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Artificial neural network modeling to evaluate and predict the deformation behavior of stainless steel type AISI 304L during hot torsion
Applied Soft Computing 9 (2009) 237–244
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Applied Soft Computing journal homepage: www.elsevier.com/locate/asoc
Artificial neural network modeling to evaluate and predict the deformation behavior of stainless steel type AISI 304L during hot torsion Sumantra Mandal a,*, P.V. Sivaprasad a, S. Venugopal a, K.P.N. Murthy b a b
Metallurgy and Materials Group, Indira Gandhi Centre for Atomic Research, Kalpakkam 603102, Tamil Nadu, India School of Physics, University of Hyderabad, Hyderabad 500046, Andhra Pradesh, India
A R T I C L E I N F O
A B S T R A C T
Article history: Received 24 January 2006 Received in revised form 19 December 2007 Accepted 7 March 2008 Available online 23 April 2008
The deformation behavior of type 304L stainless steel during hot torsion is investigated using artificial neural network (ANN). Torsion tests in the temperature range of 600–1200 8C and in the (maximum surface) strain rate range of 0.1–100 s1 were carried out. These experiments provided the required data for training the neural network and for subsequent testing. The input parameters of the model are strain, log strain rate and temperature while torsional flow stress is the output. A three layer feed-forward network was trained with standard back propagation (BP) and Resilient propagation (Rprop) algorithm. The paper makes a robust comparison of the performances of the above two algorithms. The network trained with Rprop algorithm is found to perform better and also needs less number of iterations for convergence. The developed ANN model employing this algorithm could efficiently track the work hardening, dynamic softening and flow localization regions of the deforming material. Sensitivity analysis showed that temperature and strain rate are the most significant parameters while strain affects the flow stress only moderately. The ANN model, described in this paper, is an efficient quantitative tool to evaluate and predict the deformation behavior of type 304L stainless steel during hot torsion. ß 2008 Elsevier B.V. All rights reserved.
Keywords: Artificial neural network Austenitic stainless steel Deformation behavior Hot torsion Back propagation Resilient propagation Sensitivity
1. Introduction Hot working operations are extensively used in the first step of converting a cast ingot into a wrought product. Determinations of the load to carry out these operations are of paramount importance. The load depends on flow stress (s) of the materials besides the geometry of deformation and the friction at toolwork-piece interface. Therefore, the understanding of the hot deformation behavior linking process variables such as strain (e), strain rate ðe˙ Þ and temperature (T) to the flow stress of the deforming materials is necessary. During hot working of materials, several metallurgical phenomena such as work hardening, dynamic recovery (DRV), dynamic recrystallization (DRX), flow localization etc. occur. Work hardening increases the flow stress of materials and reduces the ductility while phenomena like DRV or DRX causes dynamic softening in materials and thereby restore the ductility. On the other hand, flow localization produces shear zone/internal cracks, which become the sites for eventual failure of the components during forming as well as in service [1]. This implies that the
* Corresponding author. E-mail address: [email protected] (S. Mandal). 1568-4946/$ – see front matter ß 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.asoc.2008.03.016
deformation behavior of materials during hot working is quite complex in nature. The hot deformation behavior of materials is usually described by phenomenological [2–4] or empirical/semiempirical equations [5–7] and is most often fit using multivariate non-linear regression. Although these approaches attempt to represent the non-linear relations among s, e˙ , e and T, they are usually restricted to some limited processing domain where a specific deformation mechanism operates and breaks down across deformation mechanism domains. Therefore, separate equations and/or various equation parameters are needed to represent the complete hot deformation behavior. Furthermore, development of such phenomenological or empirical/semi-empirical equations is always time consuming and usually has low accuracy in prediction. It is precisely in this context, soft computing technique like artificial neural network (ANN) provides an efficient alternative. Neural network provides a fundamentally different approach to material modeling and material processing control techniques than statistical or numerical methods [8]. The basic advantage of ANN is that it does not need any mathematical model; an ANN learns from examples and recognizes patterns in a series of input and output values without any prior assumptions about their nature and interrelations. Since ANN does not explicitly embed the physical knowledge of the deformation mechanism, it has the ability to predict the flow stress value across deformation mechanisms
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domain. Therefore, a single ANN has the inherent capability to describe the complete hot deformation behavior. Provision of model free solutions, data error tolerance, built in dynamism and lack of any exogenous input requirement makes the network attractive [9]. Since recent times, ANN is being increasingly used to model the deformation behavior of materials under hot compression [10– 13]. However, the flow behavior of a material also depends on the state-of-stress and therefore, the deformation behavior of materials during torsion differs significantly from that in compression. For example, it is well known that axial stresses develop during torsion of a bar with end constraints [14]. The state-of-stress in torsion would therefore be biaxial and yield occurs at lower stresses than in compression. At lower temperatures and higher strain rates, the material exhibits flow instabilities which are more accentuated in torsion than in compression. In addition, the flow stresses in torsion are lower because of the absence of friction and a contribution from the reduced polar moment of inertia of the cross section due to the undeformed central region [14,15]. Though ANN models have been developed to study the flow behavior of various grades of steel during hot compression, very limited efforts have been reported to predict the deformation behavior during hot torsion. For example, an integrated phenomenological and ANN model have been developed to predict the flow stress of commercial 304L stainless steel during hot torsion [16]. However, hot torsion test is useful in many engineering application and also in laboratory studies of plastic flows of materials. In the present study, therefore, an ANN model has been suggested to understand and predict the deformation behavior of 304L austenitic stainless steel during hot torsion. Toward this end, a feed-forward neural network has been trained by standard back propagation (BP) as well as upgraded algorithm like Resilient propagation (Rprop). The paper makes a robust comparison of the performances of the above two algorithms for high temperature torsional flow stress prediction in type 304L stainless steel. Besides, sensitivity analysis has been carried out to quantify the relative importance of individual process variables (i.e. strain, strain rate and temperature) on high temperature flow stress. The technological importance of type 304L stainless steel stems from its higher resistance to stress corrosion cracking, in addition to better mechanical properties, makes it a candidate material for applications in process industries. 2. Experimental The composition of 304L material, used in the present study, is given in Table 1. The material was received in the form of 25 mm diameter rods in hot rolled and annealed condition. These rods were cold swaged to 12 mm diameter and subsequently annealed at 1050 8C for 30 min. Solid cylindrical torsion specimens of various diameters (D) to length (L) ratios were machined from these rods for obtaining different values of maximum surface strain rate. The dimensions of the specimens used in the present study are depicted in Table 2. The torsion tests were carried out in a hot torsion testing machine designed and fabricated by one of the authors (S. Venugopal). The machine is equipped with a high capacity (3.7 kW) thyristor controlled D.C. shunt motor with armature
voltage feed back control circuitry. The torsion tests were carried out under isothermal conditions, i.e. the grips and the specimen were enclosed in the split furnace, heated by a silicon carbide heating elements. Testing was carried out in the temperature range of 600–1200 8C at an interval of 50 8C and in the maximum surface strain rate range of 0.1–100 s1. The temperature of the specimen was measured with appropriate thermocouple (depending on the test temperature) and controlled using microprocessor based proportional controller. The adiabatic temperature rise in the specimen during deformation was also monitored and recorded. The torque-twist data obtained in torsion were converted to equivalent stress, strain and strain rate using the following equations [17].
s eq ¼
Mð3 þ n þ mÞ 2pr 3
ru eeq ¼ pffiffiffi
(2)
r u˙ e˙ eq ¼ pffiffiffi
(3)
3L
3L
In the above, seq is the Von Mises equivalent true stress, M is the torque, n ¼ ð@ ln M=@ ln uÞj and m ¼ ð@ ln M=@ ln u˙ Þj; u is the angle of twist, u˙ is the rate of twist, T is the temperature, r is the radius of the specimen, L is the length of the specimen, eeq is the Von Mises equivalent strain and e˙ eq is the Von Mises equivalent strain rate. 3. Model overview ANN is a highly simplified model of the structure of a biological network. The fundamental unit or building block of ANN is the processing element, also called an artificial neuron or simply a neuron. Some neurons interact with the real world to receive input, and some provide the real world with the output. Rest of the neurons remain hidden. Neurons are connected to each other by synapses; associated with each synapse is a weight factor. In this study, a multilayer perceptron (MLP) based feed-forward ANN has been used since multilayer network has greater representational power for dealing with highly non-linear, strongly coupled, multivariable system [18]. Although multilayer neural network does not ensure a global minimum solution for any given problem, it is a reasonable approximation that if the network is trained with a comprehensive database, the resulting model will approximate all of the laws of mechanics that the actual material or process obeys [8]. A general scheme of the present ANN model is given in Fig. 1. The inputs of the model are strain (e), log strain rate ðlog e˙ Þ and temperature (T). The output of the model is flow stress (s). Instead of e˙ , log e˙ has been chosen since s usually varies with log e˙ on a physical basis. Experimental data obtained from hot torsion tests were used to train and test the model. The statistical analysis of input and output data is represented in Table 3. A comprehensive database (total of 455 experimental data points) has been used to develop Table 2 Dimension of the torsion test specimen used in the present investigation Strain rate (s1)
Table 1 Chemical composition (in wt.%) of 304L stainless steel C
Cr
Ni
Mo
Mn
S
P
Si
Fe
0.028
18.6
10.3
0.07
1.7
0.005
0.035
0.58
Balance
(1)
0.1, 0.5, and 1 5 10 50 and 100
Dimensions (mm) L0.01
D0.01
28 14 7 1.6
7 7 7 8
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updated algorithm like Resilient propagation. A little description of the working of Rprop is given below. 4.1. Rprop algorithm The Rprop algorithm uses a sign-based scheme to update the weights [19]. It works on the basis of the sign of the derivative to indicate the direction of weight update. If the derivative is positive (increasing error), the weight is decreased by its update value. On the other hand, if the derivative is negative, the update value is added. The algorithm can be mathematically expressed as follow [20]: Fig. 1. Schematic of ANN for prediction of torsional flow stress in type 304L stainless steel.
the present model. All the data were normalized employing the relation given by, x xmin xmax xmin
xN ¼
if
(7)
else
w ji ðn þ 1Þ ¼ w ji ðnÞ þ Dw ji ðnÞ
4. Learning algorithms The MLP based feed-forward ANN is generally trained with back propagation algorithm. The objective is to minimize the instantaneous value of total error energy, given by, 1X n 2 ðd j ynj Þ 2
(5)
and of average error energy, given by, Eav ¼
> þD ji ðnÞ; > > > : 0;
@EðnÞ >0 @w ji @EðnÞ <0 @w ji
if
(4)
where xN is the normalized value of the parameter x (temperature, strain, logarithm of strain rate or flow stress); xmax and xmin are the maximum and minimum values of x respectively; accordingly each parameter lies in the interval 0–1. These datasets were then divided into two parts. 91 data (20%) were randomly removed and remaining 364 data (80%) were used for training. The removed 91 datasets were subsequently used for testing. A logistic sigmoid function expressed as output = (1 + einput)1) was employed as the activation function; the learning is based on gradient descent algorithm and hence requires the activation function to be differentiable.
En ¼
Dw ji ðnÞ ¼
8 > > > D ji ðnÞ; > <
D ji ðnÞ ¼
8 > > > hþ D ji ðn 1Þ; > > <
h D ji ðn 1Þ; > > > > > : D ji ðn 1Þ;
if if
@Eðn 1Þ @EðnÞ >0 @w ji @w ji @Eðn 1Þ @EðnÞ <0 @w ji @w ji
(8)
else
where E(n) is the instantaneous value of total error energy; Eav is the average error energy; w ji is the connection weight value, n denotes the number of iterations and 0 < h < 1 < h+. As can be seen from Eq. (7), the size of weight change is solely determined by the weight specific update value Dji(n). Each time the partial derivative of the corresponding weight w ji changes its sign, the Dji(n) is decreased by a factor h (Eq. (8)), since it is indicated that the last update was too large and the algorithm jumped over a local minimum. On the other hand, if the derivative retains its sign, the update value is slightly increased by the factor h+ in order to accelerate the convergence in shallow regions. 5. Results and discussion
N 1X En ; N n¼1
(6) n
where n denotes the number of iterations; E is the instantaneous value of total error energy; Eav is the average error energy; N is the total number of training pattern; dnj is the target output for neuron j and ynj is the network output of neuron j. In BP algorithm, the error between target and the network output is calculated and this will be back propagated using the steepest descent or gradient descent approach. The network weights are adjusted by moving a small step in the direction of negative gradient of error surface during each iteration. The iterations are repeated until a specified convergence is reached. Along with BP algorithm, network has also been trained with an Table 3 Statistical analysis of input and output data
One hidden layer is found to be adequate for the present problem. This observation reaffirms the universal approximation theorem that a single layer of non-linear hidden units is sufficient to approximate any continuous function. Hornik et al. [21] have also shown that a three layer ANN with sigmoid transfer function can map any function of practical interest. Neurons in the hidden layer are varied from 1 to 20. After repeated trials, it is found that a network with one hidden layer consisting of 14 hidden neurons produces best performances when BP algorithm is employed. However, for Rprop algorithm, a network with 16 hidden neurons produces the best results. The performance of the network for test data at different hidden neurons level is shown in Fig. 2. 5.1. Model performances
Variables
Maximum
Minimum
S.D.
Average
Strain (%) Strain rate (s1) Temperature (8C) Flow stress (MPa)
0.5 100 1250 413.1
0.1 0.1 750 12.9
0.142 37.99 194.72 96.94
0.3 23.8 900 157.19
A wide variety of standard statistical performance evaluation measures have been employed to evaluate the model performance. The predictability of the network is quantified in terms of correlation coefficient (R), average absolute relative error (AARE), average root mean square error (RMSE), normalized mean bias
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Table 4 Performances of the BP and Rprop algorithm for torsional flow stress prediction in type 304L stainless steel
AAREð%Þ ¼
RMSE ¼
N 1X Ei P i 100 N i¼1 Ei
2 31=2 p N X 1X 4 ðP i j Ei j Þ2 5 N i¼1 j¼1
NMBEð%Þ ¼
SI ¼
RMSE E¯
ð1=NÞ
AARE (%)
RMSE (%)
NMBE (%)
SI
During training BP Rprop
4.96 3.79
5.69 4.78
0.037 1.15
0.036 0.030
During testing BP Rprop
5.04 4.16
6.15 5.32
0.32 1.45
0.039 0.033
iterations). Therefore, the model trained with Rprop algorithm is found to be superior with respect to prediction accuracy as well as convergence. So, it could be suggested that model trained with Rprop algorithm is the most efficient model for this problem; hence this model has been applied for further simulation and application. It should be noted here that ANN model trained with Rprop algorithm has also been found to produce best performances than other BP type algorithms while predicting the flow stress of as cast 304 stainless steel during hot compression [12].
Fig. 2. Performance of the network at different hidden neurons level.
error (NMBE) and scatter index (SI), expressed as: PN ¯ ¯ i¼1 ðEi EÞðP i PÞ R ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi PN 2 PN ¯ ¯ 2 i¼1 ðEi EÞ i¼1 ðP i PÞ
Learning algorithm
(9)
(10)
(11)
PN
ð1=NÞ
i¼1 Ei P i 100 PN i¼1 Ei
(12)
(13)
where E is the experimental finding and P is the predicted value obtained from the neural network model. E¯ and P¯ are the mean values of E and P respectively. N is the total number of data employed in the investigation and p is the number of variables in the output (in this case p = 1). The correlation coefficient is a commonly used statistic and provides information on the strength of linear relationship between observed and the computed values. Sometimes higher value of R may not necessarily indicate better performance of an ANN model [22] because of the tendency of the model to be biased towards higher or lower values. The AARE and RMSE are computed through a term by term comparison of the relative error and therefore are unbiased statistics for measuring the predictability of a model [23]. The NMBE gives information on the mean bias in prediction from a model. Positive NMBE indicates over prediction whereas negative indicates under prediction from a model. The performance of BP and Rprop algorithm is depicted in Table 4. As can be seen, Rprop algorithm consistently outperforms BP with respect to almost all statistical indices. This can also be seen from Figs. 3 and 4 which show that Rprop algorithm shows better correlation as compared to the BP for training as well as test data. The scattering of the data, as revealed by scatter index in Table 4, is less when Rprop learning algorithm is employed. Further, Rprop algorithm has taken lesser computational time (4990 iterations) to converge as compared to standard BP algorithm (16,430
Fig. 3. Predicted torsional flow stress (MPa) from the neural network versus experimental values for the training data set using (a) back propagation and (b) Resilient propagation algorithm.
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where E and P has the same meaning as stated earlier. The results are reported graphically as a typical number versus error plot (Fig. 5a and b). The error shows a typical Gaussian distribution with about zero mean value. For more than 90% training and test data set, the error of prediction is shown to be within 10% This signifies that main source of prediction error is the noise in the experimental data and cannot be wholly attributed to the predictability of the neural network model. The noise in the flow stress measurements normally arises due to unavoidable variations in temperature, strain rate and interfacial frictional resistances [8]. 5.2. Effect of temperature The developed ANN model is now applied to simulate the effect of temperature on high temperature flow behavior of type 304L stainless steel during hot torsion. The result has been shown in Fig. 6. As can be seen, simulated data can track well the experimented data. With increase of temperature, flow stress decreases in both the highest and lowest strain rate. Similar kind of trend has also been obtained for other strain rates and thereby not reiterated. The predicted dependencies on temperature are in accordance with the relation between flow stress and temperature. As the temperature increases, the available thermal activation
Fig. 4. Predicted torsional flow stress (MPa) from the neural network versus experimental values for the test data set using (a) back propagation and (b) Resilient propagation algorithm.
The reason of better performances of the ANN model employing Rprop algorithm over BP now needs to be discussed. The BP algorithm uses an instantaneous estimate for the gradient of error surface in weight space. The algorithm is therefore stochastic in nature; i.e. it has a tendency to zigzag its way about the true direction to a minimum on the error surface. Indeed, BP learning is an application of a statistical method known as stochastic approximation. Consequently, it tends to converge slowly. Rprop, on the other hand, is an effective learning algorithm based on direct adaptation of the weight step. The adaptation is done based on local gradient information. It does not consider, different from standard BP, the harmful influence of the absolute value of the partial derivative for the calculation of weight changes, but only the sign of the derivative to indicate the direction of weight update. The performance of the ANN model trained with Rprop algorithm is further investigated by statistical analysis of the error of neural network predictions for both the training and testing data. Neural network predictions are compared with the corresponding experimental data and subsequently the relative errors are calculated as below: Relative error ¼
EP 100%; E
(14)
Fig. 5. Statistical analysis of the error of neural network predictions employing Resilient propagation algorithm for (a) training and (b) test data.
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Fig. 6. Effect of temperature on flow stress in type 304L stainless steel at 0.5 strain level.
energy will be more which eventually leads to higher extent of dynamic softening. This dynamic softening may arise either from DRV or DRX. DRV leads to annihilation of opposite pairs of dislocation; thus reduces the density of dislocations in the matrix. This annihilation of opposite pairs of dislocation is thermally activated process and thereby favored at higher temperature. DRX, on the other hand, is controlled by nucleation of new strain free grains and strain induced grain boundary migration. Both of them are accelerated at higher temperature. The simulated results are therefore consistent with what is expected from the fundamental theory of hot working. 5.3. Flow behavior The flow curves obtained from ANN predictions have been shown against those of experimental data in a wide range of temperature, strain and strain rate in Fig. 7. The results show that the agreement is quite good over the full range of data. The predicted curves can track well the work hardening and dynamic softening region of the deforming material. Therefore, the present ANN model can be efficiently applied to predict the complete hot deformation behavior of type 304L stainless steel with reasonable accuracy and reliability. This can be considered as a major potential of the developed model as compared to the phenomenological model or traditional empirical/semi-empirical equations which can be applied only for a limited processing domain. The model is found to be statistically accurate and a robust tool to understand and predict the flow behavior of austenitic stainless steel. The only limitation of this model is that it does not produce any empirical mathematical equation that may be used in future. However, as long as the computer model is available, that mathematical relation is no longer required.
Fig. 7. Flow behavior of type 304L stainless steel at different temperature (a) e˙ ¼ 100 s1 .
assessing the susceptibility of a material to undergo flow localization [24]. The instability in the flow behavior at 1000 8C, particularly in the strain rate regime of 10–100 s1, arises due to ferrite formation. It has already been shown that ferrite formation
5.4. Effect of strain rate Effect of strain rate on flow behavior of 304L stainless steel has been simulated employing the ANN model. The result has been shown in Fig. 8. As can be seen, flow behavior is irregular at low temperature regime (600–800 8C). This can be attributed to the flow localization that causes instability in the flow behavior of materials. The flow localization in torsion arises due to the fact that the state-of-stress in torsion is essentially shear which accentuates shear localization. In fact, torsion test is considered to be ideal for
Fig. 8. Effect of strain rate on flow behavior of 304L stainless steel at 0.5 strain level.
S. Mandal et al. / Applied Soft Computing 9 (2009) 237–244
is favored in torsion as compared to compression [25]. These ferrites are formed due to the deformation heating at this higher strain rate level. At 1200 8C, the trend is quite regular which signifies that flow behavior in this temperature regime is mainly governed by work hardening and dynamic softening. As strain rate increases, the extent of dynamic softening reduces which eventually increases the flow stress. The above discussion suggested that though the deformation behavior of 304L stainless steel during hot torsion is associated with various complicated deformation mechanisms, our model can able to evaluate and predict it with sufficient accuracy and reliability. The model also gives us useful information to choose the desired deformation domain of this material especially in order to avoid flow localization. The model can also be linked with some optimizing tool like genetic algorithm in order to evaluate the best possible processing combination to optimize the workability of stainless steel type AISI 304L during hot torsion. 5.5. Sensitivity analysis Sensitivity analysis was carried out to quantify the relative importance of the input parameters to the output. Following Olden et al. [26], ‘‘connection weight approach’’ was employed to quantify the relative importance of input variables. This approach uses raw input-hidden and hidden-output connection weights in the neural network (see Appendix A) as compared to Garson’s algorithm [27] which uses absolute value of connection weights. The relative importance of input variables on the torsional flow stress has been shown by bar plots in Fig. 9. As can be seen from the figure, both temperature and strain rate affect the flow stress significantly and more or less equally. The pronounced effect of strain rate and temperature on various grades of austenitic stainless steels during hot deformation has well been documented in literature [28,29]. In the present study, pronounced effect of temperature and strain rate on torsional flow stress could be observed in Figs. 6 and 8 respectively. Strain, on the other hand, has only moderate effect on flow stress as revealed by sensitivity analysis. The modest effect of strain is also manifested in the flow curves (Fig. 7). The contribution of strain arises mainly from the low temperature deformation region i.e. 600–800 8C where the rate of work hardening is more as compared to dynamic softening. From the above discussion, it could be suggested that the developed ANN model has correctly quantified the relative importance of individual process variables.
243
6. Conclusions An artificial neural network model has been suggested to evaluate and predict the deformation behavior of type 304L stainless steel using experimental data from hot torsion tests in the temperature range of 600–1200 8C and in the maximum surface strain rate range of 0.1–100 s1. The input parameters of the neural network were strain, log strain rate and temperature while torsional flow stress was obtained as output. The network has been trained by standard back propagation and Resilient propagation algorithm. A robust comparison of the performances of the above two algorithms was made employing a wide variety of standard statistical indices. It has been observed that Rprop algorithm performs better and also requires less computational time for convergence. The optimal configuration of the ANN model using Rprop algorithm is found to be 3-16-1. It has been shown that predicted flow stress curves could efficiently track the work hardening, dynamic softening and flow localization regions of the deforming material. The relative importance of the individual process variable on the torsional flow stress has been quantified employing connection weight approach. This analysis showed that temperature and strain rate is the most significant parameters while strain affects the flow stress moderately. The pronounced effect of temperature and strain rate on torsional flow stress has also been manifested in simulated results. It can be, therefore, concluded that the ANN model proposed in this paper is an efficient quantitative tool to evaluate and predict the deformation behavior of type 304L stainless steel during hot torsion. The model can be coupled with genetic algorithm in order to evaluate the best possible processing combination to optimize the workability of stainless steel type AISI 304L during hot torsion. Acknowledgement The authors would like to express their sincere thanks to Dr. Baldev Raj, Director Indira Gandhi Centre for Atomic Research for his constant encouragement throughout the whole work. Appendix A. Example illustrating connection weight approach This appendix describes the procedure to determine the relative importance of input variables using connection weight approach. Consider an ANN with three input neurons, two hidden neurons and one output neuron with the connection weight as shown below, as an example:
1. Matrix containing input-hidden and hidden-output neuron connection weights.
Fig. 9. Bar plots showing the percentage relative importance of individual input variable using connection weight approach.
Hidden X
Hidden Y
Input 1
wX1 ¼ 0:25884
wY1 ¼ 1:169312
Input 2
wX2 ¼ 1:038212
wY2 ¼ 0:49978
Input 3
wX3 ¼ 1:868524
wY3 ¼ 2:72308
Output
wOX ¼ 0:549743
wOY ¼ 1:08274
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2. Contribution of each input neuron to the output neuron through hidden neuron is calculated as the product of the input-hidden connection weight and the hidden-output connection weight: e.g., C X1 ¼ wX1 wOX ¼ 0:25884 0:549743. Hidden X
Hidden Y
Input 1
CX1 = 0.14229
CY1 = 1.26606
Input 2
CX2 = 0.57074
CY2 = 0.54113
Input 3
CX3 = 1.02721
CY3 = 2.94838
3. Importance of individual input neuron to the outgoing signal is P calculated by the following equation. InputA ¼ YB¼X HiddenBA . Importance Input 1
I1 = 1.12377
Input 2
I2 = 1.11187
Input 3
I3 = 3.97559
I
4. Relative importance is calculated via RIX ¼ PXI 100%. X
Relative importance (%) Input 1
RI1 = 18.09
Input 2
RI2 = 17.90
Input 3
RI3 = 64.01
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[7] K.P. Rao, E.B. Hawbolt, Development of constitutive relationships using compression testing of a medium carbon steel, Trans. ASME J. Eng. Mater. Technol. 114 (1992) 116–123. [8] M.S. Chuan, J. Biglou, J.G. Lenard, J.G. Kim, Using neural networks to predict parameters in the hot working of aluminum alloys, J. Mater. Process. Technol. 86 (1999) 245–251. [9] S. Rao, S. Mandal, Hindcasting of storm waves using neural networks, Ocean Eng. 32 (2005) 667–684. [10] K.P. Rao, Y.K.D.V. Prasad, Neural network approach to flow stress evaluation in hot deformation, J. Mater. Process. Technol. 53 (1995) 552–566. [11] J. Liu, H. Chang, T.Y. Hsu, X.J. Ruan, Prediction of the flow stress of high-speed steel during hot deformation using a BP artificial neural network, J. Mater. Process. Technol. 103 (2000) 200–205. [12] S. Mandal, P.V. Sivaprasad, S. Venugopal, Capability of a feed-forward artificial neural network to predict the constitutive flow behavior of as cast 304 stainless steel under hot deformation, Trans. ASME J. Eng. Mater. Technol. 129 (2007) 242– 247. [13] S. Mandal, P.V. Sivaprasad, S. Venugopal, K.P.N. Murthy, Constitutive flow behavior of austenitic stainless steels under hot deformation: artificial neural network modeling to understand, evaluate and predict, Model. Simulation Mater. Sci. Eng. 14 (2006) 1053–1070. [14] D. Hardwick, W.J.McG. Tegart, Structural changes during the hot deformation of Copper, Aluminium and Nickel at high temperature and high strain rates, J. Inst. Metals 90 (1961) 17. [15] M.J. Luton, C.M. Sellers, Acta Metall. 17 (1969) 1033. [16] P.D. Hodgson, L.X. Kong, C.H.J. Davies, The prediction of the hot strength of steels with an integrated phenomenological and artificial neural network, J. Mater. Process. Technol. 87 (1999) 131–138. [17] S.L. Semiatin, G.D. Lahoti, Deformation and unstable flow in hot torsion of Ti-6Al2Sn-4Zr-2Mo-0.1Si, Metall. Trans. A 12A (1981) 1719–1728. [18] S.C. Juang, Y.S. Tarang, H.R. Lii, A comparison between the back-propagation and counter-propagation networks in the modeling of the TIG welding process, J. Mater. Process. Technol. 75 (1998) 54–62. [19] A.D. Anastasiadisa, G.D. Magoulasa, M.N. Vrahatis, New globally convergent training scheme based on the resilient propagation algorithm, Neurocomputing 64 (2005) 253–270. [20] M. Reidimiller, H. Braun, A direct adaptive method for faster back propagation learning: the RPROP algorithm, in: Proc. Int. Conf. Neural Networks, San Francisco, CA, (1993), pp. 586–591. [21] K. Hornik, M. Stinchcombe, H. White, Neural Netw. 2 (5) (1989) 359–366. [22] M.P. Phaniraj, A.K. Lahiri, The applicability of neural network model to predict flow stress for carbon steel, J. Mater. Process. Technol. 141 (2003) 219–227. [23] S. Srinivasulu, A. Jain, A comparative analysis of training methods for artificial neural network rainfall-runoff models, Appl. Soft Comput. 6 (2006) 295–306. [24] S.L. Semiatin, J.J. Jonas, Formability and Workability of Metals, ASM, Metals Park, Ohio, 1984, p. 13. [25] S.Venugoapl, PhD thesis, Department of Mechanical Engineering, University of Madras, 1993. [26] J.D. Olden, M.K. Joy, R.G. Death, An accurate comparison of methods for quantifying variable importance in artificial neural networks using simulated data, Ecol. Model. 178 (2004) 389–397. [27] G.D. Garson, Interpreting neural-network connection weights, Artif. Intell. Expert 6 (1991) 47–51. [28] S.L. Mannan, K.G. Samuel, P. Rodriguez, Mater. Sci. Eng. A 68 (1984–1985) 143. [29] R.A. Varin, The effect of temperature and strain rate on the plastic flow and ductility of ultra-fined grained type 316 stainless steel, Mater. Sci. Eng. A 94 (1987) 93–107.
Contents lists available at ScienceDirect
Applied Soft Computing journal homepage: www.elsevier.com/locate/asoc
Artificial neural network modeling to evaluate and predict the deformation behavior of stainless steel type AISI 304L during hot torsion Sumantra Mandal a,*, P.V. Sivaprasad a, S. Venugopal a, K.P.N. Murthy b a b
Metallurgy and Materials Group, Indira Gandhi Centre for Atomic Research, Kalpakkam 603102, Tamil Nadu, India School of Physics, University of Hyderabad, Hyderabad 500046, Andhra Pradesh, India
A R T I C L E I N F O
A B S T R A C T
Article history: Received 24 January 2006 Received in revised form 19 December 2007 Accepted 7 March 2008 Available online 23 April 2008
The deformation behavior of type 304L stainless steel during hot torsion is investigated using artificial neural network (ANN). Torsion tests in the temperature range of 600–1200 8C and in the (maximum surface) strain rate range of 0.1–100 s1 were carried out. These experiments provided the required data for training the neural network and for subsequent testing. The input parameters of the model are strain, log strain rate and temperature while torsional flow stress is the output. A three layer feed-forward network was trained with standard back propagation (BP) and Resilient propagation (Rprop) algorithm. The paper makes a robust comparison of the performances of the above two algorithms. The network trained with Rprop algorithm is found to perform better and also needs less number of iterations for convergence. The developed ANN model employing this algorithm could efficiently track the work hardening, dynamic softening and flow localization regions of the deforming material. Sensitivity analysis showed that temperature and strain rate are the most significant parameters while strain affects the flow stress only moderately. The ANN model, described in this paper, is an efficient quantitative tool to evaluate and predict the deformation behavior of type 304L stainless steel during hot torsion. ß 2008 Elsevier B.V. All rights reserved.
Keywords: Artificial neural network Austenitic stainless steel Deformation behavior Hot torsion Back propagation Resilient propagation Sensitivity
1. Introduction Hot working operations are extensively used in the first step of converting a cast ingot into a wrought product. Determinations of the load to carry out these operations are of paramount importance. The load depends on flow stress (s) of the materials besides the geometry of deformation and the friction at toolwork-piece interface. Therefore, the understanding of the hot deformation behavior linking process variables such as strain (e), strain rate ðe˙ Þ and temperature (T) to the flow stress of the deforming materials is necessary. During hot working of materials, several metallurgical phenomena such as work hardening, dynamic recovery (DRV), dynamic recrystallization (DRX), flow localization etc. occur. Work hardening increases the flow stress of materials and reduces the ductility while phenomena like DRV or DRX causes dynamic softening in materials and thereby restore the ductility. On the other hand, flow localization produces shear zone/internal cracks, which become the sites for eventual failure of the components during forming as well as in service [1]. This implies that the
* Corresponding author. E-mail address: [email protected] (S. Mandal). 1568-4946/$ – see front matter ß 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.asoc.2008.03.016
deformation behavior of materials during hot working is quite complex in nature. The hot deformation behavior of materials is usually described by phenomenological [2–4] or empirical/semiempirical equations [5–7] and is most often fit using multivariate non-linear regression. Although these approaches attempt to represent the non-linear relations among s, e˙ , e and T, they are usually restricted to some limited processing domain where a specific deformation mechanism operates and breaks down across deformation mechanism domains. Therefore, separate equations and/or various equation parameters are needed to represent the complete hot deformation behavior. Furthermore, development of such phenomenological or empirical/semi-empirical equations is always time consuming and usually has low accuracy in prediction. It is precisely in this context, soft computing technique like artificial neural network (ANN) provides an efficient alternative. Neural network provides a fundamentally different approach to material modeling and material processing control techniques than statistical or numerical methods [8]. The basic advantage of ANN is that it does not need any mathematical model; an ANN learns from examples and recognizes patterns in a series of input and output values without any prior assumptions about their nature and interrelations. Since ANN does not explicitly embed the physical knowledge of the deformation mechanism, it has the ability to predict the flow stress value across deformation mechanisms
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domain. Therefore, a single ANN has the inherent capability to describe the complete hot deformation behavior. Provision of model free solutions, data error tolerance, built in dynamism and lack of any exogenous input requirement makes the network attractive [9]. Since recent times, ANN is being increasingly used to model the deformation behavior of materials under hot compression [10– 13]. However, the flow behavior of a material also depends on the state-of-stress and therefore, the deformation behavior of materials during torsion differs significantly from that in compression. For example, it is well known that axial stresses develop during torsion of a bar with end constraints [14]. The state-of-stress in torsion would therefore be biaxial and yield occurs at lower stresses than in compression. At lower temperatures and higher strain rates, the material exhibits flow instabilities which are more accentuated in torsion than in compression. In addition, the flow stresses in torsion are lower because of the absence of friction and a contribution from the reduced polar moment of inertia of the cross section due to the undeformed central region [14,15]. Though ANN models have been developed to study the flow behavior of various grades of steel during hot compression, very limited efforts have been reported to predict the deformation behavior during hot torsion. For example, an integrated phenomenological and ANN model have been developed to predict the flow stress of commercial 304L stainless steel during hot torsion [16]. However, hot torsion test is useful in many engineering application and also in laboratory studies of plastic flows of materials. In the present study, therefore, an ANN model has been suggested to understand and predict the deformation behavior of 304L austenitic stainless steel during hot torsion. Toward this end, a feed-forward neural network has been trained by standard back propagation (BP) as well as upgraded algorithm like Resilient propagation (Rprop). The paper makes a robust comparison of the performances of the above two algorithms for high temperature torsional flow stress prediction in type 304L stainless steel. Besides, sensitivity analysis has been carried out to quantify the relative importance of individual process variables (i.e. strain, strain rate and temperature) on high temperature flow stress. The technological importance of type 304L stainless steel stems from its higher resistance to stress corrosion cracking, in addition to better mechanical properties, makes it a candidate material for applications in process industries. 2. Experimental The composition of 304L material, used in the present study, is given in Table 1. The material was received in the form of 25 mm diameter rods in hot rolled and annealed condition. These rods were cold swaged to 12 mm diameter and subsequently annealed at 1050 8C for 30 min. Solid cylindrical torsion specimens of various diameters (D) to length (L) ratios were machined from these rods for obtaining different values of maximum surface strain rate. The dimensions of the specimens used in the present study are depicted in Table 2. The torsion tests were carried out in a hot torsion testing machine designed and fabricated by one of the authors (S. Venugopal). The machine is equipped with a high capacity (3.7 kW) thyristor controlled D.C. shunt motor with armature
voltage feed back control circuitry. The torsion tests were carried out under isothermal conditions, i.e. the grips and the specimen were enclosed in the split furnace, heated by a silicon carbide heating elements. Testing was carried out in the temperature range of 600–1200 8C at an interval of 50 8C and in the maximum surface strain rate range of 0.1–100 s1. The temperature of the specimen was measured with appropriate thermocouple (depending on the test temperature) and controlled using microprocessor based proportional controller. The adiabatic temperature rise in the specimen during deformation was also monitored and recorded. The torque-twist data obtained in torsion were converted to equivalent stress, strain and strain rate using the following equations [17].
s eq ¼
Mð3 þ n þ mÞ 2pr 3
ru eeq ¼ pffiffiffi
(2)
r u˙ e˙ eq ¼ pffiffiffi
(3)
3L
3L
In the above, seq is the Von Mises equivalent true stress, M is the torque, n ¼ ð@ ln M=@ ln uÞj and m ¼ ð@ ln M=@ ln u˙ Þj; u is the angle of twist, u˙ is the rate of twist, T is the temperature, r is the radius of the specimen, L is the length of the specimen, eeq is the Von Mises equivalent strain and e˙ eq is the Von Mises equivalent strain rate. 3. Model overview ANN is a highly simplified model of the structure of a biological network. The fundamental unit or building block of ANN is the processing element, also called an artificial neuron or simply a neuron. Some neurons interact with the real world to receive input, and some provide the real world with the output. Rest of the neurons remain hidden. Neurons are connected to each other by synapses; associated with each synapse is a weight factor. In this study, a multilayer perceptron (MLP) based feed-forward ANN has been used since multilayer network has greater representational power for dealing with highly non-linear, strongly coupled, multivariable system [18]. Although multilayer neural network does not ensure a global minimum solution for any given problem, it is a reasonable approximation that if the network is trained with a comprehensive database, the resulting model will approximate all of the laws of mechanics that the actual material or process obeys [8]. A general scheme of the present ANN model is given in Fig. 1. The inputs of the model are strain (e), log strain rate ðlog e˙ Þ and temperature (T). The output of the model is flow stress (s). Instead of e˙ , log e˙ has been chosen since s usually varies with log e˙ on a physical basis. Experimental data obtained from hot torsion tests were used to train and test the model. The statistical analysis of input and output data is represented in Table 3. A comprehensive database (total of 455 experimental data points) has been used to develop Table 2 Dimension of the torsion test specimen used in the present investigation Strain rate (s1)
Table 1 Chemical composition (in wt.%) of 304L stainless steel C
Cr
Ni
Mo
Mn
S
P
Si
Fe
0.028
18.6
10.3
0.07
1.7
0.005
0.035
0.58
Balance
(1)
0.1, 0.5, and 1 5 10 50 and 100
Dimensions (mm) L0.01
D0.01
28 14 7 1.6
7 7 7 8
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updated algorithm like Resilient propagation. A little description of the working of Rprop is given below. 4.1. Rprop algorithm The Rprop algorithm uses a sign-based scheme to update the weights [19]. It works on the basis of the sign of the derivative to indicate the direction of weight update. If the derivative is positive (increasing error), the weight is decreased by its update value. On the other hand, if the derivative is negative, the update value is added. The algorithm can be mathematically expressed as follow [20]: Fig. 1. Schematic of ANN for prediction of torsional flow stress in type 304L stainless steel.
the present model. All the data were normalized employing the relation given by, x xmin xmax xmin
xN ¼
if
(7)
else
w ji ðn þ 1Þ ¼ w ji ðnÞ þ Dw ji ðnÞ
4. Learning algorithms The MLP based feed-forward ANN is generally trained with back propagation algorithm. The objective is to minimize the instantaneous value of total error energy, given by, 1X n 2 ðd j ynj Þ 2
(5)
and of average error energy, given by, Eav ¼
> þD ji ðnÞ; > > > : 0;
@EðnÞ >0 @w ji @EðnÞ <0 @w ji
if
(4)
where xN is the normalized value of the parameter x (temperature, strain, logarithm of strain rate or flow stress); xmax and xmin are the maximum and minimum values of x respectively; accordingly each parameter lies in the interval 0–1. These datasets were then divided into two parts. 91 data (20%) were randomly removed and remaining 364 data (80%) were used for training. The removed 91 datasets were subsequently used for testing. A logistic sigmoid function expressed as output = (1 + einput)1) was employed as the activation function; the learning is based on gradient descent algorithm and hence requires the activation function to be differentiable.
En ¼
Dw ji ðnÞ ¼
8 > > > D ji ðnÞ; > <
D ji ðnÞ ¼
8 > > > hþ D ji ðn 1Þ; > > <
h D ji ðn 1Þ; > > > > > : D ji ðn 1Þ;
if if
@Eðn 1Þ @EðnÞ >0 @w ji @w ji @Eðn 1Þ @EðnÞ <0 @w ji @w ji
(8)
else
where E(n) is the instantaneous value of total error energy; Eav is the average error energy; w ji is the connection weight value, n denotes the number of iterations and 0 < h < 1 < h+. As can be seen from Eq. (7), the size of weight change is solely determined by the weight specific update value Dji(n). Each time the partial derivative of the corresponding weight w ji changes its sign, the Dji(n) is decreased by a factor h (Eq. (8)), since it is indicated that the last update was too large and the algorithm jumped over a local minimum. On the other hand, if the derivative retains its sign, the update value is slightly increased by the factor h+ in order to accelerate the convergence in shallow regions. 5. Results and discussion
N 1X En ; N n¼1
(6) n
where n denotes the number of iterations; E is the instantaneous value of total error energy; Eav is the average error energy; N is the total number of training pattern; dnj is the target output for neuron j and ynj is the network output of neuron j. In BP algorithm, the error between target and the network output is calculated and this will be back propagated using the steepest descent or gradient descent approach. The network weights are adjusted by moving a small step in the direction of negative gradient of error surface during each iteration. The iterations are repeated until a specified convergence is reached. Along with BP algorithm, network has also been trained with an Table 3 Statistical analysis of input and output data
One hidden layer is found to be adequate for the present problem. This observation reaffirms the universal approximation theorem that a single layer of non-linear hidden units is sufficient to approximate any continuous function. Hornik et al. [21] have also shown that a three layer ANN with sigmoid transfer function can map any function of practical interest. Neurons in the hidden layer are varied from 1 to 20. After repeated trials, it is found that a network with one hidden layer consisting of 14 hidden neurons produces best performances when BP algorithm is employed. However, for Rprop algorithm, a network with 16 hidden neurons produces the best results. The performance of the network for test data at different hidden neurons level is shown in Fig. 2. 5.1. Model performances
Variables
Maximum
Minimum
S.D.
Average
Strain (%) Strain rate (s1) Temperature (8C) Flow stress (MPa)
0.5 100 1250 413.1
0.1 0.1 750 12.9
0.142 37.99 194.72 96.94
0.3 23.8 900 157.19
A wide variety of standard statistical performance evaluation measures have been employed to evaluate the model performance. The predictability of the network is quantified in terms of correlation coefficient (R), average absolute relative error (AARE), average root mean square error (RMSE), normalized mean bias
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Table 4 Performances of the BP and Rprop algorithm for torsional flow stress prediction in type 304L stainless steel
AAREð%Þ ¼
RMSE ¼
N 1X Ei P i 100 N i¼1 Ei
2 31=2 p N X 1X 4 ðP i j Ei j Þ2 5 N i¼1 j¼1
NMBEð%Þ ¼
SI ¼
RMSE E¯
ð1=NÞ
AARE (%)
RMSE (%)
NMBE (%)
SI
During training BP Rprop
4.96 3.79
5.69 4.78
0.037 1.15
0.036 0.030
During testing BP Rprop
5.04 4.16
6.15 5.32
0.32 1.45
0.039 0.033
iterations). Therefore, the model trained with Rprop algorithm is found to be superior with respect to prediction accuracy as well as convergence. So, it could be suggested that model trained with Rprop algorithm is the most efficient model for this problem; hence this model has been applied for further simulation and application. It should be noted here that ANN model trained with Rprop algorithm has also been found to produce best performances than other BP type algorithms while predicting the flow stress of as cast 304 stainless steel during hot compression [12].
Fig. 2. Performance of the network at different hidden neurons level.
error (NMBE) and scatter index (SI), expressed as: PN ¯ ¯ i¼1 ðEi EÞðP i PÞ R ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi PN 2 PN ¯ ¯ 2 i¼1 ðEi EÞ i¼1 ðP i PÞ
Learning algorithm
(9)
(10)
(11)
PN
ð1=NÞ
i¼1 Ei P i 100 PN i¼1 Ei
(12)
(13)
where E is the experimental finding and P is the predicted value obtained from the neural network model. E¯ and P¯ are the mean values of E and P respectively. N is the total number of data employed in the investigation and p is the number of variables in the output (in this case p = 1). The correlation coefficient is a commonly used statistic and provides information on the strength of linear relationship between observed and the computed values. Sometimes higher value of R may not necessarily indicate better performance of an ANN model [22] because of the tendency of the model to be biased towards higher or lower values. The AARE and RMSE are computed through a term by term comparison of the relative error and therefore are unbiased statistics for measuring the predictability of a model [23]. The NMBE gives information on the mean bias in prediction from a model. Positive NMBE indicates over prediction whereas negative indicates under prediction from a model. The performance of BP and Rprop algorithm is depicted in Table 4. As can be seen, Rprop algorithm consistently outperforms BP with respect to almost all statistical indices. This can also be seen from Figs. 3 and 4 which show that Rprop algorithm shows better correlation as compared to the BP for training as well as test data. The scattering of the data, as revealed by scatter index in Table 4, is less when Rprop learning algorithm is employed. Further, Rprop algorithm has taken lesser computational time (4990 iterations) to converge as compared to standard BP algorithm (16,430
Fig. 3. Predicted torsional flow stress (MPa) from the neural network versus experimental values for the training data set using (a) back propagation and (b) Resilient propagation algorithm.
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where E and P has the same meaning as stated earlier. The results are reported graphically as a typical number versus error plot (Fig. 5a and b). The error shows a typical Gaussian distribution with about zero mean value. For more than 90% training and test data set, the error of prediction is shown to be within 10% This signifies that main source of prediction error is the noise in the experimental data and cannot be wholly attributed to the predictability of the neural network model. The noise in the flow stress measurements normally arises due to unavoidable variations in temperature, strain rate and interfacial frictional resistances [8]. 5.2. Effect of temperature The developed ANN model is now applied to simulate the effect of temperature on high temperature flow behavior of type 304L stainless steel during hot torsion. The result has been shown in Fig. 6. As can be seen, simulated data can track well the experimented data. With increase of temperature, flow stress decreases in both the highest and lowest strain rate. Similar kind of trend has also been obtained for other strain rates and thereby not reiterated. The predicted dependencies on temperature are in accordance with the relation between flow stress and temperature. As the temperature increases, the available thermal activation
Fig. 4. Predicted torsional flow stress (MPa) from the neural network versus experimental values for the test data set using (a) back propagation and (b) Resilient propagation algorithm.
The reason of better performances of the ANN model employing Rprop algorithm over BP now needs to be discussed. The BP algorithm uses an instantaneous estimate for the gradient of error surface in weight space. The algorithm is therefore stochastic in nature; i.e. it has a tendency to zigzag its way about the true direction to a minimum on the error surface. Indeed, BP learning is an application of a statistical method known as stochastic approximation. Consequently, it tends to converge slowly. Rprop, on the other hand, is an effective learning algorithm based on direct adaptation of the weight step. The adaptation is done based on local gradient information. It does not consider, different from standard BP, the harmful influence of the absolute value of the partial derivative for the calculation of weight changes, but only the sign of the derivative to indicate the direction of weight update. The performance of the ANN model trained with Rprop algorithm is further investigated by statistical analysis of the error of neural network predictions for both the training and testing data. Neural network predictions are compared with the corresponding experimental data and subsequently the relative errors are calculated as below: Relative error ¼
EP 100%; E
(14)
Fig. 5. Statistical analysis of the error of neural network predictions employing Resilient propagation algorithm for (a) training and (b) test data.
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Fig. 6. Effect of temperature on flow stress in type 304L stainless steel at 0.5 strain level.
energy will be more which eventually leads to higher extent of dynamic softening. This dynamic softening may arise either from DRV or DRX. DRV leads to annihilation of opposite pairs of dislocation; thus reduces the density of dislocations in the matrix. This annihilation of opposite pairs of dislocation is thermally activated process and thereby favored at higher temperature. DRX, on the other hand, is controlled by nucleation of new strain free grains and strain induced grain boundary migration. Both of them are accelerated at higher temperature. The simulated results are therefore consistent with what is expected from the fundamental theory of hot working. 5.3. Flow behavior The flow curves obtained from ANN predictions have been shown against those of experimental data in a wide range of temperature, strain and strain rate in Fig. 7. The results show that the agreement is quite good over the full range of data. The predicted curves can track well the work hardening and dynamic softening region of the deforming material. Therefore, the present ANN model can be efficiently applied to predict the complete hot deformation behavior of type 304L stainless steel with reasonable accuracy and reliability. This can be considered as a major potential of the developed model as compared to the phenomenological model or traditional empirical/semi-empirical equations which can be applied only for a limited processing domain. The model is found to be statistically accurate and a robust tool to understand and predict the flow behavior of austenitic stainless steel. The only limitation of this model is that it does not produce any empirical mathematical equation that may be used in future. However, as long as the computer model is available, that mathematical relation is no longer required.
Fig. 7. Flow behavior of type 304L stainless steel at different temperature (a) e˙ ¼ 100 s1 .
assessing the susceptibility of a material to undergo flow localization [24]. The instability in the flow behavior at 1000 8C, particularly in the strain rate regime of 10–100 s1, arises due to ferrite formation. It has already been shown that ferrite formation
5.4. Effect of strain rate Effect of strain rate on flow behavior of 304L stainless steel has been simulated employing the ANN model. The result has been shown in Fig. 8. As can be seen, flow behavior is irregular at low temperature regime (600–800 8C). This can be attributed to the flow localization that causes instability in the flow behavior of materials. The flow localization in torsion arises due to the fact that the state-of-stress in torsion is essentially shear which accentuates shear localization. In fact, torsion test is considered to be ideal for
Fig. 8. Effect of strain rate on flow behavior of 304L stainless steel at 0.5 strain level.
S. Mandal et al. / Applied Soft Computing 9 (2009) 237–244
is favored in torsion as compared to compression [25]. These ferrites are formed due to the deformation heating at this higher strain rate level. At 1200 8C, the trend is quite regular which signifies that flow behavior in this temperature regime is mainly governed by work hardening and dynamic softening. As strain rate increases, the extent of dynamic softening reduces which eventually increases the flow stress. The above discussion suggested that though the deformation behavior of 304L stainless steel during hot torsion is associated with various complicated deformation mechanisms, our model can able to evaluate and predict it with sufficient accuracy and reliability. The model also gives us useful information to choose the desired deformation domain of this material especially in order to avoid flow localization. The model can also be linked with some optimizing tool like genetic algorithm in order to evaluate the best possible processing combination to optimize the workability of stainless steel type AISI 304L during hot torsion. 5.5. Sensitivity analysis Sensitivity analysis was carried out to quantify the relative importance of the input parameters to the output. Following Olden et al. [26], ‘‘connection weight approach’’ was employed to quantify the relative importance of input variables. This approach uses raw input-hidden and hidden-output connection weights in the neural network (see Appendix A) as compared to Garson’s algorithm [27] which uses absolute value of connection weights. The relative importance of input variables on the torsional flow stress has been shown by bar plots in Fig. 9. As can be seen from the figure, both temperature and strain rate affect the flow stress significantly and more or less equally. The pronounced effect of strain rate and temperature on various grades of austenitic stainless steels during hot deformation has well been documented in literature [28,29]. In the present study, pronounced effect of temperature and strain rate on torsional flow stress could be observed in Figs. 6 and 8 respectively. Strain, on the other hand, has only moderate effect on flow stress as revealed by sensitivity analysis. The modest effect of strain is also manifested in the flow curves (Fig. 7). The contribution of strain arises mainly from the low temperature deformation region i.e. 600–800 8C where the rate of work hardening is more as compared to dynamic softening. From the above discussion, it could be suggested that the developed ANN model has correctly quantified the relative importance of individual process variables.
243
6. Conclusions An artificial neural network model has been suggested to evaluate and predict the deformation behavior of type 304L stainless steel using experimental data from hot torsion tests in the temperature range of 600–1200 8C and in the maximum surface strain rate range of 0.1–100 s1. The input parameters of the neural network were strain, log strain rate and temperature while torsional flow stress was obtained as output. The network has been trained by standard back propagation and Resilient propagation algorithm. A robust comparison of the performances of the above two algorithms was made employing a wide variety of standard statistical indices. It has been observed that Rprop algorithm performs better and also requires less computational time for convergence. The optimal configuration of the ANN model using Rprop algorithm is found to be 3-16-1. It has been shown that predicted flow stress curves could efficiently track the work hardening, dynamic softening and flow localization regions of the deforming material. The relative importance of the individual process variable on the torsional flow stress has been quantified employing connection weight approach. This analysis showed that temperature and strain rate is the most significant parameters while strain affects the flow stress moderately. The pronounced effect of temperature and strain rate on torsional flow stress has also been manifested in simulated results. It can be, therefore, concluded that the ANN model proposed in this paper is an efficient quantitative tool to evaluate and predict the deformation behavior of type 304L stainless steel during hot torsion. The model can be coupled with genetic algorithm in order to evaluate the best possible processing combination to optimize the workability of stainless steel type AISI 304L during hot torsion. Acknowledgement The authors would like to express their sincere thanks to Dr. Baldev Raj, Director Indira Gandhi Centre for Atomic Research for his constant encouragement throughout the whole work. Appendix A. Example illustrating connection weight approach This appendix describes the procedure to determine the relative importance of input variables using connection weight approach. Consider an ANN with three input neurons, two hidden neurons and one output neuron with the connection weight as shown below, as an example:
1. Matrix containing input-hidden and hidden-output neuron connection weights.
Fig. 9. Bar plots showing the percentage relative importance of individual input variable using connection weight approach.
Hidden X
Hidden Y
Input 1
wX1 ¼ 0:25884
wY1 ¼ 1:169312
Input 2
wX2 ¼ 1:038212
wY2 ¼ 0:49978
Input 3
wX3 ¼ 1:868524
wY3 ¼ 2:72308
Output
wOX ¼ 0:549743
wOY ¼ 1:08274
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2. Contribution of each input neuron to the output neuron through hidden neuron is calculated as the product of the input-hidden connection weight and the hidden-output connection weight: e.g., C X1 ¼ wX1 wOX ¼ 0:25884 0:549743. Hidden X
Hidden Y
Input 1
CX1 = 0.14229
CY1 = 1.26606
Input 2
CX2 = 0.57074
CY2 = 0.54113
Input 3
CX3 = 1.02721
CY3 = 2.94838
3. Importance of individual input neuron to the outgoing signal is P calculated by the following equation. InputA ¼ YB¼X HiddenBA . Importance Input 1
I1 = 1.12377
Input 2
I2 = 1.11187
Input 3
I3 = 3.97559
I
4. Relative importance is calculated via RIX ¼ PXI 100%. X
Relative importance (%) Input 1
RI1 = 18.09
Input 2
RI2 = 17.90
Input 3
RI3 = 64.01
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