- Email: [email protected]
Use of artificial neural network for prediction of ion nitrided case depth in Fe–Cr alloys
Materials and Design 24 (2003) 203–207
Use of artificial neural network for prediction of ion nitrided case depth in Fe–Cr alloys Kenan Genel* Mechanical Engineering Department, Sakarya University, Esentepe 54187, Sakarya, Turkey Received 29 August 2002; received in revised form 24 October 2002; accepted 16 December 2002
Abstract In this work, a simple artificial neural network (ANN) model using back-propagation training algorithm for ion nitriding behaviour of Fe–Cr alloys was established. The case depth data were extracted from experimental data and used in the formation of training sets of ANN in order to predict case depth of ion nitrided Fe–Cr alloys, 2.5% Cr intervals for 5–20% Cr. The modelling results confirm the feasibility of this approach and show good agreement with experimental data by Alves et al. (Mater Sci Eng 2002; 279A: 10–15) with high accuracy. A contour diagram as a function of Cr (wt.%) and ion nitriding time for Fe– Cr alloy was constructed for industrial application. It is concluded that a considerable saving in terms of cost and time could be obtained from using the trained ANN model and, it provides more useful data from relatively small experimental databases. 䊚 2003 Elsevier Science Ltd. All rights reserved. Keywords: Ion nitriding; Artificial neural networks; Fe–Cr alloys
1. Introduction Plasma or ion nitriding, owing to a number of advantages over conventional nitriding methods, have found increasing applications in industry. It is well known that ion nitriding surface treatment, which is ferritic thermochemical methods, is widely used to improve fatigue, corrosion fatigue and tribological properties of Fe-based alloys w1–6x. During ion nitriding, reaction takes place both at surface of the metal and below the surface due to diffusion of nitrogen atoms from the surface to interior region. Hence, thin iron nitride layer (compound layer) and diffusion zone form on the surface and subsurface region of steel, respectively. Both surface hardness and nitrided case depth are mainly related to chemical composition of metal for given process parameters such as time, temperature and composition of gas mixture w6,7x. Chromium is often used an alloying element due to its affinity for nitrogen. Previous studies, which were performed on nitriding of Fe–Cr alloys, indicated that chromium nitrides (CrN and Cr2N), in contrast with the iron nitrides, were precipitate at the nitriding temperature *Tel.: q90-264-346-0353x319; fax: q90-264-346-0351. E-mail address: [email protected] (K. Genel).
w8x. Micro-structural differences were observed in nitrided case depending on chromium content and ion nitriding temperature w9x. On nitriding, chromium-alloyed ferritic matrix submicroscopical, coherent, chromium nitride particles develop giving rise to high hardness of the order of 1000 HV w10x. The expansion of the nitrided case is counteracted by core region. As a result, compressive residual stress develop near the surface, while a tensile stress evolves in the core. Case depth has a favourable effect on magnitude of beneficial compressive residual stress as well as fatigue behaviour of nitrided steels w2,8,10,11x. The progress of case depth inversely affected by chromium content for specific process time. Moreover, nitrided case depth does not vary directly with the inverse of the chromium concentration w1,9x. Ion nitriding is a diffusion-controlled process and selecting different time and Cr content combinations can produce certain case depths. Therefore, the prediction of case depth before nitriding treatment is very important and useful particularly from the perspective of manufacturers in the practice and this can be achieved by selecting appropriate process parameters. Although many works had been carried out on nitriding response of chromium-alloyed steels, limited study on ion nitriding behaviour of chromium-alloyed steels as a function of Cr content w1,9,12,13x.
0261-3069/03/$ - see front matter 䊚 2003 Elsevier Science Ltd. All rights reserved. doi:10.1016/S0261-3069(03)00002-5
204
K. Genel / Materials and Design 24 (2003) 203–207
Fig. 1. The structure of three layered neural network.
Artificial neural networks (ANNs), which are relatively new modelling techniques, have shown remarkable performance when used to model complex linear and non-linear relationships. ANNs offer a fundamentally different approach to material modelling and material processing control techniques than statistical methods, and recently the interest to the ANNs modelling in the fields of the materials science and physical metallurgy has increased w14–18x. The previous studies showed that ANN modelling is superior to conventional regression method w18,19x. One of the main advantages of this technique is that, there is no need to make a prior assumption about concerned material behaviour. The objectives of present study is to investigate possibility of predicting case depth in Fe–Cr steels as a function of Cr content and process time for a given nitriding temperature by using ANN and obtain useful case depth data from neural network modelling. Using data obtained from result of ANN modelling and experimental data, a contour plot for ion nitrided case depth related to chromium content and process time is constructed. 2. Neural networks approach ANNs are computational models, which simulate the function of biological network, compose of neurons. The system has three layers of neurons: input layer, a hidden layer and an output layer. The neurons or units of the network are connected by the weights. Input layer consist of all the input factors, information from the input layer is then processed through one hidden layer, following output vector is computed in the final (output) layer, structure of the network used in this study is shown schematically in Fig. 1. In this study, a neural network approach is applied to determine case depth for various pair of Cr content and process time. Currently, there are various training algorithms available. A backpropagation (BP) algorithm was used to train a multilayered feed forward neural network, which is simple, reliable and most commonly utilised. Detailed description of the mathematical formulation of the BP algorithm
has been covered in literature extensively w20,21x. The important aspects of the BP related to this study will be described briefly. The BP training algorithm is an iterative gradient algorithm, designed to minimise the mean square error between the predicted output and the actual output. During training, an ANN is presented with the data of thousands of times, which is referred to as cycles. After each cycle, the error between the ANN output (predicted) and actual (or desired) values are propagated backward to adjust the weight in a manner mathematically guaranteed to converge w20x. Training is the act of continuously adjusting their connection weights until they reach unique values that allow the network to produce outputs that are close enough to the actual desired outputs. The accuracy of the developed model, therefore, depends on these weights. Once optimum weights are reached, the weights and biased values encode the network’s state of knowledge. The neural network was written in the Cqq programming language and all tests were performed on a personal computer. The activation and error function used in this study is a logistic sigmoid function and standard sum-of-squared error function, respectively. Neural network requires that the range of the both input values and output values should be between 0 and 1 due to restriction of sigmoid function, consequently, the data must be unified. H9s(HyHmin)y(Hmax yHmin ) is widely employed method in unification. Where Hmax and Hmin, respectively, indicate the largest and smallest value of H, and H9 the unified value of the corresponding H. In order to facilitate the comparisons between predicted values for different network parameters and measured or actual values, there is a need error evaluation. Mean relative error (MRE), which also considering effect of actual value in error term, is calculated according to following the expression MREs
1 n 100ZaiypiZ n8 ai is1
(1)
where ai is actual or measured value, pi the predicted (output) value and n is the number of data.
K. Genel / Materials and Design 24 (2003) 203–207
205
Table 1 Data used in training and test set including MRE, and network predicting values Time (h)
1 2.5 3 5.5 8 MRE (%)
Chromium content (wt.%) Experimental data w1x
Predicted data
Training set
Test set
7.5
12.5
1.5
17.5
5
20
10
150 200 234 275 288 2.2
117 138 150 162 175 2.2
137 170 210 232 270 3.4
140.9 186.0 216.3 254.1 276.2
130.4 165.3 184.4 213.5 234.9
124.7 155.9 170.0 194.7 213.9
120.7 147.4 158.9 177.9 194.7
2.1. ANN modelling The case depth data of 5, 10 and 20% Cr alloyed Fe–Cr steels, ion nitrided in gas mixture of 20% N2 and 80% H2 at 500 8C for 1, 2.5, 3, 5.5 and 8 h were considered from Alves et al. w1x. The network was trained using case depth values for 5 and 20% Cr–Fe steels for 1–8 ion nitriding time, and nitrided case depth values for 10% Cr–Fe steel were used in the test set, seen Table 1. Ion nitriding time (t) and chromium content (WCr) are input factors and predicted depth (Pd) vs. Cr content for various ion nitriding times are output factor. In order to decide the optimum structure of neural network, the rate of error convergence was checked by changing the number of hidden neuron and by adjusting the learning rate and momentum coefficient.
Fig. 2. The effect of training cycles on the network performance.
network prediction for the test set (unseen condition) with corresponding experimental data are given in Fig. 3b. It can be obviously concluded that the predictions are in good agreement with the experimental data. It is also possible to say that errors are in the range of deviation resulted from case depth measurements. Fig. 4 represents ion nitrided case depth as a function of
3. Results and discussion A learning rate of 0.7, with momentum coefficient of 0.9 resulted in the fastest convergence. The five hidden units were concluded to be most efficient design. The networks were trained up to 4000 for prediction of the thickness of ion nitrided layer. Fig. 2 presents mean squared error vs. number of cycles for training and test set. It is clearly noticed from this figure that further training cycles has no considerable effect on both network performance. The errors of the both sets decrease dramatically at the beginning of the training cycles, and then it gradually tend to go through minimum up to threshold cycles as mentioned above. The predicted data of training set together with corresponding experimental data are given in Fig. 3a. When the results of training set are considered, it was noticed from result of ANN that the experimental and predicted values are very close to each other with a MRE of 2.2% (Table 1). However, the main quality indicator of a neural network is its generalization ability, its ability to predict accurately the output of unseen test data. The
Fig. 3. The predicted ion nitrided depth from neural network vs. experimental depth for the training set (a) test set (b).
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K. Genel / Materials and Design 24 (2003) 203–207
Fig. 6. Comparisons of relative errors in each data used in test set and MRE for the regression and neural network approach.
Fig. 4. Ion nitrided case depth as a function of chromium content for various process time.
chromium content for various process times. When the trends of each curves varied with chromium contents are examined, it is noticed that decreases in slope of each curve with decreasing alloying content, especially longer process time, and results of neural network are in good agreement, as expected from the metallurgical point of view. It is known that case depth of ion nitrided steels increased with squared root of time w2,6,22x. Case progresses of ion nitrided Cr-alloyed steels vs. squared root of time are given in Fig. 5, including neural network prediction. It can also be concluded that predicted case depth values described by a power relationship for the process time, similar to trend of plotting of experimental values with time. An evaluation of the ANN capabilities can be made by comparisons with regression analysis. Empirical relationship as a Lorentzian type equation between ion nitrided case depth and chromium content as well as process time, has been developed by regression analysis.
Fig. 5. Ion nitrided case depth vs. process time for various chromium content, including network prediction.
Data, which are used in the training set, are employed in construction of empirical equation that is given Eq. (2). The correlation coefficient (R 2) and standard error of estimate were determined as 0.965 and 14.78, respectively. 341.9 ds w B W y9.75 E2zw B ty6.94 E2z F |x1qC F| x1qC Cr D G ~y D 11 7 G~ y
(2)
where d is ion nitrided case depth (mm), WCr is chromium content (wt.) and t is process time (h). Fig. 6 shows that comparisons of relative error in each datum for the data of 10% Cr alloy that is used in the test set, together with MREs for both the regression and neural network approach. It can be clearly seen that the neural network is much better than regression method and MRE value is 10 times lower than that of it. Considering experimental and neural network predicted data, plotted contour diagram enables the selection
Fig. 7. Ion nitrided depth as a functions of Cr (wt.%) and ion nitriding time for Fe–Cr alloy (500 8C, 20% N2 –80% H2).
K. Genel / Materials and Design 24 (2003) 203–207
of process time for specific Cr content in industrial application, see Fig. 7. 4. Conclusion In the present study, the effectiveness of ANN for prediction of case depth progressive in ion nitrided Fe– Cr alloys was demonstrated. ANN results show a good agreement with experimental data, and neural network modelling provides more useful data from relatively small experimental databases. It means that considerable saving cost and time. However, neuron number in hidden layer is not suggested as theoretical and it has great effect on prediction results, which constitute main limitation of networks. It is also concluded that ANN is a successful analytical tool that if properly used. References w1x Alves JC, Rodrigues JA, Martinelli AE. Growth of nitrided layers on Fe–Cr alloys. Mater Sci Eng Part A 2000;279:10 – 15. w2x Genel K, Demirkol M, Capa M. Effect of ion nitriding on fatigue behaviour of AISI 4140 steel. Mater Sci Eng Part A 2000;279:207 –216. w3x Bell T, Loh NL. The fatigue characteristics of plasma nitrided three pct Cr–Mo steel. J Heat Treat 1982;2(3):232 –237. w4x Genel K, Demirkol M, Gulmez ¨ T. Corrosion fatigue behaviour of ion nitrided AISI 4140 steel. Mater Sci Eng Part A 2000;288:91 –100. w5x Devi MU, Chakraborty TK, Mohanty ON. Wear behaviour of plasma nitrided tool steels. Surf Coat Technol 1999;116:212 – 221. w6x Robino CV, Inal OT. Ion nitriding behaviour of several low alloy steels. Mater Sci Eng 1983;59:79 –90. w7x Edenhofer B. The ion nitriding process—thermochemical treatment of steel and cast iron materials. Metallurgist Mater Technol 1976;8(8):421 –426.
207
w8x Mittemeijer EJ, Vogels ABP, Van Der Schaaf PJ. Morphology and lattice distortions of nitrided iron and iron–chromium alloys and steels. J Mater Sci 1980;15:3129 –3140. w9x Granito N, Kuwahara H, Aizawa T. Normal and abnormal microstructure of plasma nitrided Fe–Cr alloys. J Mater Sci 2002;37(4):835 –844. w10x Van Wiggen PC, Rozendaal HCF, Mittemeijer EJ. The nitriding behaviour of iron–chromium–carbon alloys. J Mater Sci 1985;20:4561 –4582. w11x Mittemeijer EJ. Fatigue of case-hardened role of residual macro- and microstress. J Heat Treat 1983;3(2):114 –119. w12x Mridha S, Jack DH. Characterization of nitrided 3% chromium steel. Metal Sci 1982;16:398 –404. w13x Hekker PM, Rozendaal HCF, Mittemeijer EJ. Excess nitrogen and discontinuous precipitation in nitrided iron–chromium alloys. J Mater Sci 1985;20:718 –729. w14x Kang JY, Song JH. Neural network applications in determining the fatigue crack opening load. Int J Fatigue 1998;20:57 –69. w15x Malinova T, Malinov S, Pantev N. Simulation of microhardness profiles for nitrocarburized surface layers by artificial neural network. Surf Coat Technol 2001;135:258 –267. w16x Zeng Q, Zu J, Zhang L, Dai G. Designing expert system with artificial neural networks for in situ toughened Si3N4. Mater Des 2002;23:287 –290. w17x Yescas MA, Bhadieshia HKDH, MacKay DJ. Estimation of the amount of retained austenite in austempered ductile irons using neural networks. Mater Sci Eng Part A 2001;311:162 – 173. w18x Jain RK, Jain VK, Kalra PK. Modelling of abrasive flow machining process: a neural network approach. Wear 1999;231:242 –248. w19x Genel K, Ozbek I, Kurt A, Bindal C. Boriding response of AISI W1 steel and use of artificial neural network for prediction of borided layer properties. 2002;160(1):38–43. w20x Haykin S. Neural networks—a comprehensive foundation. New York: Macmillan, 1994. p. 138 –153. w21x Rumelhart DE, Hinton G, Williams R. In: Rumelhart DE, et al, editor. Parallel distributed processing, vol. I. Cambridge: MIT press, 1986. p. 318 –362. w22x Seybolt AU. Some observations on the metallurgy of ion nitriding. Trans Met Soc AIME 1969;245(4):769 –778.
Use of artificial neural network for prediction of ion nitrided case depth in Fe–Cr alloys Kenan Genel* Mechanical Engineering Department, Sakarya University, Esentepe 54187, Sakarya, Turkey Received 29 August 2002; received in revised form 24 October 2002; accepted 16 December 2002
Abstract In this work, a simple artificial neural network (ANN) model using back-propagation training algorithm for ion nitriding behaviour of Fe–Cr alloys was established. The case depth data were extracted from experimental data and used in the formation of training sets of ANN in order to predict case depth of ion nitrided Fe–Cr alloys, 2.5% Cr intervals for 5–20% Cr. The modelling results confirm the feasibility of this approach and show good agreement with experimental data by Alves et al. (Mater Sci Eng 2002; 279A: 10–15) with high accuracy. A contour diagram as a function of Cr (wt.%) and ion nitriding time for Fe– Cr alloy was constructed for industrial application. It is concluded that a considerable saving in terms of cost and time could be obtained from using the trained ANN model and, it provides more useful data from relatively small experimental databases. 䊚 2003 Elsevier Science Ltd. All rights reserved. Keywords: Ion nitriding; Artificial neural networks; Fe–Cr alloys
1. Introduction Plasma or ion nitriding, owing to a number of advantages over conventional nitriding methods, have found increasing applications in industry. It is well known that ion nitriding surface treatment, which is ferritic thermochemical methods, is widely used to improve fatigue, corrosion fatigue and tribological properties of Fe-based alloys w1–6x. During ion nitriding, reaction takes place both at surface of the metal and below the surface due to diffusion of nitrogen atoms from the surface to interior region. Hence, thin iron nitride layer (compound layer) and diffusion zone form on the surface and subsurface region of steel, respectively. Both surface hardness and nitrided case depth are mainly related to chemical composition of metal for given process parameters such as time, temperature and composition of gas mixture w6,7x. Chromium is often used an alloying element due to its affinity for nitrogen. Previous studies, which were performed on nitriding of Fe–Cr alloys, indicated that chromium nitrides (CrN and Cr2N), in contrast with the iron nitrides, were precipitate at the nitriding temperature *Tel.: q90-264-346-0353x319; fax: q90-264-346-0351. E-mail address: [email protected] (K. Genel).
w8x. Micro-structural differences were observed in nitrided case depending on chromium content and ion nitriding temperature w9x. On nitriding, chromium-alloyed ferritic matrix submicroscopical, coherent, chromium nitride particles develop giving rise to high hardness of the order of 1000 HV w10x. The expansion of the nitrided case is counteracted by core region. As a result, compressive residual stress develop near the surface, while a tensile stress evolves in the core. Case depth has a favourable effect on magnitude of beneficial compressive residual stress as well as fatigue behaviour of nitrided steels w2,8,10,11x. The progress of case depth inversely affected by chromium content for specific process time. Moreover, nitrided case depth does not vary directly with the inverse of the chromium concentration w1,9x. Ion nitriding is a diffusion-controlled process and selecting different time and Cr content combinations can produce certain case depths. Therefore, the prediction of case depth before nitriding treatment is very important and useful particularly from the perspective of manufacturers in the practice and this can be achieved by selecting appropriate process parameters. Although many works had been carried out on nitriding response of chromium-alloyed steels, limited study on ion nitriding behaviour of chromium-alloyed steels as a function of Cr content w1,9,12,13x.
0261-3069/03/$ - see front matter 䊚 2003 Elsevier Science Ltd. All rights reserved. doi:10.1016/S0261-3069(03)00002-5
204
K. Genel / Materials and Design 24 (2003) 203–207
Fig. 1. The structure of three layered neural network.
Artificial neural networks (ANNs), which are relatively new modelling techniques, have shown remarkable performance when used to model complex linear and non-linear relationships. ANNs offer a fundamentally different approach to material modelling and material processing control techniques than statistical methods, and recently the interest to the ANNs modelling in the fields of the materials science and physical metallurgy has increased w14–18x. The previous studies showed that ANN modelling is superior to conventional regression method w18,19x. One of the main advantages of this technique is that, there is no need to make a prior assumption about concerned material behaviour. The objectives of present study is to investigate possibility of predicting case depth in Fe–Cr steels as a function of Cr content and process time for a given nitriding temperature by using ANN and obtain useful case depth data from neural network modelling. Using data obtained from result of ANN modelling and experimental data, a contour plot for ion nitrided case depth related to chromium content and process time is constructed. 2. Neural networks approach ANNs are computational models, which simulate the function of biological network, compose of neurons. The system has three layers of neurons: input layer, a hidden layer and an output layer. The neurons or units of the network are connected by the weights. Input layer consist of all the input factors, information from the input layer is then processed through one hidden layer, following output vector is computed in the final (output) layer, structure of the network used in this study is shown schematically in Fig. 1. In this study, a neural network approach is applied to determine case depth for various pair of Cr content and process time. Currently, there are various training algorithms available. A backpropagation (BP) algorithm was used to train a multilayered feed forward neural network, which is simple, reliable and most commonly utilised. Detailed description of the mathematical formulation of the BP algorithm
has been covered in literature extensively w20,21x. The important aspects of the BP related to this study will be described briefly. The BP training algorithm is an iterative gradient algorithm, designed to minimise the mean square error between the predicted output and the actual output. During training, an ANN is presented with the data of thousands of times, which is referred to as cycles. After each cycle, the error between the ANN output (predicted) and actual (or desired) values are propagated backward to adjust the weight in a manner mathematically guaranteed to converge w20x. Training is the act of continuously adjusting their connection weights until they reach unique values that allow the network to produce outputs that are close enough to the actual desired outputs. The accuracy of the developed model, therefore, depends on these weights. Once optimum weights are reached, the weights and biased values encode the network’s state of knowledge. The neural network was written in the Cqq programming language and all tests were performed on a personal computer. The activation and error function used in this study is a logistic sigmoid function and standard sum-of-squared error function, respectively. Neural network requires that the range of the both input values and output values should be between 0 and 1 due to restriction of sigmoid function, consequently, the data must be unified. H9s(HyHmin)y(Hmax yHmin ) is widely employed method in unification. Where Hmax and Hmin, respectively, indicate the largest and smallest value of H, and H9 the unified value of the corresponding H. In order to facilitate the comparisons between predicted values for different network parameters and measured or actual values, there is a need error evaluation. Mean relative error (MRE), which also considering effect of actual value in error term, is calculated according to following the expression MREs
1 n 100ZaiypiZ n8 ai is1
(1)
where ai is actual or measured value, pi the predicted (output) value and n is the number of data.
K. Genel / Materials and Design 24 (2003) 203–207
205
Table 1 Data used in training and test set including MRE, and network predicting values Time (h)
1 2.5 3 5.5 8 MRE (%)
Chromium content (wt.%) Experimental data w1x
Predicted data
Training set
Test set
7.5
12.5
1.5
17.5
5
20
10
150 200 234 275 288 2.2
117 138 150 162 175 2.2
137 170 210 232 270 3.4
140.9 186.0 216.3 254.1 276.2
130.4 165.3 184.4 213.5 234.9
124.7 155.9 170.0 194.7 213.9
120.7 147.4 158.9 177.9 194.7
2.1. ANN modelling The case depth data of 5, 10 and 20% Cr alloyed Fe–Cr steels, ion nitrided in gas mixture of 20% N2 and 80% H2 at 500 8C for 1, 2.5, 3, 5.5 and 8 h were considered from Alves et al. w1x. The network was trained using case depth values for 5 and 20% Cr–Fe steels for 1–8 ion nitriding time, and nitrided case depth values for 10% Cr–Fe steel were used in the test set, seen Table 1. Ion nitriding time (t) and chromium content (WCr) are input factors and predicted depth (Pd) vs. Cr content for various ion nitriding times are output factor. In order to decide the optimum structure of neural network, the rate of error convergence was checked by changing the number of hidden neuron and by adjusting the learning rate and momentum coefficient.
Fig. 2. The effect of training cycles on the network performance.
network prediction for the test set (unseen condition) with corresponding experimental data are given in Fig. 3b. It can be obviously concluded that the predictions are in good agreement with the experimental data. It is also possible to say that errors are in the range of deviation resulted from case depth measurements. Fig. 4 represents ion nitrided case depth as a function of
3. Results and discussion A learning rate of 0.7, with momentum coefficient of 0.9 resulted in the fastest convergence. The five hidden units were concluded to be most efficient design. The networks were trained up to 4000 for prediction of the thickness of ion nitrided layer. Fig. 2 presents mean squared error vs. number of cycles for training and test set. It is clearly noticed from this figure that further training cycles has no considerable effect on both network performance. The errors of the both sets decrease dramatically at the beginning of the training cycles, and then it gradually tend to go through minimum up to threshold cycles as mentioned above. The predicted data of training set together with corresponding experimental data are given in Fig. 3a. When the results of training set are considered, it was noticed from result of ANN that the experimental and predicted values are very close to each other with a MRE of 2.2% (Table 1). However, the main quality indicator of a neural network is its generalization ability, its ability to predict accurately the output of unseen test data. The
Fig. 3. The predicted ion nitrided depth from neural network vs. experimental depth for the training set (a) test set (b).
206
K. Genel / Materials and Design 24 (2003) 203–207
Fig. 6. Comparisons of relative errors in each data used in test set and MRE for the regression and neural network approach.
Fig. 4. Ion nitrided case depth as a function of chromium content for various process time.
chromium content for various process times. When the trends of each curves varied with chromium contents are examined, it is noticed that decreases in slope of each curve with decreasing alloying content, especially longer process time, and results of neural network are in good agreement, as expected from the metallurgical point of view. It is known that case depth of ion nitrided steels increased with squared root of time w2,6,22x. Case progresses of ion nitrided Cr-alloyed steels vs. squared root of time are given in Fig. 5, including neural network prediction. It can also be concluded that predicted case depth values described by a power relationship for the process time, similar to trend of plotting of experimental values with time. An evaluation of the ANN capabilities can be made by comparisons with regression analysis. Empirical relationship as a Lorentzian type equation between ion nitrided case depth and chromium content as well as process time, has been developed by regression analysis.
Fig. 5. Ion nitrided case depth vs. process time for various chromium content, including network prediction.
Data, which are used in the training set, are employed in construction of empirical equation that is given Eq. (2). The correlation coefficient (R 2) and standard error of estimate were determined as 0.965 and 14.78, respectively. 341.9 ds w B W y9.75 E2zw B ty6.94 E2z F |x1qC F| x1qC Cr D G ~y D 11 7 G~ y
(2)
where d is ion nitrided case depth (mm), WCr is chromium content (wt.) and t is process time (h). Fig. 6 shows that comparisons of relative error in each datum for the data of 10% Cr alloy that is used in the test set, together with MREs for both the regression and neural network approach. It can be clearly seen that the neural network is much better than regression method and MRE value is 10 times lower than that of it. Considering experimental and neural network predicted data, plotted contour diagram enables the selection
Fig. 7. Ion nitrided depth as a functions of Cr (wt.%) and ion nitriding time for Fe–Cr alloy (500 8C, 20% N2 –80% H2).
K. Genel / Materials and Design 24 (2003) 203–207
of process time for specific Cr content in industrial application, see Fig. 7. 4. Conclusion In the present study, the effectiveness of ANN for prediction of case depth progressive in ion nitrided Fe– Cr alloys was demonstrated. ANN results show a good agreement with experimental data, and neural network modelling provides more useful data from relatively small experimental databases. It means that considerable saving cost and time. However, neuron number in hidden layer is not suggested as theoretical and it has great effect on prediction results, which constitute main limitation of networks. It is also concluded that ANN is a successful analytical tool that if properly used. References w1x Alves JC, Rodrigues JA, Martinelli AE. Growth of nitrided layers on Fe–Cr alloys. Mater Sci Eng Part A 2000;279:10 – 15. w2x Genel K, Demirkol M, Capa M. Effect of ion nitriding on fatigue behaviour of AISI 4140 steel. Mater Sci Eng Part A 2000;279:207 –216. w3x Bell T, Loh NL. The fatigue characteristics of plasma nitrided three pct Cr–Mo steel. J Heat Treat 1982;2(3):232 –237. w4x Genel K, Demirkol M, Gulmez ¨ T. Corrosion fatigue behaviour of ion nitrided AISI 4140 steel. Mater Sci Eng Part A 2000;288:91 –100. w5x Devi MU, Chakraborty TK, Mohanty ON. Wear behaviour of plasma nitrided tool steels. Surf Coat Technol 1999;116:212 – 221. w6x Robino CV, Inal OT. Ion nitriding behaviour of several low alloy steels. Mater Sci Eng 1983;59:79 –90. w7x Edenhofer B. The ion nitriding process—thermochemical treatment of steel and cast iron materials. Metallurgist Mater Technol 1976;8(8):421 –426.
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