Modeling medium carbon steels by using artificial neural networks

Materials Science and Engineering A 508 (2009) 93–105

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Materials Science and Engineering A journal homepage: www.elsevier.com/locate/msea

Modeling medium carbon steels by using artificial neural networks N.S. Reddy a,∗ , J. Krishnaiah b , Seong-Gu Hong c , Jae Sang Lee a a Alternative Technology Laboratory, Graduate Institute of Ferrous Technology (GIFT), Pohang University of Science and Technology, Pohang (POSTECH), Kyungbuk 790-784, Republic of Korea b Formerly scholar at Department of Mechanical Engineering, Indian Institute of Technology, Kharagpur 721302, India c Center for New and Renewable Energy Measurement, Division of Industrial Metrology, Korea Research Institute of Standards and Science, 209 Gajeong-Ro, Yuseong-Gu, Daejeon 305-340, Republic of Korea

a r t i c l e

i n f o

Article history: Received 7 October 2008 Received in revised form 11 December 2008 Accepted 11 December 2008 Keywords: Artificial neural networks Low alloys steels Mechanical properties Heat treatment parameters Alloy design

a b s t r a c t An artificial neural network (ANN) model has been developed for the analysis and simulation of the correlation between the mechanical properties and composition and heat treatment parameters of low alloy steels. The input parameters of the model consist of alloy compositions (C, Si, Mn, S, P, Ni, Cr, Mo, Ti, and Ni) and heat treatment parameters (cooling rate and tempering temperature). The outputs of the ANN model include property parameters namely: ultimate tensile strength, yield strength, percentage elongation, reduction in area and impact energy. The model can be used to calculate the properties of low alloy steels as a function of alloy composition and heat treatment variables. The individual and the combined influence of inputs on properties of medium carbon steels is simulated using the model. The current study achieved a good performance of the ANN model, and the results are in agreement with experimental knowledge. Explanation of the calculated results from the metallurgical point of view is attempted. The developed model can be used as a guide for further alloy development. © 2008 Elsevier B.V. All rights reserved.

1. Introduction The properties of alloy steels mainly depend on the mechanical treatment, alloying elements and heat treatment variables. These three affect the microstructure, which in turn affects the properties. All alloys have defects such as intrusions, micro-blowholes and cracks. Apart from these defects, when alloys are made under industrial conditions, some amount of inhomogeneity and inconsistencies invariably creep in. These make the system-model chaotic. Besides this, the input variables that are considered are in reality fuzzy in nature. The first hurdle to overcome during modeling of steels is to acquire a reliable database. The industries that manufacture these alloys (or steels) tend to classify their process variables for obvious reasons. It is therefore very difficult to get a data set or information that would consist all of the above details. As the relationships between these outputs and inputs are nonlinear and complex in nature, it is impossible to develop them in the form of mathematical equations. Techniques such as linear regression are not well suited for accurate modeling of data which exhibits considerable ‘noise’, which is usually the case. Regression analysis to model non-linear data necessitates the use of an equation to attempt to transform the data into a linear form [1]. This represents an approximation that inevitably introduces a significant

degree of error. Similarly, it is not easy to use statistical methods to relate multiple inputs to multiple process outputs. The method using neural networks (NN), on the other hand, has been identified as a suitable way for overcoming these difficulties. NN are mathematical models and algorithms that imitate certain aspects of the information-processing and knowledge-gathering methods of the human nervous system [2–5]. A NN can perform highly complex mappings on nonlinearly related data by inferring subtle relationships between input and output parameters [6–8]. It can, in principle, generalize from a limited quantity of training data to overall trends in functional relationships [9]. Although several network architectures and training algorithms are available, the feed-forward neural network with the back-propagation (BP) learning algorithm is more commonly used [8]. Therefore, within the last decade, the application of neural networks in the materials science research has steadily increased. A number of reviews carried out recently have identified the application of neural networks to a diverse range of materials science applications [10,11]. The objective of the present work is, therefore, to develop a neural network model, which can predict the properties for a given composition and heat treatment, and the relationship of the properties with respect to these input variables. 2. Back-propagation neural networks (BPNN) modeling

∗ Corresponding author. Tel.: +82 54 279 5035; fax: +82 54 279 4499. E-mail address: [email protected] (N.S. Reddy). 0921-5093/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.msea.2008.12.022

A BPNN model consists of an input layer and an output layer with as many units as their respective number of variables. In-between

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Fig. 1. Cooling rates for (a) oil quenching and (b) water quenching from ASM Metals handbook.

the input and output layers are the hidden layers, each having a certain number of nodes (or units). The actual number of nodes depends heavily on the system and the acceptable error level of the model. The error-correction step takes place after a pattern is presented at the input layer and the forward propagation is completed. Each processing unit in the output layer produces a single real number as its output, which is compared with the targeted output specified in the training set. Based on this difference, an error value is calculated for each unit in the output layer. The weight of each of these units is adjusted for all of the interconnections that are established with the output layer. After this, the second sets of weights are adjusted for all the interconnections coming into the hidden layer that is just beneath the output layer. The process is continued until the weights of the first hidden layer are adjusted. As, the correction mechanism starts with the output units and propagates backward through each internal hidden layer to the input layer, the algorithm is called as back-error propagation or back propagation (BP). A BP network is trained using supervised learning mechanism. The network is presented with patterns, (inputs and targeted outputs), in the training phase. Upon each presentation, the weights are adjusted to decrease the error between the networks’ output and the targeted output [3,12]. A detailed description on BP algorithm used in the present study is beyond the scope of the present paper, and it can be found in literature [3,13]. The training of the neural network has been carried out by using software developed in programming language C, and sigmoid function was used as an activation function with the BP training algorithm.

3. The steel under consideration Typical applications of low alloy steels are general engineering components, high tensile bolts and studs, crankshafts, gears, boring bars, lead screws, shafting, milling and boring cutter bodies, collects. In the softened condition, the machinability of these steels is approximately 60% of that of mild steel, while in the hardened and tempered condition; the machinability is 45–55% of that of mild steel. The data on this low alloy steel has been collected from the handbook on Standard EN Steels [14]. The data consists of the composition, i.e., the percentage of carbon, silicon, manganese, sulphur, phosphorous, nickel and chromium, as well as the section size, soaking temperature and type of quenching. The properties considered are the yield strength (YS), ultimate tensile strength (UTS), % elongation (EL), % reduction in area (RA) and impact strength (IS) in Joules. The various heat treatment processes applied to low alloy steels are softening (sub-critical annealing) at 650–690 ◦ C followed by air cooling or oil quench, hardening at 830–860 ◦ C followed by oil quench and tempering at 550–660 ◦ C followed by air cooling or oil quenching. The data consists of different section sizes and their heat treatment temperatures and type of quenching. So, the cooling rate varies for different section sizes, which is not uniform; furthermore, the variations of cooling rates are not linear. Based on ASM Metal handbook [15] (as shown in Fig. 1), the cooling rate equations were developed for different section sizes (Fig. 2). The equations obtained based on the best fits are shown below.

Fig. 2. Cooling rates variation with diameter of the sample for (a) oil quenching and (b) water quenching.

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Table 1 Statistics of data used for neural network modeling. Variables

C Si Mn S P Ni Cr Mo Mn/s Cooling rate Tempering temperature YS (MPa) UTS (MPa) EL (%) RA (%) Impact strength (J)

Training data (112 data sets)

Testing data (28 data sets)

Minimum

Maximum

Mean

Standard deviation

Minimum

Maximum

Mean

Standard deviation

0.32 0.19 0.33 0.01 0.02 0.56 0.21 0.11 7.86 2.8 400 542.8 707 13 32 15

0.44 0.37 1.51 0.042 0.038 1.08 0.57 0.25 150 118 700 1193.6 1284.8 28.5 67.5 94

0.37 0.27 0.91 0.033 0.031 0.82 0.44 0.17 37.27 24.9 588.8 798.3 923.6 21.4 56.9 58.5

0.030 0.044 0.520 0.010 0.005 0.131 0.073 0.030 34.32 29.79 66.85 147.7 127.3 3.430 6.190 21.96

0.32 0.19 0.33 0.01 0.02 0.56 0.21 0.12 7.86 2.8 450 547.4 732.9 14 31 15

0.44 0.36 1.51 0.042 0.038 1.08 0.56 0.25 145 54 700 1161.7 1295.5 29 64 93

0.38 0.26 1.27 0.03 0.031 0.82 0.42 0.18 52.6 24.3 592.7 850.1 972 20.25 54.3 53.6

0.032 0.053 0.340 0.010 0.006 0.127 0.109 0.034 33.90 17.49 69.00 155.5 125.00 3.310 7.210 21.50

Fig. 3. Variation of (a) MSE and average error in output prediction (Etr ) of (b) YS, (c) UTS, (d) EL, (e) RA and (f) IS of the train data as a function of the number of hidden neurons with single hidden layer and two hidden layers.

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N.S. Reddy et al. / Materials Science and Engineering A 508 (2009) 93–105 Table 2 Variation of MSE of the trained data as a function of number of iterations ( = 0.55 and ˛ = 0.65, number of hidden neurons = 22 and number of hidden layers = 2).

For water quenching, Y = 5.77 − 0.01X − 0.002X 2 + 2.19E − 5 X 3 − 1.11E − 7 X 4 + 1.41E − 10 X 5

(1)

For oil quenching, Y = 6.16 − 0.15X + 0.003X 2 − 2.72E − 5 X 3 + 9.33E − 8 X 4 + 7.31E − 11 X 5

(2)

where, X and Y are diameter of the specimen and cooling rate, respectively. Based on the above equations, the cooling rate has been calculated for different section sizes and used as one of input parameters for NN training. After coding, the total sets available for modeling are 140 and shown in Appendix. Among the 140 data sets available, 112 data sets have been used for NN training. When training a model, the choice of input variables is of great importance. Also, when a combination of these variables is believed to be of particular importance, the model can be improved by adding the combination as an explicit variable. The model was trained on the ratio of Mn and S, where the concentrations are in wt%. This is because sulphur reacts with manganese and forms MnS. To avoid biasing of the model, the individual variables making up the Mn/S ratio are also included, so that the direct influence of any of them can also be detected. The statistics of the steels data used for neural network modeling is presented in Table 1.

Number of iterations

MSE

500 1,000 5,000 10,000 20,000 40,000 50,000 60,000 75,000 100,000 125,000 150,000 175,000 180,000 185,000 190,000 195,000 200,000

0.09592 0.07067 0.01442 0.00943 0.00611 0.00464 0.00479 0.00461 0.00411 0.00237 0.00268 0.00149 0.00145 0.00104 0.00109 0.00107 0.00102 0.00102

4. The analysis 4.1. The training behavior of BPNN model Results obtained from modeling studies are presented and discussed in detail in this section. In this study, the optimal parameters of the network are determined based on the minimum mean sum square (MSE) error [3] and the average error in output prediction

Table 3 The variation of MSE as a function of hidden neurons. (: 0.25, ˛: 0.9 and the number of iterations executed are 200,000). Neurons in layer 1

Neurons in layer 2

MSE

Etr (YS)

Etr (UTS)

Etr (%EL)

Etr (%RA)

Etr (IS)

7 7 8 8 8 8 9 10 15 16 16 16 18 20 24 25 25 30 32 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 25 30

6 8 6 7 16 32 8 20 25 18 24 32 16 10 16 15 30 25 8 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 25 30

0.04178 0.05658 0.04140 0.02832 0.02613 0.00828 0.03171 0.01160 0.00723 0.00860 0.00585 0.00559 0.00848 0.01204 0.00685 0.00672 0.00568 0.00470 0.01215 0.02045 0.02313 0.02351 0.01517 0.01576 0.01272 0.01165 0.01039 0.00981 0.00880 0.00747 0.00672 0.00652 0.00677 0.00406 0.00596 0.00433 0.00359

45.50 47.62 51.19 36.66 39.05 23.87 41.38 24.32 28.70 19.56 23.73 31.50 23.89 29.08 22.81 21.21 23.29 29.71 28.00 42.69 38.07 32.30 28.19 31.87 24.75 24.01 25.51 26.37 26.67 26.55 21.19 32.33 23.40 16.21 19.53 21.59 22.45

32.12 38.12 39.26 30.28 29.91 21.06 30.72 24.62 24.38 15.88 20.48 22.24 22.17 23.20 20.95 19.13 18.09 20.10 25.37 31.99 28.60 27.25 22.44 24.75 22.62 19.44 22.39 22.48 21.87 21.96 16.96 26.12 17.68 13.91 17.34 20.67 21.57

1.08 1.23 1.17 0.93 0.95 0.52 0.84 0.59 0.41 0.51 0.53 0.40 0.56 0.64 0.41 0.50 0.31 0.37 0.72 0.69 0.83 0.96 0.84 0.72 0.59 0.58 0.55 0.64 0.50 0.54 0.47 0.47 0.36 0.28 0.34 0.36 0.36

3.93 2.92 2.35 1.79 2.03 0.99 2.40 1.32 0.54 0.83 0.57 0.64 0.71 1.23 0.64 0.81 0.37 0.61 1.20 1.65 1.50 1.90 1.32 1.42 1.21 1.11 0.90 0.92 0.73 0.77 0.67 0.58 0.50 0.41 0.60 0.58 0.59

7.73 7.70 6.95 6.84 5.84 3.06 6.14 3.97 2.36 2.33 2.46 2.73 2.35 3.01 2.17 3.20 1.73 2.46 2.70 5.53 6.10 5.86 4.27 4.27 3.46 2.92 3.45 2.97 2.59 2.77 2.46 2.16 2.18 1.52 1.97 2.09 2.21

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(Etr ) of the trained data. 1 |(Ti (x) − Oi (x))| N N

Etr (x) =

(3)

i=1

where Etr (x) = average error in prediction of training data set for output parameter x. N = number of data sets, Ti (x) = targeted output, Oi (x) = output calculated. In applying the BPNN for the proposed estimation work, the following key design issues, i.e. number of hidden layers and hidden neurons, neural network parameters (learning rate and momentum rate) and number of iterations are discussed. 4.1.1. Choice of number of hidden layers and neurons in the hidden layer The influence of the number of hidden layers and the number of neurons in each hidden layer on the convergence criterion is studied extensively [16]. In order to decide the suitable number of hidden layers, two basic structures of the neural networks are examined: one with single hidden layer and the other with two hidden layers. In the first case, the BPNN structure of one hidden layer with a learning rate of 0.25 and a momentum rate of 0.9 was trained with different hidden neurons starting from 8 to 30. Eight hidden neu-

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rons in single layer yield a value of MSE of 0.02045 after 200,000 iterations. Fig. 3 shows the variation of MSE and Etr for different properties with the number of neurons in the hidden layer. With an increase in the number of neurons, the MSE value increases up to 10 hidden neurons and then gradually decreases with further increase of hidden neurons. In order to examine the effect of number of layers, the NN structure is also designed with two hidden layers, each containing 8–30 neurons in each layer, and the results are shown in Table 2. The effect of different combination of hidden neurons on two hidden layers is also shown in Table 3. Excellent convergence was observed with 22 hidden neurons in each layer, and a value of MSE of 0.00427 was obtained after 200,000 iterations with  = 0.25 and ˛ = 0.9. From Fig. 3, it is clear that the convergence is better for the two hidden layers than the single hidden layer with different number of hidden neurons in the layer. Hence, 22 hidden neurons with two hidden layers have been selected for further training to optimise other parameters. Even though the MSE is 0.00359 with 30 hidden neurons, which is far less than that of 22 hidden neurons, the average error in output predictions is greater in all the mechanical properties. Hence, 22 hidden neurons in two layers were selected for analysis. One more advantage with the 22 hidden neurons selection is its less complexity in comparison to if 30 hidden neurons were to be selected.

Fig. 4. Variation of (a) MSE and average error in output prediction (Etr ) of (b) YS, (c) UTS, (d) EL, (e) RA and (f) IS of the train data as a function of learning rate and momentum rate in the steps of 0.05.

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N.S. Reddy et al. / Materials Science and Engineering A 508 (2009) 93–105 Table 4 Composition and heat treatment variables of sample 91. S. no. C Si Mn P Ni Cr Mo Mn/S CR Temp.

Fig. 5. The distribution of weights for best trained BPNN architecture.

4.1.2. Choice of neural networks parameters For back-propagation learning algorithm with fixed values of learning rate, , and momentum rate, ˛, the optimum values are obtained by simulation with different values of  and ˛. Two values,  = 0.25 and ˛ = 0.9, are initially chosen and varied to get the optimum value for each parameter. Fig. 4 shows the variation of MSE and Etr for mechanical properties of the training data for different

S91 0.4 0.3 1.51 0.034 0.89 0.21 0.16 41.94 3.1 550

values of  and ˛ with two hidden layers consisting of 22 hidden neurons in each layer. From Fig. 4, it is clearly evident that larger value of learning rate results in higher MSE and Etr , which is an expected behavior [16,17]. This phenomenon is usually referred to as over training. However, it was observed that with higher learning rates, the convergence takes place faster than it does at lower learning rates. The figure also indicates that MSE and Etr are not quite sensitive to the momentum rate. The best combination is found to be when  is 0.55 and ˛ is 0.65 (Fig. 4). For the above combination, the MSE of 0.001015 is obtained after 200,000 iterations. 4.1.3. Number of iterations The number of iterations executed to obtain optimum values is an important parameter in the case of BPNN training. The MSE

Fig. 6. The comparison of actual (A) and predicted (P) values and the respective percentage errors for 15 test samples of EN100 steels. (a) and (b) YS, (c) and (d) EL and (e) and (f) IS.

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Fig. 7. Effect of carbon variation on mechanical properties of sample 91: (a) YS, (b) UTS, (c) EL and (d) RA.

Fig. 8. Effect of silicon variation on mechanical properties of sample 91: (a) YS, (b) UTS, (c) EL and (d) RA.

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achieved from different iterations is shown in Table 3. It is evident that the value of MSE gradually decreases during the progress of training, as expected. An error of 0.07067 after 1000 iterations reduces to 0.00237 after 100,000 iterations and falls to 0.00104 after 180,000 iterations, and no change has been found with further increase in the number of iterations. It has been observed that the MSE does not change significantly beyond 200,000 iterations and hence further processing has been stopped after the abovementioned iterations. Finally, the 11 units in input layer, 22 hidden neurons in two hidden layers and 5 units in output layer (11-22-225) architecture with a learning rate of 0.55 and a momentum rate of 0.65 are used for prediction and analysis of the EN100 steels.

114, 115, 116, 118, 119, 123, 125, 127, 128, 131, 133, 136, 137, 138 and 140 data sets (a total of 15 sets) shown in Appendix are presented in Fig. 6(a), (c), and (e). The comparison between the actual and predicted values and the respective percentage error for 15 testing data sets is also shown in Fig. 6(b), (d), and (f). It has been observed that most (nearly 95%) of the outputs predicted by the model are within 4% of the error band (indicated in Fig. 6(b), (d) and (f)). For the testing data that has never been seen by the model, the NN gives a reasonably accurate prediction. In the case of sample 11 (data set of 133 in Appendix), it can be seen that there is a less variation between actual and predicted values as shown in Fig. 6(e), but the percentage error is higher because the actual values are very small.

4.1.4. Range of weights for EN100 steels training The sigmoid function is used as the activation function in the present network. From Fig. 5, it is very much clear that the distribution of weights of the trained network, with 11-22-22-5 architecture, takes sigmoid shape. The initial values for the weights are randomly generated between −0.5 and 0.5, but as the training progresses the weights varied between a minimum value of −56.87 to a maximum value of 57.61. The magnitude of the variation of the weights indicates the complexity of the present system. A total of 885 weights will be available ((11 + 1) × 22 + 23 × 22 + 23 × 5). On completion of the training phase, the mechanical properties for the test data are estimated simply by passing the input data in forward path of the network and using updated weights of the network obtained.

4.3. Hypothetical alloys

4.2. Predictions of test data sets

The developed model can be used in practice to manipulate the mechanical properties of EN100 steels, in general, by altering chemical composition and heat treatment variables. The effects of all of these can be estimated quantitatively by using the BPNN model to observe whether they make any metallurgical sense. The combined effect of a single variable or any two variables on the properties has also been studied. Here we present some hypothetical alloys by varying the composition and heat treatment variables and their effect on mechanical properties namely: (a) yield strength (YS), (b) ultimate tensile strength (UTS), (c) % elongation (EL), (d) % reduction in area (RA) and (e) impact strength (IS) for sample number 91 of Appendix is presented in Figs. 7–9. The effect of combined variation of two elements on properties is presented in Figs. 10 and 11. The composition of sample 91 is given in Table 4.

Total data sets available for BPNN Model training is 140. The randomly selected 112 data sets have been used for training, and the remaining 28 data sets are used for testing. The predictions of

4.3.1. Effect of carbon on properties Carbon is an essential non-metallic element in iron to make steel. None of the other elements so dramatically alters the strength

Fig. 9. Effect of Mn/S variation on mechanical properties of sample 5: (a) YS, (b) UTS, (c) EL and (d) RA.

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Fig. 11. Combined effect of cooling rate and tempering temperature variation on properties for sample 91: (a) YS, (b) UTS and (c) EL. Fig. 10. Combined effect of Ni and Cr variation on mechanical properties for sample 91: (a) YS, (b) UTS and (c) EL.

and the hardness as do small changes in the carbon content does. Carbon is the principal hardening element, influencing the level of hardness or the strength attainable by quenching. The model predictions as indicated in Fig. 7 show that the YS and UTS increase while the EL and RA decrease with increase in C content. As carbon increases from 0.32% to 0.44%, (Fig. 7), there will be approximately a 15% increase in the formation of pearlite accompanied by a reduction of ferrite by the same amount, which results in an increase in strength and a decrease in ductility. The increase in strength and the decrease in ductility with the increase in carbon content are also due to the increase in the formation of various carbides with the other alloying elements as the carbon percentage is increased from 0.3 to 0.44. The impact strength would also decrease, as the area under the stress–strain curve would decrease owing to the reduction in ductility. Thus, the predictions of Fig. 7 are clearly is in agreement with the expected experimental trend. 4.3.2. Effect of silicon on properties Silicon is one of the principal deoxidizers used in steel making and therefore the amount of silicon present is related to the type of steel. Usually only small amounts (0.20%) are present in rolled steel when it is used as a deoxidizer. Silicon dissolves in iron and tends to strengthen it. In the low alloy steels, Si is present in the range of 0.2–0.4%. Fig. 8 shows the variation of mechanical properties with silicon content in the range of 0.2–0.4%. The variation in mechanical properties as observed in Fig. 8 cannot be explained based on the effect of Si alone. It is possible that a number of other elements present might be responsible for the behavior represented in Fig. 8. Fig. 8 shows that a significant change occurs in mechanical prop-

erties at around 0.3% Si, which needs to be explained based on metallurgical principles involving the complex combined effects of a number of elements. However, it is important to note that a significant increase in the UTS at around 0.3% Si brings a significant reduction in EL and RA exactly at the same composition. This suggests that the model is able to establish strong correlations between the individual output parameters, though such information is not fed to the model a prior. 4.3.3. Effect of Mn/S ratio on properties Manganese is a deoxidizer and degasifier, which reacts favorably with sulphur to improve forging ability and surface quality as it converts sulphur to manganese sulphide. Manganese increases tensile strength, hardness, hardenability, and resistance to wear and increases the rate of carbon penetration during carburizing. There is a tendency nowadays to increase the manganese content and reduce the carbon content in order to achieve steels with an equal tensile strength but improved ductility. Fig. 9 shows the effect of Mn/S variation on mechanical properties. As Mn/S ratio increases strength increases with a corresponding reduction in ductility, which is clearly brought out by the model in Fig. 9. 4.3.4. Effect of nickel and chromium on properties Nickel is used to stabilise alloy steels. Nickel and manganese exhibit very similar behavior and both lowers the eutectoid temperature. It strengthens ferrite by solid solution but is a powerful graphitiser. Nickel increases the impact strength of engineering steels considerably. It increases the strength and hardness without sacrificing ductility and toughness. Steels with 0.5% nickel is similar to carbon steel but is stronger because of the forma-

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Table 5 Comparison of actual and predicted properties for hypothetical alloy of sample 91. Hypothetical alloy

Sample 91 (experimental) Carbon (0.4) Silicon (0.3) Manganese (1.51) Phosphorous (0.034) Nickel (0.89) Chromium (0.21) Molybdenum (0.16) Mn/S ratio (41.94) Cooling rate (3.1 ◦ C/s) Tempering temperature (550 ◦ C)

Fig. no.

7 8

9

Mechanical properties (experimental/predicted) YS

UTS

%EL

%RA

IS

760 753 737 738 736 738 737 734 739 724 737

931 921 906 905 901 906 904 904 907 916 904

19.50 18.93 19.78 19.67 19.89 19.78 19.00 19.49 19.45 19.19 19.64

49.50 47.73 48.90 48.8 48.95 48.90 48.00 48.45 48.38 49.82 48.64

20.00 17.80 20.80 20.00 21.94 20.80 11.36 16.00 18.75 22.37 20.06

tion of finer pearlite and the presence of nickel in solution in the ferrite. Among all of the common alloying elements, chromium ranks near the top in promoting hardenability. It makes the steel apt for oil or air hardening as it reduces the critical cooling rate required for the formation of martensite. Fig. 10(a)–(c) shows combined effect of Cr and Ni on the YS, UTS and EL, respectively. The figure clearly points out that the YS and UTS decrease with an increase in Cr content at lower concentrations for a given Ni concentration, which can be attributed to the ferrite stabilisation. However, at higher Cr content, the strength starts to increase again due to the formation of carbides. The percentage elongation shows exactly the opposite trend, as expected. Thus, the model is able to predict the combined effect of Ni and Cr quite successfully. 4.3.5. Combined effect of cooling rate and tempering temperature on properties The effect of cooling rate on properties is non-linear and complex, and it is difficult to explain. Fig. 11(a)–(c) shows the combined effect of cooling rate and tempering temperature on the YS, UTS and EL, respectively. Increase in tempering temperature brings in an initial increase in the strength in the temperature range of 450–550 ◦ C, which could be attributed to the secondary hardening. Further increase in the tempering temperature leads to a decrease in the strength, as expected. The EL shows a reverse trend as expected. It is also important to note that the variation in the mechanical properties is larger with different heat treatment parameters (cooling rate and tempering temperature), in comparison to that with Cr and Ni variation. This clearly suggests that the mechanical properties are more sensitive to heat treatment parameters than the concentration of alloying elements such as Cr and Ni, which is clearly predicted by the model. Table 5 shows actual and model predicted mechanical properties for different hypothetical alloys for sample 91. Though it is difficult to explain each and every behavior of hypothetical predictions, the model predicted a sudden increase in ductility, whenever there is a sudden decrease in the strength and vice versa, which has to be noted. It is very much clear that the error in the model predictions

is less than 4% in all cases. This shows the efficiency of the model in understanding the relationships between input parameters and output parameters. This will greatly help the designing of alloy to achieve certain target properties. 5. Conclusions • A comparison of the modeled and experimental results indicated that NN can very well be employed for the estimation and analysis of mechanical properties as a function of chemical composition and/or heat treatment for low alloy steels. • Two hidden layers converged better than the single hidden layer. At higher learning rate, the MSE and Etr significantly increased as expected. • Even though Rumelhurt et al. [12] suggested that a learning rate,  = 0.25, and a momentum rate, ˛ = 0.9, yield good results for most applications, the present work indicated it is not always so, and that the optimum values vary for different steels. This fact was in agreement with the finding of Satish and Zaengl [18] and Chakravorti and Mukherjee [19]. The best combination of ANN parameters for the best results in each case studied was identified. • Present model was able to map the relation between the output parameters, even though such a relation was not explicitly fed to the model. For example, the relationship between ductility and strengths was clearly brought out by the model (for example, Figs. 7–11). • Results from the current study demonstrated that the neural network model can be used to examine the effects of individual input variables on the output parameters, which is incredibly difficult to do experimentally. (Development of hypothetical alloys.) Acknowledgements One of the authors (N.S. Reddy) acknowledges Dr. N.N. Acharya, Department of Metallurgical and Materials Engineering, Indian Institute of Technology, Kharagpur for useful discussions on determining cooling rate equations and Yandra Kiran Kumar for helping to develop GUI design for ANN Model.

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103

Appendix A. Low alloy steels used in modeling: chemical composition, heat treatment parameters (CR: cooling rate in ◦ C/s and TT: tempering temperature in ◦ C) and mechanical properties. Sl. no. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75

C

Si

Mn

P

Ni

Cr

Mo

Mn/S

0.32 0.32 0.33 0.33 0.35 0.35 0.35 0.35 0.35 0.35 0.35 0.35 0.35 0.35 0.35 0.35 0.35 0.35 0.35 0.35 0.35 0.35 0.35 0.35 0.35 0.35 0.35 0.35 0.35 0.35 0.35 0.35 0.35 0.35 0.35 0.35 0.35 0.35 0.35 0.35 0.35 0.35 0.35 0.35 0.35 0.35 0.35 0.35 0.35 0.35 0.35 0.35 0.35 0.35 0.35 0.35 0.36 0.36 0.36 0.36 0.36 0.36 0.37 0.38 0.38 0.38 0.38 0.38 0.38 0.38 0.38 0.38 0.38 0.38 0.38

0.23 0.23 0.19 0.19 0.19 0.19 0.19 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.26 0.2 0.2 0.2 0.2 0.2 0.2 0.25 0.33 0.33 0.33 0.33 0.33 0.33 0.33 0.33 0.33 0.33 0.33

1.28 1.28 1.45 1.45 1.5 1.5 1.33 0.33 0.33 0.33 0.33 0.33 0.33 0.33 0.33 0.33 0.33 0.33 0.33 0.33 0.33 0.33 0.33 0.33 0.33 0.33 0.33 0.33 0.33 0.33 0.33 0.33 0.33 0.33 0.33 0.33 0.33 0.33 0.33 0.33 0.33 0.33 0.33 0.33 0.33 0.33 0.33 0.33 0.33 0.33 0.33 0.33 0.33 0.33 0.33 0.33 1.36 1.28 1.28 1.28 1.28 1.28 1.3 1.26 1.26 1.26 1.26 1.26 1.26 1.26 1.26 1.26 1.26 1.26 1.26

0.028 0.028 0.026 0.026 0.026 0.026 0.026 0.032 0.032 0.032 0.032 0.032 0.032 0.032 0.032 0.032 0.032 0.032 0.032 0.032 0.032 0.032 0.032 0.032 0.032 0.032 0.032 0.032 0.032 0.032 0.032 0.032 0.032 0.032 0.032 0.032 0.032 0.032 0.032 0.032 0.032 0.032 0.032 0.032 0.032 0.032 0.032 0.032 0.032 0.032 0.032 0.032 0.032 0.032 0.032 0.032 0.03 0.036 0.036 0.036 0.036 0.036 0.03 0.025 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02

0.85 0.85 0.89 0.89 0.93 0.93 0.91 0.91 0.91 0.91 0.91 0.91 0.91 0.91 0.91 0.91 0.91 0.91 0.91 0.91 0.91 0.91 0.91 0.91 0.91 0.91 0.91 0.91 0.91 0.91 0.91 0.91 0.91 0.91 0.91 0.91 0.91 0.91 0.91 0.91 0.91 0.91 0.91 0.91 0.91 0.91 0.91 0.91 0.91 0.91 0.91 0.91 0.91 0.91 0.91 0.91 0.79 0.74 0.74 0.74 0.74 0.74 0.84 0.56 0.56 0.56 0.56 0.56 0.56 0.56 0.56 0.56 0.56 0.56 0.56

0.46 0.46 0.56 0.56 0.57 0.57 0.43 0.43 0.43 0.43 0.43 0.43 0.43 0.43 0.43 0.43 0.43 0.43 0.43 0.43 0.43 0.43 0.43 0.43 0.43 0.43 0.43 0.43 0.43 0.43 0.43 0.43 0.43 0.43 0.43 0.43 0.43 0.43 0.43 0.43 0.43 0.43 0.43 0.43 0.43 0.43 0.43 0.43 0.43 0.43 0.43 0.43 0.43 0.43 0.43 0.43 0.46 0.46 0.46 0.46 0.46 0.46 0.43 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5

0.16 0.16 0.12 0.12 0.18 0.18 0.15 0.15 0.15 0.15 0.15 0.15 0.15 0.15 0.15 0.15 0.15 0.15 0.15 0.15 0.15 0.15 0.15 0.15 0.15 0.15 0.15 0.15 0.15 0.15 0.15 0.15 0.15 0.15 0.15 0.15 0.15 0.15 0.15 0.15 0.15 0.15 0.15 0.15 0.15 0.15 0.15 0.15 0.15 0.15 0.15 0.15 0.15 0.15 0.15 0.15 0.19 0.25 0.25 0.25 0.25 0.25 0.11 0.21 0.21 0.21 0.21 0.21 0.21 0.21 0.21 0.21 0.21 0.21 0.21

38 38 73 73 75 75 67 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 43 49 49 49 49 49 43 42 79 79 79 79 79 79 79 79 79 79 79

CR 38 16 16 7 7 3 3 38 38 38 38 38 38 38 12 12 12 12 12 7 7 7 7 7 3 3 3 3 3 3 3 118 118 118 118 34 34 34 34 34 15 15 15 15 15 15 15 8 8 8 8 8 8 8 8 8 9 38 38 7 7 7 38 54 118 118 118 118 3 23 23 23 8 8 3

TT 580 620 605 625 650 650 550 450 500 550 600 650 675 700 450 500 550 600 625 500 550 600 625 675 400 500 575 600 625 675 700 400 500 550 600 400 500 550 600 625 450 500 550 600 625 650 675 400 450 500 550 600 625 650 675 700 610 550 600 550 600 650 600 625 500 550 600 650 620 550 600 620 580 600 580

YS

UTS

EL

RA

IS

934 736 785 693 681 730 683 1163 1069 906 786 718 692 575 727 645 622 579 582 639 614 623 543 870 683 661 620 588 558 750 667 941 855 745 710 985 813 742 663 605 850 762 692 643 611 596 1194 906 830 760 744 626 634 588 903 858 1110 821 1037 760 888 744 715 839 1137 1080 946 801 788 1089 943 821 967 924 871

1019 845 888 824 839 870 850 1222 1131 993 906 821 763 753 888 821 783 757 737 800 777 777 707 1008 850 829 772 754 713 903 835 1002 912 810 778 1078 920 853 800 766 973 899 823 777 733 728 1285 1007 947 894 855 794 785 730 1026 976 1165 926 1107 886 1000 889 851 946 1250 1216 1046 912 943 1192 1077 967 1070 1034 1049

18.0 22.0 21.0 21.5 21.5 24.0 21.0 15.0 18.5 19.0 24.0 25.5 27.0 27.5 21.0 24.0 26.0 27.0 27.5 23.0 25.0 24.0 27.0 14.0 21.0 23.0 24.0 25.0 27.0 18.5 22.0 20.0 22.5 24.5 26.0 17.0 21.0 24.0 25.0 26.0 18.5 20.0 25.0 26.0 28.0 28.5 13.0 19.0 20.0 21.0 23.0 24.0 25.0 28.0 17.5 19.0 16.5 21.5 16.5 22.0 20.0 24.0 25.0 23.0 15.5 18.0 19.0 23.0 22.0 15.0 18.0 20.0 18.0 20.0 19.0

54.0 59.0 60.0 57.0 58.0 56.0 57.0 50.0 55.0 55.5 61.5 66.0 64.5 64.5 56.5 62.5 63.0 66.0 61.0 58.0 60.0 63.0 64.5 46.5 57.0 57.5 60.0 61.5 64.5 52.0 56.5 56.0 61.0 63.5 64.0 52.5 61.0 63.0 65.0 65.5 52.5 58.0 63.5 64.5 67.5 66.5 46.0 57.0 59.0 55.5 63.0 61.0 63.5 64.5 48.0 54.0 52.0 64.0 52.0 64.0 48.0 55.0 63.5 56.0 45.0 47.0 55.0 57.0 61.5 45.0 52.5 59.5 52.5 57.0 55.0

35 42 72 43 89 87 46 15 26 49 75 93 94 90 23 52 70 91 80 36 53 70 79 93 19 47 74 61 75 82 20 24 42 79 84 21 45 55 81 80 35 36 52 60 70 79 93 29 43 58 66 70 81 86 82 20 60 69 78 65 76 60 72 48 17 18 53 82 69 15 44 64 45 60 38

104

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Appendix A (Continued ) Sl. no.

C

Si

Mn

P

Ni

Cr

Mo

Mn/S

76 77 78 79 80 81 82 83 84 85 86 87 88 89 90

0.38 0.38 0.38 0.39 0.39 0.39 0.39 0.39 0.39 0.4 0.4 0.4 0.4 0.4 0.4

0.33 0.33 0.33 0.25 0.33 0.3 0.3 0.29 0.29 0.2 0.32 0.3 0.3 0.3 0.3

1.26 1.26 1.26 1.37 1.47 1.41 1.36 1.45 1.45 1.36 1.46 1.51 1.51 1.51 1.51

0.02 0.02 0.02 0.025 0.02 0.025 0.036 0.035 0.035 0.035 0.033 0.034 0.034 0.034 0.034

0.56 0.56 0.56 0.73 0.73 0.9 1.08 0.87 0.87 1 0.77 0.89 0.89 0.89 0.89

0.5 0.5 0.5 0.48 0.49 0.38 0.35 0.35 0.35 0.53 0.45 0.21 0.21 0.21 0.21

0.21 0.21 0.21 0.18 0.17 0.18 0.21 0.17 0.17 0.23 0.2 0.16 0.16 0.16 0.16

79 79 79 86 82 47 40 47 47 54 58 42 42 42 42

3 3 3 7 3 7 7 38 7 38 54 38 9 9 9

91

0.4

0.3

1.51

0.034

0.89

0.21

0.16

42

3

550

760

0.4 0.4 0.41 0.41 0.42 0.42 0.42 0.42 0.42 0.42 0.42 0.42 0.42 0.42 0.42 0.43 0.43 0.44 0.44 0.44 0.44 0.32 0.33 0.35 0.35 0.35 0.35 0.35 0.36 0.36 0.37 0.38 0.38 0.38 0.39 0.39 0.39 0.4 0.4 0.4 0.4 0.4 0.4 0.42 0.42 0.42 0.42 0.44 0.44

0.3 0.3 0.25 0.2 0.19 0.37 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.35 0.36 0.36 0.36 0.36 0.23 0.19 0.19 0.19 0.25 0.25 0.25 0.2 0.2 0.2 0.33 0.33 0.33 0.25 0.25 0.3 0.2 0.3 0.3 0.3 0.3 0.3 0.27 0.25 0.25 0.25 0.36 0.36

1.51 1.51 1.43 1.37 1.48 1.29 1.4 1.4 1.4 1.4 1.4 1.4 1.4 1.4 1.4 1.34 1.31 1.43 1.43 1.43 1.43 1.28 1.45 1.28 1.33 0.33 0.33 0.33 1.28 1.28 1.25 1.26 1.26 1.26 1.41 1.4 1.36 1.36 1.51 1.51 1.51 1.51 1.51 1.36 1.4 1.4 1.4 1.43 1.43

0.034 0.034 0.038 0.03 0.028 0.035 0.033 0.033 0.033 0.033 0.033 0.033 0.033 0.033 0.033 0.032 0.038 0.038 0.038 0.038 0.038 0.028 0.026 0.02 0.026 0.032 0.032 0.032 0.036 0.036 0.03 0.02 0.02 0.02 0.025 0.036 0.036 0.035 0.034 0.034 0.034 0.034 0.034 0.035 0.033 0.033 0.033 0.038 0.038

0.89 0.89 0.65 0.8 0.64 0.65 0.77 0.77 0.77 0.77 0.77 0.77 0.77 0.77 0.77 0.97 0.64 0.68 0.68 0.68 0.68 0.85 0.89 0.85 0.91 0.91 0.91 0.91 0.74 0.74 0.76 0.56 0.56 0.56 0.9 0.81 1.08 1 0.89 0.89 0.89 0.89 0.89 0.87 0.77 0.77 0.77 0.68 0.68

0.21 0.21 0.48 0.5 0.42 0.43 0.49 0.49 0.49 0.49 0.49 0.49 0.49 0.49 0.49 0.56 0.42 0.5 0.5 0.5 0.5 0.46 0.56 0.46 0.43 0.43 0.43 0.43 0.46 0.46 0.47 0.5 0.5 0.5 0.38 0.43 0.35 0.53 0.21 0.21 0.21 0.21 0.21 0.49 0.49 0.49 0.49 0.5 0.5

0.16 0.16 0.23 0.2 0.19 0.14 0.18 0.18 0.18 0.18 0.18 0.18 0.18 0.18 0.18 0.21 0.14 0.17 0.17 0.17 0.17 0.16 0.12 0.16 0.15 0.15 0.15 0.15 0.25 0.25 0.21 0.21 0.21 0.21 0.18 0.24 0.21 0.23 0.16 0.16 0.16 0.16 0.16 0.21 0.18 0.18 0.18 0.17 0.17

42 42 41 39 46 34 40 40 40 40 40 40 40 40 40 41 36 41 41 41 41 38 73 79 79 8 8 8 49 49 42 79 79 79 47 41 40 54 42 42 42 42 42 36 40 40 40 41 41

3 3 5 16 54 45 38 38 38 38 38 38 3 3 3 9 45 38 38 7 7 7 42 38 12 7 3 34 38 7 7 23 8 8 54 54 38 45 38 38 38 9 3 7 38 38 3 38 7

600 650 640 630 670 550 560 590 650 560 620 650 590 620 650 630 550 550 600 550 650 625 580 580 600 450 550 450 650 550 650 450 570 650 650 590 640 580 550 650 700 500 700 660 620 590 560 650 600

730 642 815 821 791 1011 981 932 763 935 803 734 807 756 693 778 889 1064 976 1046 775 660 961 934 853 692 645 914 912 760 889 1162 1004 791 712 1095 778 1052 1016 797 547 1016 566 684 882 880 858 832 914

92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140

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CR

TT

YS

UTS

EL

RA

IS

600 620 650 650 650 650 630 640 630 600 600 600 550 600 650

824 788 684 778 625 778 791 730 836 943 955 964 955 858 739

985 943 876 885 847 876 909 865 918 1025 1034 1016 1064 958 836

22.0 22.0 21.0 22.0 24.0 24.0 22.0 23.0 20.0 20.5 20.0 19.5 17.0 19.5 24.5

61.5 61.5 57.0 58.0 56.0 57.0 56.0 58.0 59.0 55.0 52.5 51.0 48.5 57.0 62.5

65 69 50 81 81 94 72 74 43 65 42 66 29 63 83

931

19.5

49.5

20

891 806 931 943 897 1093 1046 1032 874 1046 932 882 1019 943 858 891 1004 1168 1072 1136 900 803 1043 1019 958 844 827 1020 1014 886 1016 1295 1125 924 867 1131 894 1119 1070 839 912 1125 733 821 979 985 994 938 1037

20.0 23.5 22.0 21.0 21.0 17.5 15.0 17.0 20.5 15.5 18.5 18.5 17.5 20.5 23.0 23.5 23.0 16.5 18.5 17.5 23.0 21.0 20.0 18.0 20.5 21.5 22.0 19.0 20.5 22.0 21.0 14.0 19.0 22.0 22.0 17.5 21.0 19.0 17.5 26.0 19.0 16.5 29.0 27.0 18.5 18.5 15.0 22.5 17.5

53.5 60.5 50.0 57.0 52.0 53.5 47.0 47.0 55.0 44.0 55.0 59.0 47.0 50.0 62.0 64.0 54.5 32.0 52.0 48.0 60.0 59.0 54.0 53.0 59.0 57.5 56.5 55.0 59.0 64.0 57.0 45.0 55.0 59.5 56.0 55.0 57.0 54.0 47.0 61.5 40.5 47.0 63.5 61.5 55.0 50.0 31.0 58.5 49.0

32 74 56 71 67 30 31 47 81 42 68 77 57 66 66 69 53 27 48 27 86 81 51 35 54 30 60 46 26 65 43 15 40 74 69 42 60 38 44 93 63 15 93 84 56 58 30 80 57

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