to produce precipitations hardening properties or to enhance the corrosion resistance. Present model predicts that copper does not have any significant effect on DIM formation in austenitic stainless steels (Fig. 14(g)). The effect of copper on Ms temperature is not as clear as the austenite stabilising elements (i.e. Mn, Ni etc.). Capdevilla et al. [43] has suggested that for copper concentration up to 1.0 wt%, this element does not influence Ms temperature. On the other hand, Hong and Koo [47] suggested that the addition of copper can suppress the formation of DIM during tensile testing to prevent strain hardening. The effect of cobalt content on DIM formation is shown in Fig. 14(i). It is predicted that with the increase in cobalt concentration, the tendency of DIM formation increases. Present result is fully consistent with those reported in the literature demonstrating the role of cobalt increasing Ms temperature [43]. Capdevilla et al. [43] also suggested that the additions of cobalt change the tendency of Ms temperature depending on the chromium concentration. Co–Cr alloys are well suited to high temperature creep and fatigue resistance applications. The effect of adding micro alloying elements on the stability of the austenite (i.e. DIM fraction) as predicted by the model is shown in Fig. 14(h and j). Since the error bars in Fig. 14(h) are large because of the insufficient experimental data, it cannot be concluded that niobium really affects DIM transformation. In austenitic stainless steels, it is generally added to improve the resistance to intergranular corrosion but it also enhances the mechanical properties at high temperatures. From Fig. 14(j), it is predicted that titanium enhances the martensitic transformation. The addition of titanium can influence the stability of austenite phase especially under strained condition. Titanium is generally added for carbide stabilisation especially when austenitic stainless steels are to be welded. 2.4.2. Effect of strain rate on martensitic transformation Most investigations carried out in order to clarify the effect of strain rate on DIM formation, which has indicated that the transformation is suppressed with increasing strain rate. This has been mostly explained in terms of the adiabatic heating, which decreases the chemical driving force of the transformation. However,

18

A. Das et al. / Materials Science and Engineering A 529 (2011) 9–20

Staudhammer et al. [48] suggested that high strain rate may promote more irregular shear band arrays compared to the low strain rate. This may lead to a reduced probability of the formation of ˛ (bcc) martensite embryos of the critical size, and thus, suppresses the formation of ˛ (bcc) martensite. On the other hand, it has been found that high strain rate, (103 s−1 ) promoted shear band formation in 304 stainless steel compared to low strain rate, (10−3 s−1 ). This led to an increased number of shear band intersections and higher volume fraction of ˛ (bcc) martensite at the early stages of tensile 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) martensite transformation was suppressed at high strain rate. This was attributed to the rise in temperature (i.e. adiabatic heating) at high strain rates which would suppress the martensitic transformation. Present authors [40,41] have already investigated the role of strain rate on the formation of DIM in 304LN stainless steels experimentally. In the present model, we have predicted the influence of strain rate on DIM formation (Fig. 14(k)). It has been found that with the increase in strain rate, martensitic transformation suppresses drastically. 2.4.3. Effect of testing temperature on martensitic transformation Martensitic transformation is affected by two composition dependent parameters (i.e. stacking fault energy and GCHEM ). Since both of these parameters are temperature dependent, the tendency to the DIM formation is sensitive to the temperature, as well. It is well known that the DIM formation is suppressed with increasing temperature [4]. The behaviour is normally attributed to the decrease in the GCHEM with increasing temperature. In the present model, we have demonstrated the influence of temperature on DIM formation (Fig. 14(l)). It is noted that with the increase in temperature, DIM transformation decreases. As it is well known that stability of austenite increases with increase in temperature and on increasing the temperature beyond a limit Md , no transformation takes place [50]. The predicted DIM fraction, which is shown in Fig. 14(l) decreases with increase in testing temperature from −75 to 100 ◦ C (approximately) and beyond that there is uncertainty in prediction. Sugimoto et al. [51] found that on increasing the temperature further to 300 ◦ C, the predicted stability of austenite decreases for TRIP aided dual phase steel, which has been reported elsewhere [51], where the authors have found that beyond a certain temperature, austenite undergoes strain induced bainitic transformation, which results in a decrease in stability. It is noted from (Fig. 14(l)) that the error bars associated with these high temperature predictions (i.e. beyond 100 ◦ C) are very large. This is a reflection of the limited number of high temperature data in the training data set, which result in rather uncertain predictions. Nevertheless, it is noteworthy that in spite of the limited data the network is able to capture quite accurately the effect of temperature. 2.4.4. Effect of austenite grain size on martensitic transformation Gonzales et al. [52] studied the effect of austenite grain size on martensitic transformation in 304 stainless steel. The transformation was found to be enhanced by large grain size. Varma et al. [7] also found that the large grain size promoted DIM formation during tensile and cold rolling of 304 and 316 stainless steels both. In contrast, Srinivas et al. [53] found that the formation of DIM during cold rolling increases with decreasing grain size in 304 stainless steel and is grain size independent in 316 stainless steel. The predicted effect of grain size on the martensitic transformation is shown in Fig. 14(m). From this figure, it is noted that with the increase in austenite grain size, DIM formation enhances drastically. According to Yang and Bhadeshia [54], the extent of DIM in the early stages of transformation is proportional to the cube of the austenite grain size. They have clearly explained that there is a large dependence

of Ms temperature as a function of austenite grain size. According to Guimarães et al. [55], a fine austenitic grain size should be expected to shift MS and MS to a lower temperature and a higher stress value, respectively, as observed experimentally. Hence, the observation strongly conforms to the published theory. 2.4.5. Effect of stress on martensitic transformation Patel and Cohen [56] have described the criterion for the application of external stress on martensitic transformation in their elegant study. According to them, when the external force is acting, the resulting effect on the Ms temperature is calculated from the mechanical work done (i.e. GMECH ) on or by the transforming region as the resolved shear stress and normal components of the applied stress are carried through the corresponding transformation strains. This work done (U) on or by the transformation due to the action of applied stress is comprised of: ( 0 ), the shear stress resolved along a potential habit plane times the transformation shear strain, and (∈0 ), the normal stress resolved perpendicular to the habit plane times the normal component of the transformation strain. Thus U = 0 + ε0

(12)

 is numerically positive when the normal stress is tensile, and negative when this component is compressive.  is always taken to be positive because the many habit permutations (±{259} in these alloys) virtually permit shearing in either sense [56]. Hence in 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, the transformation is aided by both the shear and (positive) normal components of stress, and therefore the Ms is raised even more than in the case of uniaxial compression. At temperatures just above Ms , transformation can be induced via stress assisted nucleation on the same heterogeneous sites responsible for the transformation on cooling according to Stringfellow et al. [57]. Hong-zhuang et al. [58] also concluded in their study that as the applied stress,  is increased, the maximum magnitude of the transformation induced plasticity effect along the longitudinal direction decreases. Recently we have demonstrated that a large amount of published data relating the fraction of DIM to plastic strain can in fact be described in terms of the pure thermodynamic effect of applied stress [59]. In the present model, we have demonstrated the effect of true stress on DIM formation (Fig. 14(n)). It shows that with the increase in true stress, the formation of DIM increases drastically. Hence, the role of stress on martensitic transformation is convincingly revealed. 2.4.6. Effect of stress-state and strain on martensitic transformation In the temperature range above MS and below Md , the transformation is dominated by strain induced nucleation on potent nucleation sites created by plastic strain [60]. Iwamoto et al. [61] found that in compression the transformation rate was initially higher than in tension, but at higher strains the relation was reversed. Hecker et al. [6] also found that more ˛ (bcc) martensite was formed in biaxial tension than in uniaxial tension. Fig. 15 demonstrates the influence of stress-state on the formation of DIM in cold rolled and cold drawing 316 stainless steel [62]. Due to lack of data in the published domain, we could not incorporate stress-state as of input parameter, which plays a significant role on martensitic transformation. We have also predicted the influence of true strain on DIM formation (Fig. 14(o)). It has been found that strain alone is having very little influence on martensitic transformation. As strain is increasing, DIM volume fraction increases slightly. The independent effect of strain is seen to be minor and more uncertain. These results confirm the earlier conclusion that in experiments where martensite is stimulated during a tensile test, it

A. Das et al. / Materials Science and Engineering A 529 (2011) 9–20

is the mechanical driving force, which plays a dominant role rather than strain induced transformation. This empirical analysis of a wide range of data supports the predominant role of stress over plastic strain. 2.4.7. Effect of microtexture of parent austenite on martensitic transformation Initial microtexture of austenite is an important parameter to control the amount of DIM while application of external stress. According to Wang et al. [63], cold rolling generates residual stress, which is dependent not only on the specimen sections but also on the grain orientations. Nakamura and Wakasa [64] have also investigated the variation of DIM in a two phase steel having textures. In the present model, it was not possible to include the microtexture data of the initial austenite as one of the input parameters because of lack of data availability in the published domain. Recently Kundu and Bhadeshia [65] have clearly demonstrated in their elegant transformation texture model that the initial microtexture of austenite strongly influence the variant selection of DIM transformation in stainless steels. 2.4.8. Significance of input variables on martensitic transformation In the present model, it has been possible to show the isolated influence of input parameters. The Bayesian neural network modelling has an excellent advantage to calculate the significance of the input variables, which has been clearly demonstrated by MacKay [29–33] in his pioneer studies. Fig. 16 shows the perceived significance of the inputs for the best model. The parameter,  w is rather like a partial correlation coefficient in linear regression analysis in that it represents the amount of variation in the output that can be attributed to any particular input parameter and does not necessarily represent the sensitivity of the output to each of the inputs. It should be noted that it does not indicate the sensitivity of the output to the inputs. The sign of the effect is indicated in Fig. 16 for each variable that has been perceived to have a high significance. It is clearly understood that the effect of temperature and stress are more predominant than the rest. From these graph, we can draw the importance of each input variables on DIM formation. 3. Conclusions Estimation of DIM has been investigated with its influencing parameters in a variety of austenitic grade stainless steels through Bayesian neural network modelling. The obtained results are summarised as follows: a) A neural network model has been developed to predict the nature of variation of DIM with its influencing parameters which can be applied for designing the nuclear power plant components, automobile industries etc. The model has been extensively applied for recent published data, which matches almost accurately. b) The model predictions confirm the important effect of C, Mn, Ni, Mo, N, Co, Ti on the variation of DIM formation. It also interestingly brings out the insignificant effect of few alloying elements (i.e. Cu, Nb, etc.). c) The increase in austenite grain size enhances martensitic transformation. DIM formation suppresses while increasing strain rate, which is mainly attributed to the increase in adiabatic heat. With the increase in temperature, DIM formation triggers drastically. d) The remarkable result is that the role of stress is convincingly revealed, whereas the independent effect of strain is seen to be minor and more uncertain. These results confirm the earlier conclusion that in experiments where martensite is stimulated

19

during a tensile test, it is the mechanical driving force, which plays 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 the BOYSCAST Fellowship. The authors are extremely grateful to Professor H. K. D. H. Bhadeshia, University of Cambridge, UK, for the provision of Neuromat Neural Network software for the present analyses. The authors would like to thank the respected reviewers for their positive and constructive comments for this manuscript.

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