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Prediction of age hardening parameters for 17-4PH stainless steel by artificial neural network and genetic algorithm
Materials Science & Engineering A 675 (2016) 147–152
Contents lists available at ScienceDirect
Materials Science & Engineering A journal homepage: www.elsevier.com/locate/msea
Prediction of age hardening parameters for 17-4PH stainless steel by artificial neural network and genetic algorithm Sayyed Ali Razavi n, Fakhreddin Ashrafizadeh, Saghar Fooladi Department of Materials Engineering, Isfahan University of Technology, Isfahan 8415683111, Iran
art ic l e i nf o
a b s t r a c t
Article history: Received 5 May 2016 Received in revised form 11 August 2016 Accepted 12 August 2016 Available online 13 August 2016
17-4PH alloy is a precipitation hardening stainless steel with a desirable combination of high corrosion resistance and good mechanical properties at temperatures up to 300 °C in engineering applications. Ageing mechanism in this alloy is rather complicated, hence, for optimum hardness, selection of heat treatment parameters is critical. In this investigation, first, Artificial Neural Network (ANN) was used to model the relationship between ageing times and temperatures with the corresponding hardness. R2 value was 0.9747 in the final ANN model indicating this model could predict hardness values, appropriately. Then, Genetic Algorithm (GA) was used to find optimum aging time and temperature for the target of maximum hardness value. ANN model was used as the fitness function in GA. According to the GA-ANN simulation, a hardness of 44.9 HRC would be achieved by ageing 129 min at 464 °C. Finally, the model was validated by conducting heat treatment experiments carried out using the predicted parameters. A maximum hardness of 44.1 HRC was obtained in the experimental work, showing a difference of 1.8% with the proposed model. & 2016 Elsevier B.V. All rights reserved.
Keywords: Hardness 17-4PH stainless steel Age hardening Ageing Artificial neural network Genetic algorithm
1. Introduction 17-4PH alloy (AISI 630) is among precipitation hardenable stainless steels with attractive properties in terms of fabrication, high hardness and strength together with high ductility, especially at temperatures below 300 °C, and excellent corrosion resistance [1–3]. Precipitation hardening steels are divided into two groups; semiaustenitic type and martensitic type where the martensitic grades are used in many industrial sectors with more applications than semi-austenitic grades [2]. 17-4PH is the most common alloy in the family of precipitation hardening stainless steels; it has a good combination of corrosion resistance and mechanical properties obtained by proper heat treatment and is used for engineering components in chemical process equipment, pump shafts and gears in marine environments and pressure vessels [4–6]. The heat treatment process normally applied to 17-4PH stainless steel is age-hardening for which the alloy is first solution annealed at 1050 °C, followed by air or oil quenching and aged at the final step. Optimum mechanical properties are associated with precipitation of copper rich phase and formation of carbides and reverted austenite in the martensitic matrix during ageing treatment [1,4,6–9]. Ageing response in this alloy is very sensitive to heat treatment cycle and, accordingly, for maximum hardening, n
Corresponding author. E-mail address: [email protected] (S.A. Razavi).
http://dx.doi.org/10.1016/j.msea.2016.08.049 0921-5093/& 2016 Elsevier B.V. All rights reserved.
ageing parameters must be carefully selected; in some cases, a long procedure would be required to set these parameters. ANN models and genetic algorithms have been used to predict and optimize several properties of materials [10–12]. Song et al. [13] applied these techniques to find optimum conditions for heat treatment of aluminum alloy 7175. Nada et al. [14] used an ANN and back propagation algorithm to model the relation between hardness and ageing temperature in Al-3 wt% Mg alloy. Konieczny [15] applied an ANN model to predict the influence of chemical composition on the properties of copper alloys. Zakeri et al. [3] used GA-ANN combined algorithm to assign conditions for heat treatment of 17-4PH stainless steel on the basis of yield strength. In the present work, Artificial Neural Network (ANN) was used to predict the hardness of 17-4PH stainless steel at different ageing times and temperatures up to 240 min in the temperature range 400–600 °C. Then, Genetic Algorithm (GA) and ANN model as the fitness function were employed to find maximum hardness and the corresponding ageing time and temperatures. The results were compared with experimental data and are discussed in this paper. 1.1. Artificial neural network (ANN) Artificial neural network (ANN) is an established numerical computerized system that can model complex relationship between random experimental inputs and their relative outputs [14,16–18]. ANN works according to biological neural networks; biological neural networks are constructed from small parts that
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S.A. Razavi et al. / Materials Science & Engineering A 675 (2016) 147–152
called neurons and electrochemical signals are received through synapses to the neuron. Each synapse adds the special weight to the received signal and modifies weights during learning. A group of neurons form subsystem and subsystem collection form brain [3]. Artificial neural networks consist of neurons, too. Input data to each neuron are weighted and these weights are corrected during learning. It can be said that ANN works as a brain, ANN layers work as subsystems and neurons have the same role at two networks. ANN generally contains three layers. The input layer that input data are entered into the network, hidden layer that connects input layer to the output layer, and output layer that output data are recognized to the network [14,16,17]. Between different types of ANN models, multilayer perceptron (MLP) neural networks are more common and used for the nonlinear functions. MLP neural network is a feed-forward ANN and formed from an input layer, an output layer and some hidden layers. The activation function in output layer at MLP ANN must be linear and in hidden layers can be linear or nonlinear. A sigmoid function, Tansig or Logsig, is mostly used as nonlinear function in MLP ANN models (Fig. 1)[3]. The more common ANN algorithm is back-propagation. In this algorithm error value of a hidden layer is determined with back propagating of the error according to output layer [3,14,18]. Generally, feed-forward back propagation has the best performance to find a nonlinear relationship between inputs and output data and, therefore, it has been widely used in a variety of applications. On the other hand, feed forward back propagation networks have been recommended for problems with too many changeable weights and too many training cycles. In back propagation method, if the network gives an incorrect output, the weights are corrected by the model; this means the results would be more reliable with fewer errors [19–21]. In this study, a feed-forward back propagation MLP neural network was used and Levenberg-Marquardt method was applied as the training algorithm. 1.2. Genetic algorithm Genetic algorithm (GA) is a powerful tool for optimizing mathematical models. GA is created according to genetics and natural selection [12,22]. In GA, points are known as individuals or population. Each individual gains the special value according to the related function that is known as fitness function. Good or bad individual selected according to individual value that gain from the fitness function. Strong individuals can go to next generation and weak individuals die. There are two kinds of operation in GA, crossover and mutation. In crossover operation, two strong individual build new individual. New individual has their parent's characteristics. On the other hand, in mutation operation a random individual is selected, then two elements of this individual exchanged. Mutation helps GA reach its aim rapidly. GA with these operators tries to reach from initial population to optimum individual.
Fig. 1. Equation and shape of sigmoid functions.
The material was solution annealed at 1050 °C for 30 min followed by oil quenching at room temperature. Solutionizing temperature is another process parameter that can be verified as an input parameter in a separate model. An isothermal section of FeC-Cu ternary diagram of 17-4PH alloy at 1050 °C, shown in Fig. 2 (a) suggests that appropriate temperature could not be selected far from 1050 °C. It is not possible to experience large variations in the solutionizing temperature since too low a temperature would reduce the stability of austenite leading to formation of copper and α-iron phases (Fig. 2(b)) and too high temperatures would result in grain growth phenomenon and, thus, decrease in mechanical properties of the alloy. To study the relationship between ageing time and temperature with hardness, small specimens of 2 cm length were aged at 400, 450, 500, 550, and 600 °C in a salt bath furnace for specified time periods. The bulk hardness was measured by Rockwell C hardness testing with KOOPA (model: UV1) universal hardness tester, thus, more than 100 data was collected including pairs of ageing time-temperature and their related hardness values. For modeling of age hardening process, ANN (artificial neural network) was employed. In ANN, a feed-forward back propagation MLP was used the number of neurons in hidden layer was selected by try and error progress. Ageing times and temperatures were used as input parameters in the ANN model, whereas the hardness was output of the model. ANN model was performed in MATLAB software. The best ANN model was selected as the fitness function in the Genetic Algorithm. GA-ANN combination was used to find the optimum heat treatment cycle for obtaining maximum hardness value in age hardening of 17-4PH stainless steel. In the final step, in order to validate the prediction of ANN model and GA, samples of 17-4PH were heat treated according to GA output parameters followed by hardness measurements.
3. Results and discussion 3.1. Age-Hardening in 17-4PH stainless steel
2. Experimental procedure The material used in this investigation was 17-4PH stainless steel with chemical composition shown in Table 1 as obtained by optical spectrometer. The material was in the form of a rod of 10 mm diameter. Table 1 Chemical composition of 17-4PH stainless steel. Element
C
Cr
Ni
Cu
Si
Mn
Mo
Nb
Fe
wt%
0.02
15.86
4.05
3.34
0.5
0.68
0.16
0.24
bal.
Variation of hardness in age hardenable alloys is closely related to the formation of precipitates, hence, it is vital to understand the kinetics of precipitation at different temperatures. Fig. 3 shows the effect of temperature on ageing response of 17-4PH stainless steel. Hardness of solution treated material was measured 33 HRC and age hardening curves indicate that peak hardness of the samples decreased as the ageing temperature was increased. The lower the process temperature, the higher is the hardness value. It is observed that at lower temperatures, peak hardness value is achieved at longer ageing time, however, prolonged ageing at low temperatures did not produce any significant decrease in the peak hardness. This trend is shown in Table 2. Ageing test at 400 °C was
S.A. Razavi et al. / Materials Science & Engineering A 675 (2016) 147–152
149
Fig. 2. Isothermal section of Fe-C-Cu ternary diagram of 17-4PH; a) 1050 °C, b) 850 °C [23].
46 44
Hardness (HRC)
42 40 38 400 450
36
500 34
550 600
32
0
50
100
150
200
250
300
Time (min)
Fig. 3. Age-Hardening behavior of 17-4PH stainless steel at different temperatures. Table 2 Ageing time-temperature pairs and corresponding peak hardness values. Ageing temperature (°C)
Ageing time (min)
Peak hardness (HRC)
400 450 500 550 600
300 180 20 10 5
40.2 70.9 44.8 7 0.4 43.4 70.4 42.17 0.8 39.6 70.8
continued to 300 min, although hardness had an increasing trend. The trend of variations in hardness is associated with interaction between tempering of martensite, precipitation of particles and formation of reverted austenite. Reverted austenite is mostly formed at temperatures above 550 °C and the decrease of bulk hardness at high temperatures, dominantly after peak hardness, is attributed to this phase. It has been reported [1,4,7] that hardness and strength decrease with the formation of reverted austenite. While reverted austenite formation and martensite tempering can reduce hardness value, precipitation of copper rich phase would significantly increase the hardness; thus, the net result would be an increase in the hardness of 17–4PH stainless steel. As it is evident in Fig. 4, solubility of Cu atoms in iron matrix at solution temperature (1050 °C) is about 6 wt%
Fig. 4. Fe-Cu phase diagram [23].
and at room temperature is nearly zero. As a result, oil quenching produced a supersaturated solid solution and ageing treatment persuades copper atoms to exit from the matrix and precipitate as the copper rich particles. Interaction between these particles and dislocations would increase both the hardness and strength of the steel. Such a phenomenon is well established in the literature [1,4,8] although the mechanisms involved are under debate. 3.2. Artificial neural network (ANN) model In the process of age hardening, hardness is a function of ageing
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R2=1−
SSres SStot
(2)
where SSres is the residual sum of squares and SStot is the total sum of squares which can be calculated by Eqs. (3) and (4); n
SSres =
∑ i=1
n
SStot =
∑ i=1
time and temperature and, in this investigation, ageing timetemperature pairs were set as input variables of ANN model, while hardness values are output of the modelling. In practical situation, the multi-layer perceptron (MLP) is known as an appropriate network architecture for approximate nonlinear functions [3,18,24]. Since the function of hardness versus ageing time and temperature is nonlinear, MLP ANN was used. In this work, a feedforward back propagation MLP was employed; 100 data sets were used to make and train ANN model and 15 data sets were selected randomly to verify the prediction capability of the ANN model. (Fig. 5). In order to investigate the best ANN model, number of neurons from 1 to 20 and sigmoid activation functions (Tansig and Logsig) in hidden layer have been changed. The best parameters of ANN were selected on the basis of minimum mean square error (MSE). The calculated MSE for different numbers of neurons and sigmoid activation functions are shown in Fig. 6. The mean square error (MSE) has been calculated as follows [17]; n
∑i = 1
(ai − fi )2 (1)
n
(3)
( ( y − y̅ ) )
(4)
2
i
i
2
i
n shows the number of data compared, yi is the experimental data, fi is the predicted data by ANN model and y̅ is the mean value of experimental data. In the final ANN model, R2 value is 0.9747. R2 becomes unity if experimental data is equal to simulated data. Fig. 7 reveals that in this simulation, some of random real hardness values are approximately close to simulated values. It shows that the trained ANN model simulates hardness value during ageing process at different ageing times and temperatures. According to Fig. 7 and R2 value, it can be concluded that made and trained ANN model predicts age-hardening behavior, perfectly. Fig. 8 presents the simulated age hardening curves of 17-4PH alloy at different temperatures. Comparison of Fig. 3 and Fig. 8 indicates that simulated age hardening and experimental age hardening follow similar trends, although some differences exist. For instance, after 7 min ageing at 400 °C, an unpredictable reduction in hardness is evident. Using this simulation, the hardness of other ageing time-temperature pairs can be predicted. It appears that the trained ANN model can simulate hardness at different ageing times and temperatures, properly. Fig. 9 shows the predicted results of ANN model; the maximum hardness achieved at lower 490 °C is indicated by dark red area. In Fig. 9(a), ageing times and temperatures and the related hardness values are presented by different colors; the color bar displays relative hardness value corresponding to each color. Fig. 9 (b) shows the same data in the 3-dimension mode surface; at higher temperatures and initial ageing times, the minimum value of hardness is shown at the lower level. According to these plots (Fig. 9(a) and (b)), maximum hardness in this alloy will be achieved if the samples are aged between 450 °C and 490 °C for 90 min–160 min. It should be noted that small variation in the copper and chromium content of the alloy would change the extent of supersaturation in the solution annealed condition. Although such variations for 17-4PH are normally around 1%–2%, it is expected that the final hardness of age hardened alloy is affected few percent. Accordingly, if the model is applied to compositions
Fig. 5. Schematic model of Back Propagation Neural network.
MSE =
((y − f ) )
where ai is the actual experimental data, fi is the predicted data by ANN model and n is the number of data points used. As shown in Fig. 6, minimum of mean square error was achieved with Tansig activation function of 7 and 18 neurons, but MSE value of 7 neurons was lower than that of 18 neurons. The minimum MSE value in the hidden layer is 0.625. After construction of network, it was trained until a network with maximum R2 was achieved. R2 is the coefficient of determination and is calculated as follows;
Tansig
Logsig
9 8 7
MSE
6 5 4 3 2 1 0 1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
Numbers of Neurons Fig. 6. Diagram of MSE versus number of neurons for sigmoid functions.
18
19
20
S.A. Razavi et al. / Materials Science & Engineering A 675 (2016) 147–152
400
450
500
550
151
600
46
Simulated Hardness
44 42 40 38 36 34 32 32
34
36
38
40
42
44
46
Experimental Hardness
Fig. 7. Comparison of the simulated hardness data and experimental values. 46
Fig. 10. Plot of Fitness value - Generation for the proposed GA. 44
Hardness (HRC)
42
40
38
400 450
36
500 550
34
600 32 0
50
100
150
200
250
300
Time (min)
Fig. 8. Simulated age-hardening curves of 17-4PH stainless steel at different temperatures.
near to the one in the present work, small error must be considered in the results of hardness values. 3.3. GA results In order to find the best age hardening cycle, a scattered crossover was used; the fitness function in this GA is the best ANN model. The number of initial population was selected 30 and the maximum generation was 300. The probability of crossover and probability of mutation were selected 0.8 and 0.1, respectively. In this study, the aim was maximizing of the hardness and in GA, maximizing of the fitness function is equivalent to minimizing of
the negative fitness function [22]. Therefore, in Fig. 10 that shows the fitness value in GA development, hardness values are negative. As it can be seen, in first generation, the values of mean fitness are far from the best fitness, but with generation enhancement, this difference is decreased and means fitness tends toward best fitness value. On the other hand, with generation enhancement, the mean fitness scattering is reduced and tends to the best fitness value. Table 3 shows the probable heat treatment cycle to attain the maximum value of hardness in 17-4PH stainless steel. As indicated in Table 3, according to the GA-ANN simulation, a hardness of 44.9 HRC would be achieved by ageing 129 min at 464 °C after solution anneal process at 1050 °C for 30 min. In experiments, these heat treatment parameters lead to a maximum hardness value of 44.1 HRC. By comparing simulated hardness and experimental hardness, a relative error of 1.8% was calculated. In conclusion, the ANN model can predict hardness of the 17-4PH stainless steel at different ageing times and temperatures fairly accurately. The model can predict hardness values at all ageing temperatures between 400 °C and 600 °C and ageing times up to 240 min.
4. Conclusions In this paper, ANN model was developed to predict hardness values at different ageing times and temperatures for 17-4PH stainless steel. The following points are concluded based on the
Fig. 9. Simulated ageing hardness; a) hardness contour, b) 3D surface.
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Table 3 The optimum heat treatment process obtained by GA-ANN. Solution temperature (°C) Solution time (min)
Ageing temperature (°C) Ageing time (min)
Predicted hardness (HRC)
Max. measured hardness (HRC)
Relative Error (%)
1050
464
44.9
44.1
1.8
30
129
results of modelling and experimental work.
An MLP ANN model with one hidden layer and 7 neurons in the
hidden layer is a powerful method for prediction of hardness value of 17-4PH stainless steel. For the final ANN model, R2 value is 0.9747 confirming that the model can predict hardness values with sufficient accuracy. With this model, all hardness values for ageing temperatures between 400 and 600 °C and ageing times up to 240 min, can be predicted. GA-ANN combination affords the maximum hardness value with a deviation of 1.8% from experimental results.
References [1] R. Bhambroo, S. Roychowdhury, V. Kain, V. Raja, Mater. Sci. and Eng.: A, 568, pp. 127–133. [2] W.F. Smith, Structure and properties of Engineering Alloys, McGraw-Hill, 1993. [3] M. Zakeri, A. Bahrami, S.M. Anijdan, Mater. Des. 28 (2007) 2034–2039. [4] C. Hsiao, C. Chiou, J. Yang, Mater. Chem. Phys. 74 (2002) 134–142. [5] S. Mohd, M.S. Bhuiyan, D. Nie, Y. Otsuka, Y. Mutoh, Int. J. Fatigue 82 (2016) 371–378. [6] L.E. Murr, E. Martinez, J. Hernandez, S. Collins, K.N. Amato, S.M. Gaytan, P. W. Shindo, J. Mater. Res. Technol. 1 (2012) 167–177.
[7] U. Viswanathan, S. Banerjee, R. Krishnan, Mater. Sci. Eng.: A 104 (1988) 181–189. [8] J. Wang, H. Zou, C. Li, R. Zuo, S. Qiu, B. Shen, J. Univ. Sci. Technol. Beijing Miner., Metall., Mater. 13 (2006) 235–239. [9] A. Ziewiec, A. Zielińska-Lipiec, E. Tasak, Arch. Metall. Mater. 59 (2014) 965–970. [10] E. Alibeiki, J. Rajabi, J. Rajabi, J. Asian Sci. Res. 2 (2012) 742. [11] S. Etesami, M. Enayati, A. Taherizadeh, B. Sadeghian, Trans. Indian Inst. Met. (2016) 1–8. [12] L. Gu, L.-y WANG, G.-m CHEN, S.-c HUA, J. Iron Steel Res., Int. 18 (2011) 57–64. [13] R. Song, Q. Zhang, J. Mater. Process. Technol. 117 (2001) 84–88. [14] R. Nada, D. Habashy, F.A. EL-Salam, M.Y. EL-Bakry, A.A. EL-Khalek, E.A. ELReheim, Mater. Sci. Eng.: A 567 (2013) 80–83. [15] J. Konieczny, J. Achiev. Mater. Manuf. Eng. 55 (2012) 529–535. [16] A.N. Celik, T. Muneer, Energy Convers. Manag. 67 (2013) 117–124. [17] R. Haghbakhsh, H. Adib, P. Keshavarz, M. Koolivand, S. Keshtkari, Thermochim. Acta 551 (2013) 124–130. [18] T. Mareš, E. Janouchová, A. Kučerová, Adv. Eng. Softw. 95 (2016) 68–81. [19] S.S. Behzadi, C. Prakasvudhisarn, J. Klocker, P. Wolschann, H. Viernstein, Powder Technol. 195 (2009) 150–157. [20] C. Chen, H. Yin, I.S. Humail, Y. Wang, X. Qu, Int. J. Refract. Met. Hard Mater. 25 (2007) 411–416. [21] B. Kermanshahi, H. Iwamiya, Int. J. Electr. Power Energy Syst. 24 (2002) 789–797. [22] L.D. Chambers, Practical Handbook of Genetic Algorithms: Complex coding Systems, CRC press, Chapman & Hall/CRC, 1998. [23] A. Handbook, ASM International, Materials Park, OH, USA, 1992, p. 2.48. [24] S.M. Anijdan, A. Bahrami, Mater. Sci. Eng.: A 396 (2005) 138–142.
Contents lists available at ScienceDirect
Materials Science & Engineering A journal homepage: www.elsevier.com/locate/msea
Prediction of age hardening parameters for 17-4PH stainless steel by artificial neural network and genetic algorithm Sayyed Ali Razavi n, Fakhreddin Ashrafizadeh, Saghar Fooladi Department of Materials Engineering, Isfahan University of Technology, Isfahan 8415683111, Iran
art ic l e i nf o
a b s t r a c t
Article history: Received 5 May 2016 Received in revised form 11 August 2016 Accepted 12 August 2016 Available online 13 August 2016
17-4PH alloy is a precipitation hardening stainless steel with a desirable combination of high corrosion resistance and good mechanical properties at temperatures up to 300 °C in engineering applications. Ageing mechanism in this alloy is rather complicated, hence, for optimum hardness, selection of heat treatment parameters is critical. In this investigation, first, Artificial Neural Network (ANN) was used to model the relationship between ageing times and temperatures with the corresponding hardness. R2 value was 0.9747 in the final ANN model indicating this model could predict hardness values, appropriately. Then, Genetic Algorithm (GA) was used to find optimum aging time and temperature for the target of maximum hardness value. ANN model was used as the fitness function in GA. According to the GA-ANN simulation, a hardness of 44.9 HRC would be achieved by ageing 129 min at 464 °C. Finally, the model was validated by conducting heat treatment experiments carried out using the predicted parameters. A maximum hardness of 44.1 HRC was obtained in the experimental work, showing a difference of 1.8% with the proposed model. & 2016 Elsevier B.V. All rights reserved.
Keywords: Hardness 17-4PH stainless steel Age hardening Ageing Artificial neural network Genetic algorithm
1. Introduction 17-4PH alloy (AISI 630) is among precipitation hardenable stainless steels with attractive properties in terms of fabrication, high hardness and strength together with high ductility, especially at temperatures below 300 °C, and excellent corrosion resistance [1–3]. Precipitation hardening steels are divided into two groups; semiaustenitic type and martensitic type where the martensitic grades are used in many industrial sectors with more applications than semi-austenitic grades [2]. 17-4PH is the most common alloy in the family of precipitation hardening stainless steels; it has a good combination of corrosion resistance and mechanical properties obtained by proper heat treatment and is used for engineering components in chemical process equipment, pump shafts and gears in marine environments and pressure vessels [4–6]. The heat treatment process normally applied to 17-4PH stainless steel is age-hardening for which the alloy is first solution annealed at 1050 °C, followed by air or oil quenching and aged at the final step. Optimum mechanical properties are associated with precipitation of copper rich phase and formation of carbides and reverted austenite in the martensitic matrix during ageing treatment [1,4,6–9]. Ageing response in this alloy is very sensitive to heat treatment cycle and, accordingly, for maximum hardening, n
Corresponding author. E-mail address: [email protected] (S.A. Razavi).
http://dx.doi.org/10.1016/j.msea.2016.08.049 0921-5093/& 2016 Elsevier B.V. All rights reserved.
ageing parameters must be carefully selected; in some cases, a long procedure would be required to set these parameters. ANN models and genetic algorithms have been used to predict and optimize several properties of materials [10–12]. Song et al. [13] applied these techniques to find optimum conditions for heat treatment of aluminum alloy 7175. Nada et al. [14] used an ANN and back propagation algorithm to model the relation between hardness and ageing temperature in Al-3 wt% Mg alloy. Konieczny [15] applied an ANN model to predict the influence of chemical composition on the properties of copper alloys. Zakeri et al. [3] used GA-ANN combined algorithm to assign conditions for heat treatment of 17-4PH stainless steel on the basis of yield strength. In the present work, Artificial Neural Network (ANN) was used to predict the hardness of 17-4PH stainless steel at different ageing times and temperatures up to 240 min in the temperature range 400–600 °C. Then, Genetic Algorithm (GA) and ANN model as the fitness function were employed to find maximum hardness and the corresponding ageing time and temperatures. The results were compared with experimental data and are discussed in this paper. 1.1. Artificial neural network (ANN) Artificial neural network (ANN) is an established numerical computerized system that can model complex relationship between random experimental inputs and their relative outputs [14,16–18]. ANN works according to biological neural networks; biological neural networks are constructed from small parts that
148
S.A. Razavi et al. / Materials Science & Engineering A 675 (2016) 147–152
called neurons and electrochemical signals are received through synapses to the neuron. Each synapse adds the special weight to the received signal and modifies weights during learning. A group of neurons form subsystem and subsystem collection form brain [3]. Artificial neural networks consist of neurons, too. Input data to each neuron are weighted and these weights are corrected during learning. It can be said that ANN works as a brain, ANN layers work as subsystems and neurons have the same role at two networks. ANN generally contains three layers. The input layer that input data are entered into the network, hidden layer that connects input layer to the output layer, and output layer that output data are recognized to the network [14,16,17]. Between different types of ANN models, multilayer perceptron (MLP) neural networks are more common and used for the nonlinear functions. MLP neural network is a feed-forward ANN and formed from an input layer, an output layer and some hidden layers. The activation function in output layer at MLP ANN must be linear and in hidden layers can be linear or nonlinear. A sigmoid function, Tansig or Logsig, is mostly used as nonlinear function in MLP ANN models (Fig. 1)[3]. The more common ANN algorithm is back-propagation. In this algorithm error value of a hidden layer is determined with back propagating of the error according to output layer [3,14,18]. Generally, feed-forward back propagation has the best performance to find a nonlinear relationship between inputs and output data and, therefore, it has been widely used in a variety of applications. On the other hand, feed forward back propagation networks have been recommended for problems with too many changeable weights and too many training cycles. In back propagation method, if the network gives an incorrect output, the weights are corrected by the model; this means the results would be more reliable with fewer errors [19–21]. In this study, a feed-forward back propagation MLP neural network was used and Levenberg-Marquardt method was applied as the training algorithm. 1.2. Genetic algorithm Genetic algorithm (GA) is a powerful tool for optimizing mathematical models. GA is created according to genetics and natural selection [12,22]. In GA, points are known as individuals or population. Each individual gains the special value according to the related function that is known as fitness function. Good or bad individual selected according to individual value that gain from the fitness function. Strong individuals can go to next generation and weak individuals die. There are two kinds of operation in GA, crossover and mutation. In crossover operation, two strong individual build new individual. New individual has their parent's characteristics. On the other hand, in mutation operation a random individual is selected, then two elements of this individual exchanged. Mutation helps GA reach its aim rapidly. GA with these operators tries to reach from initial population to optimum individual.
Fig. 1. Equation and shape of sigmoid functions.
The material was solution annealed at 1050 °C for 30 min followed by oil quenching at room temperature. Solutionizing temperature is another process parameter that can be verified as an input parameter in a separate model. An isothermal section of FeC-Cu ternary diagram of 17-4PH alloy at 1050 °C, shown in Fig. 2 (a) suggests that appropriate temperature could not be selected far from 1050 °C. It is not possible to experience large variations in the solutionizing temperature since too low a temperature would reduce the stability of austenite leading to formation of copper and α-iron phases (Fig. 2(b)) and too high temperatures would result in grain growth phenomenon and, thus, decrease in mechanical properties of the alloy. To study the relationship between ageing time and temperature with hardness, small specimens of 2 cm length were aged at 400, 450, 500, 550, and 600 °C in a salt bath furnace for specified time periods. The bulk hardness was measured by Rockwell C hardness testing with KOOPA (model: UV1) universal hardness tester, thus, more than 100 data was collected including pairs of ageing time-temperature and their related hardness values. For modeling of age hardening process, ANN (artificial neural network) was employed. In ANN, a feed-forward back propagation MLP was used the number of neurons in hidden layer was selected by try and error progress. Ageing times and temperatures were used as input parameters in the ANN model, whereas the hardness was output of the model. ANN model was performed in MATLAB software. The best ANN model was selected as the fitness function in the Genetic Algorithm. GA-ANN combination was used to find the optimum heat treatment cycle for obtaining maximum hardness value in age hardening of 17-4PH stainless steel. In the final step, in order to validate the prediction of ANN model and GA, samples of 17-4PH were heat treated according to GA output parameters followed by hardness measurements.
3. Results and discussion 3.1. Age-Hardening in 17-4PH stainless steel
2. Experimental procedure The material used in this investigation was 17-4PH stainless steel with chemical composition shown in Table 1 as obtained by optical spectrometer. The material was in the form of a rod of 10 mm diameter. Table 1 Chemical composition of 17-4PH stainless steel. Element
C
Cr
Ni
Cu
Si
Mn
Mo
Nb
Fe
wt%
0.02
15.86
4.05
3.34
0.5
0.68
0.16
0.24
bal.
Variation of hardness in age hardenable alloys is closely related to the formation of precipitates, hence, it is vital to understand the kinetics of precipitation at different temperatures. Fig. 3 shows the effect of temperature on ageing response of 17-4PH stainless steel. Hardness of solution treated material was measured 33 HRC and age hardening curves indicate that peak hardness of the samples decreased as the ageing temperature was increased. The lower the process temperature, the higher is the hardness value. It is observed that at lower temperatures, peak hardness value is achieved at longer ageing time, however, prolonged ageing at low temperatures did not produce any significant decrease in the peak hardness. This trend is shown in Table 2. Ageing test at 400 °C was
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149
Fig. 2. Isothermal section of Fe-C-Cu ternary diagram of 17-4PH; a) 1050 °C, b) 850 °C [23].
46 44
Hardness (HRC)
42 40 38 400 450
36
500 34
550 600
32
0
50
100
150
200
250
300
Time (min)
Fig. 3. Age-Hardening behavior of 17-4PH stainless steel at different temperatures. Table 2 Ageing time-temperature pairs and corresponding peak hardness values. Ageing temperature (°C)
Ageing time (min)
Peak hardness (HRC)
400 450 500 550 600
300 180 20 10 5
40.2 70.9 44.8 7 0.4 43.4 70.4 42.17 0.8 39.6 70.8
continued to 300 min, although hardness had an increasing trend. The trend of variations in hardness is associated with interaction between tempering of martensite, precipitation of particles and formation of reverted austenite. Reverted austenite is mostly formed at temperatures above 550 °C and the decrease of bulk hardness at high temperatures, dominantly after peak hardness, is attributed to this phase. It has been reported [1,4,7] that hardness and strength decrease with the formation of reverted austenite. While reverted austenite formation and martensite tempering can reduce hardness value, precipitation of copper rich phase would significantly increase the hardness; thus, the net result would be an increase in the hardness of 17–4PH stainless steel. As it is evident in Fig. 4, solubility of Cu atoms in iron matrix at solution temperature (1050 °C) is about 6 wt%
Fig. 4. Fe-Cu phase diagram [23].
and at room temperature is nearly zero. As a result, oil quenching produced a supersaturated solid solution and ageing treatment persuades copper atoms to exit from the matrix and precipitate as the copper rich particles. Interaction between these particles and dislocations would increase both the hardness and strength of the steel. Such a phenomenon is well established in the literature [1,4,8] although the mechanisms involved are under debate. 3.2. Artificial neural network (ANN) model In the process of age hardening, hardness is a function of ageing
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R2=1−
SSres SStot
(2)
where SSres is the residual sum of squares and SStot is the total sum of squares which can be calculated by Eqs. (3) and (4); n
SSres =
∑ i=1
n
SStot =
∑ i=1
time and temperature and, in this investigation, ageing timetemperature pairs were set as input variables of ANN model, while hardness values are output of the modelling. In practical situation, the multi-layer perceptron (MLP) is known as an appropriate network architecture for approximate nonlinear functions [3,18,24]. Since the function of hardness versus ageing time and temperature is nonlinear, MLP ANN was used. In this work, a feedforward back propagation MLP was employed; 100 data sets were used to make and train ANN model and 15 data sets were selected randomly to verify the prediction capability of the ANN model. (Fig. 5). In order to investigate the best ANN model, number of neurons from 1 to 20 and sigmoid activation functions (Tansig and Logsig) in hidden layer have been changed. The best parameters of ANN were selected on the basis of minimum mean square error (MSE). The calculated MSE for different numbers of neurons and sigmoid activation functions are shown in Fig. 6. The mean square error (MSE) has been calculated as follows [17]; n
∑i = 1
(ai − fi )2 (1)
n
(3)
( ( y − y̅ ) )
(4)
2
i
i
2
i
n shows the number of data compared, yi is the experimental data, fi is the predicted data by ANN model and y̅ is the mean value of experimental data. In the final ANN model, R2 value is 0.9747. R2 becomes unity if experimental data is equal to simulated data. Fig. 7 reveals that in this simulation, some of random real hardness values are approximately close to simulated values. It shows that the trained ANN model simulates hardness value during ageing process at different ageing times and temperatures. According to Fig. 7 and R2 value, it can be concluded that made and trained ANN model predicts age-hardening behavior, perfectly. Fig. 8 presents the simulated age hardening curves of 17-4PH alloy at different temperatures. Comparison of Fig. 3 and Fig. 8 indicates that simulated age hardening and experimental age hardening follow similar trends, although some differences exist. For instance, after 7 min ageing at 400 °C, an unpredictable reduction in hardness is evident. Using this simulation, the hardness of other ageing time-temperature pairs can be predicted. It appears that the trained ANN model can simulate hardness at different ageing times and temperatures, properly. Fig. 9 shows the predicted results of ANN model; the maximum hardness achieved at lower 490 °C is indicated by dark red area. In Fig. 9(a), ageing times and temperatures and the related hardness values are presented by different colors; the color bar displays relative hardness value corresponding to each color. Fig. 9 (b) shows the same data in the 3-dimension mode surface; at higher temperatures and initial ageing times, the minimum value of hardness is shown at the lower level. According to these plots (Fig. 9(a) and (b)), maximum hardness in this alloy will be achieved if the samples are aged between 450 °C and 490 °C for 90 min–160 min. It should be noted that small variation in the copper and chromium content of the alloy would change the extent of supersaturation in the solution annealed condition. Although such variations for 17-4PH are normally around 1%–2%, it is expected that the final hardness of age hardened alloy is affected few percent. Accordingly, if the model is applied to compositions
Fig. 5. Schematic model of Back Propagation Neural network.
MSE =
((y − f ) )
where ai is the actual experimental data, fi is the predicted data by ANN model and n is the number of data points used. As shown in Fig. 6, minimum of mean square error was achieved with Tansig activation function of 7 and 18 neurons, but MSE value of 7 neurons was lower than that of 18 neurons. The minimum MSE value in the hidden layer is 0.625. After construction of network, it was trained until a network with maximum R2 was achieved. R2 is the coefficient of determination and is calculated as follows;
Tansig
Logsig
9 8 7
MSE
6 5 4 3 2 1 0 1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
Numbers of Neurons Fig. 6. Diagram of MSE versus number of neurons for sigmoid functions.
18
19
20
S.A. Razavi et al. / Materials Science & Engineering A 675 (2016) 147–152
400
450
500
550
151
600
46
Simulated Hardness
44 42 40 38 36 34 32 32
34
36
38
40
42
44
46
Experimental Hardness
Fig. 7. Comparison of the simulated hardness data and experimental values. 46
Fig. 10. Plot of Fitness value - Generation for the proposed GA. 44
Hardness (HRC)
42
40
38
400 450
36
500 550
34
600 32 0
50
100
150
200
250
300
Time (min)
Fig. 8. Simulated age-hardening curves of 17-4PH stainless steel at different temperatures.
near to the one in the present work, small error must be considered in the results of hardness values. 3.3. GA results In order to find the best age hardening cycle, a scattered crossover was used; the fitness function in this GA is the best ANN model. The number of initial population was selected 30 and the maximum generation was 300. The probability of crossover and probability of mutation were selected 0.8 and 0.1, respectively. In this study, the aim was maximizing of the hardness and in GA, maximizing of the fitness function is equivalent to minimizing of
the negative fitness function [22]. Therefore, in Fig. 10 that shows the fitness value in GA development, hardness values are negative. As it can be seen, in first generation, the values of mean fitness are far from the best fitness, but with generation enhancement, this difference is decreased and means fitness tends toward best fitness value. On the other hand, with generation enhancement, the mean fitness scattering is reduced and tends to the best fitness value. Table 3 shows the probable heat treatment cycle to attain the maximum value of hardness in 17-4PH stainless steel. As indicated in Table 3, according to the GA-ANN simulation, a hardness of 44.9 HRC would be achieved by ageing 129 min at 464 °C after solution anneal process at 1050 °C for 30 min. In experiments, these heat treatment parameters lead to a maximum hardness value of 44.1 HRC. By comparing simulated hardness and experimental hardness, a relative error of 1.8% was calculated. In conclusion, the ANN model can predict hardness of the 17-4PH stainless steel at different ageing times and temperatures fairly accurately. The model can predict hardness values at all ageing temperatures between 400 °C and 600 °C and ageing times up to 240 min.
4. Conclusions In this paper, ANN model was developed to predict hardness values at different ageing times and temperatures for 17-4PH stainless steel. The following points are concluded based on the
Fig. 9. Simulated ageing hardness; a) hardness contour, b) 3D surface.
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Table 3 The optimum heat treatment process obtained by GA-ANN. Solution temperature (°C) Solution time (min)
Ageing temperature (°C) Ageing time (min)
Predicted hardness (HRC)
Max. measured hardness (HRC)
Relative Error (%)
1050
464
44.9
44.1
1.8
30
129
results of modelling and experimental work.
An MLP ANN model with one hidden layer and 7 neurons in the
hidden layer is a powerful method for prediction of hardness value of 17-4PH stainless steel. For the final ANN model, R2 value is 0.9747 confirming that the model can predict hardness values with sufficient accuracy. With this model, all hardness values for ageing temperatures between 400 and 600 °C and ageing times up to 240 min, can be predicted. GA-ANN combination affords the maximum hardness value with a deviation of 1.8% from experimental results.
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