Commercial steels are nowadays sophisticated alloys formed by a large number of alloying elements. The martensite start ( Ms) temperature of such steels is of vital engineering importance, and its prediction through models allows us to enhance the design and development of industrial products. In the present work, Ms temperature dependence on chemical composition has been examined by neural network analysis. Neural networks represent powerful methods of non-linear regression modelling. The network is a mathematical function which is fitted to experimental data. The influence of alloying elements such as C, Mn, Si, Cr, Ni, Mo, V, Co, W, Al, Nb, Cu, B and N on Ms temperature was analysed. Finally, a new empirical equation for Ms temperature was derived based on the neural network results.Peer Reviewe

Friede, M. E.

Gütle, D.

Hornef, Mathias

Kveberg, L.

Leibelt, S.

Rohe, C.

Rutkowski, E.

Steinle, A.

Vaage, J. T.

Weigert, A.

S. Leibelt

M.E. Friede

C. Rohe

D. Gütle

E. Rutkowski

A. Weigert

L. Kveberg

J.T. Vaage

M.W. Hornef

A. Steinle

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Mucosal Immunology

Dedicated immunosensing of the mouse intestinal epithelium facilitated by a pair of genetically coupled lectin-like receptors

Capdevila, Carlos

García Caballero, Francisca

García de Andrés, Carlos

Digital.CSIC

Prediction of Martensite Start Temperature by Neural Network Analysis  C. Capdevila, F.G. Caballero and C. García De Andrés  Departament of Physical Metallurgy Centro Nacional de Investigaciones Metalúrgicas (CENIM - CSIC) Avda. Gregorio del Amo, 8. 28040 Madrid. SPAIN Tel. 34 915538900; Fax. 34 915347425; e-mail: ccm@cenim.csic.es  2             Abstract. Commercial steels are nowadays sophisticated alloys formed by a large number of alloying elements. The martensite start (Ms) temperature of such steels is of vital engineering importance, and its prediction through models allows us to enhance the design and development of industrial products. In the present work, Ms temperature dependence on chemical composition has been examined by neural network analysis. Neural networks represent powerful methods of non-linear regression modelling. The network is a mathematical function which is fitted to experimental data. The influence of alloying elements such as C, Mn, Si, Cr, Ni, Mo, V, Co, W, Al, Nb, Cu, B and N on Ms temperature was analysed. Finally, a new empirical equation for Ms temperature was derived based on the neural network results.  1. INTRODUCTION  The martensite start temperature, Ms, is a design parameter of vital importance for deciding on many process conditions, for example, the exact welding conditions necessary to avoid cracking. Hence great efforts have been made in predicting the Ms of steels. Obviously, chemical composition of steel is the main factor affecting its Ms, although the austenitising state, external stresses and stored deformation energy may sometimes play an important role as well. Martensite start temperatures are usually relatively easy to calculate as long as the steels have a low alloy content [1-6]. Even though empirical equations exists for high alloy steels, they are not sufficiently general and are known to provide inaccurate answers for the new steels which contain different alloying elements, or their compositional range are out of bounds of those used to formulate the equations.  It has been reported that additions of Mo, Co and W to the chemical composition increase the strength of steel with molybdenum (Mo2C) and tungsten-cobalt (WC-Co) cemented carbides [7-9]. These new developed steels are materials that are widely used in applications where a good combination between abrasion resistance and corrosion resistance is required (tool steels) [10]. Likewise, new copper-bearing low carbon steels have been developed. Additions of copper increase the strength of steel by precipitation of fine copper precipitates during ageing instead of carbide particles, which considerably improve mechanical properties. Moreover, in the last decade the use of microalloying elements such as V and Nb has increase spectacularly. Bearing in mind that Ms temperature is a parameter of considerable importance for engineering steels, to find out how Co, W, Cu, Mo, V or Nb can affect the Ms temperature is an important issue to be investigated. The aim of this work is to develop an artificial neural network model to predict the Ms temperature of steels and interpret the influence of the chemical composition.   2. BRIEF DESCRIPTION OF THE NEURAL NETWORKS  The network consisted of 14 input nodes (Table 1), a number of hidden nodes, and an output node representing the Ms temperature. Meanwhile in a traditional linear regression each input is multiplied by a random weight wi and the products are summed together with a constant θ to give the output node ∑ +=iii xwy θ , in neural network analysis the input data xj are also multiplied by weights ()1(jw ), but the sum of all these products forms the argument of a hyperbolic tangent. The output is linear combinations of these hyperbolic tangent functions:   3 )2()2()1()1(θθ+=+= ∑hwyxwtanhh jjjj      (1) where w(2) is a weight and θ(2) another constant. The weights were determined by training the network and the details are described elsewhere [11]. The model developed in this work consists of 7 hidden nodes and it is developed under a Bayesian framework [12], which allows the calculation of error bars representing the uncertainty in the fitting parameters. The method recognises that there are many functions which can be fitted or extrapolated into uncertain regions of the input space, without unduly compromising the fit in adjacent regions which are rich in accurate data. Instead of calculating a unique set of weights, a probability distribution of sets of weights is used to define the fitting uncertainty. The error bars therefore become larger when data are spare or locally noisy.  Table 1. Variables that influence Ms temperature. The values are in wt.-%. S.D. is the standard deviation. P.T. is the number of points in the trainning.  C Mn Si Cr Ni Mo V Co Al W Cu Nb B N Min. 0.001 0 0 0 0 0 0 0 0 0 0 0 0 0 Max. 1.62 3.76 3.40 17.90 27.20 5.10 4.55 30.00 1.10 12.90 0.98 0.23 0.01 0.06 Mean 0.36 0.889 0.343 1.182 1.379 0.298 0.073 0.474 0.012 0.111 0.050 0.002 0.001 0.002 S.D 0.20 0.53 0.41 1.44 3.81 0.57 0.25 2.78 0.08 0.87 0.10 0.01 0.001 0.001 PT 360 339 260 287 235 254 165 50 90 20 112 28 65 25  3. USE OF THE MODEL  The main advantage of the neural network model as compared with other empirical models is the ability of analysing separately the influence on Ms temperature of each one of the alloying elements. In this sense, the role of alloying elements such as Cu, W, Co, Mo, Nb and V on Ms has been analysed in this section and the results are presented in Fig. 1. Figure 1(a) shows that copper addition has not influence on Ms although the increase in error bars indicate a lack of data for high copper concentrations. On the other hand, Fig. 1(b) shows that tungsten increases the value of Ms. Likewise, Fig. 1(c) shows that cobalt concentrations lower than 3 wt.-% does not affect Ms temperature. However, the higher cobalt concentration up to 30 wt.-%, the higher Ms temperature. From Figs. 1(d), 1(e) and 1(f) it is shown the evolution of Ms as the content of strong carbide-formers such as Mo, Nb and V increases. It is clear from these figures that the strongest carbide former, the highest increases on Ms temperature. This behaviour could be attributed to the influence of alloying elements on the activity of carbon in the solid solution. Keeping this in mind, we can expect that interactions between carbon and carbide-forming elements tend to weaken the role of carbon, and give rise to an increase in Ms temperature.  Table 2. Fitting parameters estimated by non linear regression analysis to a byaxyM os ++=  equation type  y0 a b R Mn 768.40 -301.49 -30.61 0.99812820 Si 769.84 -311.31 -14.46 0.99867290 Cr 759.55 -290.79 -8.99 0.99832584 Ni 759.22 -299.09 -16.63 0.99939313 Mo 769.85 -306.08 2.37 0.99821097 V 761.12 -286.77 5.35 0.99712654 Co 738.63 -281.20 8.58 0.98232872 Al 797.36 -328.45 40.40 0.99283463 W 770.85 -312.88 7.42 0.99876925 Cu 777.31 -318.52 -11.34 0.99740703  4 Nb 772.70 -322.66 510.40 0.99864636   Figure 1. Effect of (a) Cu, (b) W, (c) Co, (d) Mo, (e)V, and (e) Nb on Ms temperature. 3005007009000 0,05 0,1 0,15 0,2Nb, wt.-%Ms, KC=0,1 w t.-%C=0,4 w t.-%C=0,8 w t.-%3005007009000 1 2 3 4 5V, wt.-%Ms, KC=0,1 w t.-%C=0,4 w t.-%C=0,8 w t.-%3005007009000,0 1,0 2,0 3,0 4,0 5,0Mo, wt.-%Ms, KC=0,1 w t.-%C=0,4 w t.-%C=0,8 w t.-%3005007009000,00 0,25 0,50 0,75 1,00Cu, wt.-%Ms, KC=0,1 w t.-%C=0,4 w t.-%C=0,8 w t.-%300500700900110013000 10 20 30Co, wt.-%Ms, KC=0,1 w t.-%C=0,4 w t.-%C=0,8 w t.-%3005007009000,0 3,0 6,0 9,0 12,0W, wt.-%Ms, KC=0,1 w t.-%C=0,4 w t.-%C=0,8 w t.-%(a) (b) (c) (d) (e) (f)  5  With the aim of finding an empirical equation for Ms temperature upon the steel chemistry, the neural network predictions are plotted by pairs of elements (C-Mn, C-Ni, etc). For instance, Fig. 2 shows the evolution of Ms with carbon and chromium concentrations ranging from 0.001 to 0.9 wt.%, and from 0 to 17 wt.-%, respectively. These values are fitted to a plane regression equation ( byaxyM os ++= , where x corresponds to carbon concentration values in wt.-%, and y corresponds to the alloying element) by non-linear regression analysis. The regression coefficients a and b for the different alloying elements are listed in Table 2. R in Table 2 is the correlation factor between the neural network data and the parameters of the byaxyM os ++=  fitting equation. According to that procedure, the relative effect of alloying elements is indicated in the following empirical relationship for predicting the Ms temperature of steels:   NbCuWAlCoVMoNiCrSiMnCswwwwwwwwwwwwKM4.5103.114.74.4058.83.54.26.169.85.146.304.3057.767)(+−+++++−−−−−=         (2)  The effect of carbon has been considered as the mean value of the ‘a’ values listed in Table 2. Likewise, it should be notice the large value corresponding to Nb, which is consistent with the stronger carbide former, highest Ms temperature.   Figure 2. Effect of binary interaction C-Cr on Ms temperature. Figure 3. Comparison between experimental and estimated Ms values for the steels listed in Table 3.  Figure 3 shows a comparison between the results predicted by equation (2) and the experimental Ms temperature in six very different steels whose actual compositions are listed in Table 3. The steels listed in Table 3 have not been used in the fitting procedure. It could be concluded from this figure that the neural network model provides an excellent accuracy on Ms temperature predictions. All of these steels are used for commercial purposes and, therefore, the accurate determination of Ms temperature is very important from an industrial point of view.  Table 3. Chemical composition of the six steels analysed.  C Mn Si Cr Ni W Co Mo V Al Cu Nb 373473573673773373 473 573 673 773Measured Ms, KCalculated Ms, K S1S2S5S6S4S3 6 S1 0.45 0.4 0.32 13.0 0.38 0 0 0 0 0 0 0 S2 0.8 3.52 1.67 1.1 0 0.99 1.44 0.24 0 0.01 0 0 S3 0.07 1.5 0.37 0.039 0.49 0 0 0.021 0.004 0.045 0.039 0.03 S4 0.31 1.22 0.253 0.138 0.098 0 0 0.03 0.004 0 0 0 S5 0.09 1.03 0.16 9.1 0.99 0 0 0.99 0.19 0 0 0.04 S6 0.09 0.99 0.18 8.94 0 0.98 1.87 0.96 0.18 1.87 0 0.05  4. CONCLUSIONS  A neural network method based within a Bayesian framework has been used to rationalise an enormous quantity of published experimental data on Ms temperature of steels. It is now possible, therefore, to predict the Ms temperature as a function of the chemical composition. The influence of the alloying elements is considered by means of the C-X pair interactions. The results predicted by the neural network model were compared with the experimental Ms temperature of six very different commercial steels. It is concluded that an excellent agreement between experimental and predicted Ms temperature was found.  Acknowledgements  The authors acknowledge financial support from Consejería de Educación y Cultura de la Comunidad Autónoma de Madrid (CAM 07N-0065-1998). C. Capdevila would like to thank the Consejería de Educación. D.G. de Investigación de la Comunidad Autónoma de Madrid (CAM) for the financial support in the form of a Postdoctoral Research Grant.  References  1 Capella, L.A., Metal. Prog. 46 (1944) 108-118. 2 Wang, J., van der Wolk, P.J.,Van der Zwaag, S., Mater. Trans. JIM 41 (2000) 761-768. 3 Grange, R.A., Stewart, H.M., Trans. AIME 167 (1946) 467-494. 4 Payson, P., Savage, H., Trans ASM 33 (1944) 261-280. 5 Kung, C.Y., Rayment, J.J., Hardenability Concepts with Applications to Steels, (TMS-AIME, Warrendale, PA, 1978), pp. 229-248. 6 Andrews, K.W., JISI 203 (1965) 721-727. 7 Gustafson, A., Agren, J, ISIJ Int. 41 (2001) 356-360 8 Abe, F., Mater. Sci. Engineer. A A319-321 (2001) 770-773. 9 Lisovskii, A.F., Powder Metall. Met. Ceram. 39 (2001) 428-433. 10 Gant, A.J., Gee, M.G., Wear 250 (2001) 908-915. 11 Makay, D.J.C., Neural Comput., 4 (1992) 698-705. 12 Makay, D.J.C., Neural Comput., 4 (1992) 415-422. 

Prediction of martensite start temperature by neural network analysis

Hornef, Mathias Walter

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Prediction of martensite start temperature by neural network analysis

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Digital.CSIC

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