Prediction of high temperature deformation characteristics of an Fe-based shape memory alloy using constitutive and artificial neural network modelling

Journal Pre-proof Prediction of High Temperature Deformation Characteristics of an Fe-Base Shape Memory Alloy using Constitutive and Artificial Neural Network Modelling S.H. Adarsh, V. Sampath

PII:

S2352-4928(19)30992-4

DOI:

https://doi.org/10.1016/j.mtcomm.2019.100841

Reference:

MTCOMM 100841

To appear in:

Materials Today Communications

Received Date:

1 October 2019

Revised Date:

10 December 2019

Accepted Date:

10 December 2019

Please cite this article as: Adarsh SH, Sampath V, Prediction of High Temperature Deformation Characteristics of an Fe-Base Shape Memory Alloy using Constitutive and Artificial Neural Network Modelling, Materials Today Communications (2019), doi: https://doi.org/10.1016/j.mtcomm.2019.100841

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Prediction of High Temperature Deformation Characteristics of an Fe-Base Shape Memory Alloy using Constitutive and Artificial Neural Network Modelling

S. H. Adarsh, V. Sampath* [email protected]

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*Corresponding

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Department of Metallurgical and Materials Engineering, Indian Institute of Technology Madras, Chennai – 600036 INDIA

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Abstract: The high temperature deformation characteristics of an Fe-28Ni-17Co-11.5Al-2.5Ta0.05B (at.%) shape memory alloy (SMA) were studied by high temperature compression testing under large temperature (1323-1473 K) and strain rate (0.01-10 s-1) ranges. These were predicted

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by applying the Arrhenius-type and strain-compensated Arrhenius-type constitutive models, and the artificial neural network (ANN) model to the results obtained from the experiments. The

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capability of the models for prediction was assessed as a function of the correlation coefficient (R) and the relative percentage error. The results reveal that the true stress prediction by the straincompensated Arrhenius-type constitutive model is more precise at a lower strain rate (0.01 s-1)

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than at a higher strain rate (10 s-1). Moreover, it yields better results in comparison with those obtained from Arrhenius-type model. They further reveal ANN model shows higher efficiency

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and preciseness in forecasting the high temperature flow characteristics of the SMA as compared to the strain-compensated Arrhenius-type and Arrhenius-type models. Keywords: Shape memory alloys, High temperature deformation characteristics, Straincompensated Arrhenius-type constitutive equation, Artificial neural network. 1. Introduction Although NiTi is the only commercially available shape memory alloy on the market, it has its own disadvantages too, such as poor cold workability and use of expensive raw materials. 1

Copper-based alloys, on the other hand, show lower strain recovery, poor corrosion resistance and biocompatibility. Researchers are therefore in search of viable alternatives to Cu-based and NiTi SMAs. Ferrous-based SMAs have been on the scenario for a few decades now. Their lower strain recovery and shape memory properties make them poor competitors to SMAs based on Cu and NiTi. A complex ferrous-based FeNiCoAlTaB SMA exhibiting large superelasticity (～13%) and high yield strength (1 GPa) was discovered by Tanaka et al. [1] in 2010. This opened up a new pathway in the domain of SMAs.

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The material was processed by hot rolling to reduce its thickness by 28% of its original thickness, followed by cold rolling to 98% of its initial thickness. High texture evolutions during rolling and γ′ precipitate particles resulting from aging are the key factors for obtaining such significant properties. Vacuum induction melting and vacuum arc remelting were used to prepare the Fe-base alloy leading to samples with different microstructural features [2] and thereby different

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transformation temperatures. In order to achieve therefore recrystallized grains, a better

understanding on the strain involved in high temperature deformation, strain and strain behaviour

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of the SMA subjected to different processing conditions.

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Nowadays bulk metal processing involving plastic deformation at elevated temperature to obviate fracture is used as the most promising and common approach to forecast the flow stress and thereby optimize the process parameters (strain, strain rate and temperature) to control the

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microstructure during the processing stage [3]. Also one of the major challenges to be overcome in this context is to understand plastic flow behaviour as ultimately scaling up has to be done if the alloy is to be produced in large quantities. The processing parameters, such as temperature,

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strain and strain rate play a crucial role in hot deformation. In metallic materials, the elevated temperature deformation involves many complex metallurgical processes, namely, hardening,

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recovery, recrystallization and flow instability [4-8]. Moreover, these processes show a non-linear relationship with respect to the flow stress. Consequently, modelling to forecast elevated temperature deformation becomes quite complex in nature. Constitutive relationships involve relationships between true stress, strain, strain rate and temperature under a large range of processing parameters [9, 10]. Researchers have over the years developed constitutive equations using the results obtained from experiments to discuss the hot deformation characteristics of materials [11-15]. A constitutive model of Arrhenius-type is

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extensively used to explain the elevated temperature deformation characteristics of materials. In recent times, improvements to this model have been proposed to increase its capability and preciseness of prediction [12-17]. Slooff et al. [15] incorporated the parameters that are dependent on strain in strain-compensated Arrhenius-type equation to evaluate the flow stress in a wrought Mg alloy. But this model has drawbacks, such as poor adaptability to new experimental data [17, 18] and poor accuracy in forecasting the relationships between processing variables and flow stress. Considering the disadvantages of the constitutive models and the advantages of the ANN model, the ANN can be used as a substitute for materials modelling and processing techniques [19].The

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main benefits of ANN is that it does not use any mathematical model. Based on the nature and inter-relations between the input and output variables, ANN predicts the behaviour without

assumptions. The ANN has the capability to forecast the flow stress across domains since it does not incorporate a knowledge of deformation mechanisms. Thus an ANN has the innate ability to

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forecast the complete elevated temperature deformation characteristics. ANN models have already been used with much success to forecast elevated temperature deformation characteristics of

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magnesium alloys [25], titanium alloys [23], steels [20-22], and metal matrix composites [24]. It is therefore relevant to use them on the present SMA as well. Presently there is wealth of literature available to predict the hot deformation behaviour of different materials using constitutive

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equations as well as ANN. However, there is a dearth of literature to predict the hot deformation behaviour of the complex Fe-28Ni-17Co-11.5Al-2.5Ta-0.05B (at.%) shape memory alloy in a

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wide range of temperatures and strain rates.

This work compares the capability of constitutive equations in forecasting the elevated temperature behavior of Fe-28Ni-17Co-11.5Al-2.5Ta-0.05B (at.%) SMA. The major objective of

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the present work is to come up with an appropriate model for the deformation characteristics of the alloy. Isothermal high temperature compression experiments were done using a Gleeble-3800

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thermomechanical simulator. The flow curves obtained from the experiments were then used to calculate the material constants. The different constitutive models, namely strain-compensated Arrhenius-type and Arrhenius-type, and ANN, were evaluated and compared with the experimental results using correlation coefficient and relative percentage error, for the entire range of experimental parameters. 2. Experimental Procedure

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The Fe-base alloys with the composition Fe-28Ni-17Co-11.5Al-2.5Ta-0.05B (at.%) were prepared in the form of buttons measuring 10 mm diameter, 2 mm height and weighing 25 g by vacuum arc remelting and subsequently melted in a suction casting machine under an inert gas atmosphere into bars of 10 mm diameter. The samples were solutionized at 1200℃ for 2 h under an inert atmosphere. The isothermal compression tests were carried out on samples using Gleeble – 3800 thermomechanical simulator adhering to ASTM E209 (L = 9 mm, D = 6 mm, L/D = 1.5).

To determine the temperature in the midsection of the sample a thermocouple was welded to it by spot welding. With a view to maintaining the same temperature across the cross- section and to

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decrease the friction level between the two tungsten carbide anvils, two thin graphite sheets were interposed between them. These graphite sheets act as a lubricant and also prevent the sample

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from sticking to the anvils during quenching.

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Fig. 1: Dimension details of the sample (a) Before hot deformation; (b) After hot deformation

Fig 2: Gleeble – 3800 thermomechanical simulator test conditions

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The sample dimension of the sample are shown in Fig. 1.The samples were solutionized by heating them at the rate of 5℃/s to different temperatures and holding them at these temperatures for 2 min in order to maintain the same temperature right through as shown in Fig. 2. The alloy samples were deformed at 𝜀̇ = 0.01, 0.1, 1.0, 10.0 s-1 and T = 1323, 1373, 1423 and 1473 K. The tests were performed under vacuum so as to eliminate the specimens from undergoing oxidation. Simultaneously the displacement, temperature, stress, strain and time were measured by the Gleeble-3800 thermomechanical simulator. 3. Results and discussion

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3.1 Materials flow behavior The true stress and true strain values emerging from the high temperature compression tests for various strain rates and temperatures have been plotted in the form of curves in Fig. 3. It is evident from the curves that the strain rate, strain and temperature influence the flow curves and their shapes. All flow curves show a similar trend. During the initial stages, with an increase in the true

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strain, the true stress increases. This is attributed to strain hardening. Subsequently, it remains constant or starts declining due to the onset of dynamic recrystallization [26]. True stress bears a

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non-linear relationship with the temperature, strain and strain rate.

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Fig. 3 The flow curves for NCATB shape memory alloy at various temperatures and strain rates: (a) 1323 K, (b) 1373 K, (c) 1423 K, and (d) 1473 K.

3.2 Arrhenius-type model The Arrhenius equation for hot deformation is given by equation (1). Based on the stress value, the correlation between the strain rate (ε̇ ), temperature (T) and true stress can be represented by a power law (Eq. 2) equation, an exponential law (Eq. 3) equation and a hyperbolic-sine equation

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(Eq. 4), respectively. Q

ε̇ = A𝑓(σ) exp (− RT)

when ασ > 1.2,

𝑓(σ) = exp (βσ)

For all values of ασ ,

𝑓(σ) = [sinh(ασ)]𝑛

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𝑓(σ) = σ𝑛1

when ασ < 0.8,

(1) (2) (3) (4)

β

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Where α, β, n1, n and A are material constants, while α = 𝑛 . 1

constant (8.314 J/mol/K).

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Here Q refers to the energy for activation of deformation (kJ/mol), whereas R, the universal gas

The effect of strain rate and temperature on material flow is represented by Zener - Hollomon

𝑄

(5)

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𝑍 = 𝜀̇ exp ( ) 𝑅𝑇

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parameter (Z) (Eq. 5).

Eq.4 can be modified to Eq.6 to represent it for all stress values. 𝑄

𝑍 = 𝜀̇ exp ( ) = 𝐴[sinh(𝛼𝜎)]𝑛 𝑅𝑇

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(6)

The true stress for any specific strain value may be represented in terms of Zener - Hollomon parameter using the hyberbolic sine law and is given by Eq. 7.

𝜎=

1 𝛼

𝑍 1/𝑛

{( ) 𝐴

𝑍 2/𝑛

+ [( ) 𝐴

1/2

+ 1]

}

(7)

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The material constants α, n, A and Q values are, respectively, 0.007, 3.8878, 3.19×1014 and 412 kJ/mol, respectively. On substituting these values into Eq. 6 and Eq. 7, we obtain Eq. 8. This equation may be used to forecast the flow behaviour at different strain rates and temperatures.

𝜎=

1 0.007

ln {(

Z 3.19×1014

)

412000

where Z = 𝜀̇ exp (

1/3.8878

8.314T

+ [(

Z

3.19×1014

)

2/3.8878

1/2

+ 1]

}

(8)

)

Using the above equations, the flow curves are plotted (Fig. 4) for experimental values and

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Arrhenius-type model.

Fig. 4 Comparison of flow stress values from experiments (solid lines) with those predicted (symbols) by Arrhenius-type model for NCATB alloy at different temperatures and strain rates: a. 1323 K; b. 1373 K; c. 1423 K; and d. 1473 K.

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3.3 Strain-compensated Arrhenius-type model The Arrhenius-type equation (Eq.8) does not consider the influence of strain, which results in less accurate results, as shown in Fig. 4. Strain was therefore built into the model by Lin et al. [27] by expressing A, Q, n, and α as a function of strain. The modified model is therefore known as straincompensated Arrhenius-type model. The α, n, Q and A values are determined for various values of strain (ϵ = 0.05-0.65).Polynomial equations (Fig. 5) are fitted for these material constants. The change in α, n, Q and ln A with true strain could be expressed by an eighth order polynomial as in Eq. 9. Table 1 gives the values of

α = g(ε) = α0+ α1ε+ α2ε2+ α3ε3+ α4ε4+ α5ε5+ α6ε6+ α7ε7+ α8ε8 n= h(ε) = n0+n1ε+n2ε2+n3ε3+n4ε4+n5ε5+n6ε6+n7ε7+n8ε8

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Q= j(ε) = Q0+Q1ε+Q2ε2+Q3ε3+Q4ε4+Q5ε5+Q6ε6+Q7ε7+Q8ε8

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the coefficients for polynomial equations of the eight order.

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ln A= ln f(ε) = A0+A1ε+A2ε2+A3ε3+A4ε4+A5ε5+A6ε6+A7ε7+A8ε8

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(9)

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Fig. 5 Change of material constants with strain: a. α; b. n; c. Q; and d. ln A.

Table 1 Polynomial coefficients for α, n, ln A and Q for the Fe-base SMA

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α α0 = 0.0108 α1 = -0.0863 α2 = 1.0280 α3 = -6.9712 α4 = 28.5912 α5 = -71.6467 α6 = 107.1140 α7 = -87.6515 α8 = 30.1763 0.9966

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n n0 = 3.60 n1 = 20.96 n2 =-363.23 n3 = 2903.36 n4 = -12555.86 n5 = 31416.75 n6 = -45624.28 n7 = 35764.52 n8 = -11712.75 0.9124

Q Q0 = 400.59 Q1 = -1691.4 Q2 = 46503.38 Q3 = -467932.81 Q4 = 2445320 Q5 = -7221190 Q6 = 12120000 Q7 = -10776900 Q8 = 3941570 0.9797

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ln A A0 = 32.17 A1 = -139.51 A2 = 3942.97 A3 = -39693.45 A4 = 206827.12 A5 = -608927.65 A6 =1019310 A7 = -904262.58 A8 = 330064.43 0.9814

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Fig. 6 Comparison of flow stress values from experiments (solid lines)with those predicted (symbols) by strain-compensated Arrhenius-type equation for NCATB alloy for different temperatures and strain rates: a. 1323 K; b. 1373 K; c. 1423 K; and d. 1473 K.

g(ε)

ln {(

𝑍

f(ε)

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𝜎=

1

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When the change of material constants with strain is considered, Eq. 8 gets modified to give rise to Eq. 11. 1/ℎ(𝜀)

)

where 𝑍 = 𝜀̇ exp (

+ [(

j(ε) 8.314T

𝑍

f(ε)

2/ℎ(𝜀)

)

1/2

+ 1]

}

(11)

)

By inserting these constants into Eq. 11, the true stress-true strain values can be evaluated for the relevant strain, strain rate and temperature ranges of the experiments. Fig.6 compares the results from experiments with those forecast by the strain-compensated Arrhenius-type equation for different temperatures (1323 K, 1373 K, 1423 K and 1473 K) and strain rates (0.01s-1, 0.1 s-1, 1 s-1

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and 10 s-1). As is evident, the proposed constitutive equation gives better results compared to that by Arrhenius-type equation almost under most test conditions. 3.4 Artificial neural network model In this investigation, The back-propagation (BP) algorithm is widely used to process complex non-linear relationships among several variables. The back-propagation (BP) algorithm is used to understand the relationship between the inputs and outputs, since it is a means to adjusting the biases and weights by using declining gradient to decrease the target error [28]. It possesses a better representational capability to deal with complex and non-linear relationships [29]. A typical artificial neural network architecture employed in this work is illustrated in Fig. 7. The

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network has three layers: input, hidden and output layers. The input layer embodies information related to temperature (T), strain (ε), strain rate (ε̇ ), while the output layer stress (σ). In the present study, the architecture of ANN becomes 3-10-1, 3 refers to the input values, 10 refers to the number of hidden layer neurons, and 1 refers to the output values.

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Normalization is carried out for the input and output variables in ANN in the range 0-1. Eq.10 is commonly used for unification of the corresponding data in the neural network modelling [30]. XMax −XMin

)

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X−XMin

X ′ = 0.1 +0.8× (

(10)

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where X refers to the original values, while X ′ the unified values corresponding to X. This equation is employed in the present work to bring together the values of stress, temperature and strain. However, it was found from the study that influence of strain rate is more predominant

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compared with those of stress and temperature. The lowest value of strain observed after unification was found to be much smaller by the ANN. As a consequence, the following equation

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is used to unify the strain rate values. 𝜀̇′ = 0.1 + 0.8× (

log ε̇ −log ε̇ Min

)

(11)

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log ε̇ Max −log ε̇ Min

From among total experimental values, 208 data sets were chosen from the flow curves (for ε = 0.05-0.65 at an interval of 0.05). Among these, 187 random values (90%) were used to train the ANN model, while the remaining 21 (10%) values were subsequently used to test the performance of the ANN model. The study was carried out using the neural network toolbox available in the MATLAB R2018a software.

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Fig. 7 Schematic artificial neural network architecture

The structural variables, i.e. the number of hidden layers, transfer function, training function and number of neurons for hidden layer, play a significant role in accuracy and convergence speed of the model. In the present study, one hidden layer is used for higher accuracy. The functions “Tan-

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sigmoid” and “pure linear” were used as the associated transfer functions for the hidden layer and output layer, respectively. Additionally, the training function used in the present study is

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“TrainLM”. The presence of less number of neurons makes the ANN model inadequate to predict the flow behaviour accurately. In contrast, many neurons may slow down the convergence rates or

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over-fit the data. The number of neurons set for the present work is 10, which was optimized between the accuracy and the convergence rate. The parameters used for the present study were tabulated in Table 2. Fig. 8 illustrates the compares the experimental values and those in ANN

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model drawn in true stress-true strain curves for different strain rates and temperatures. Table 3 compares the accuracy of prediction of flow stress obtained by the ANN model among training

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data, testing data and overall data points through correlation coefficient (R).

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Table 2 Training variables for the neural network Contents Feed-forward back propagation TrainLM MSE Learngdm 10000 1×10-6 Tan-sigmoid Liner (purelin) 10

Table 3 Linear fitting results of data points in ANN model

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Value of intercept 0.86468 5.4576 1.3221

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Training data point Testing data point Overall data point

Value of slope 0.9936 0.9585 0.9903

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Source of data

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Name of Network Parameters Network Training function Performance function Adaption learning function Training epoch Goal Transfer function of hidden layer Transfer function of output No. of neurons

R

0.9966 0.9964 0.9965

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Fig. 8 Comparison of flow stresses obtained from experiments (solid lines) with those predicted (symbols – Rectangular symbols for testing and remaining symbols for training) by ANN model for NCATB alloy for various temperatures and strain rates: a. 1323 K; b. 1373 K; c. 1423 K; and d.1473 K. 3.5 Comparison of constitutive models and ANN model

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The capability for prediction of the constitutive models (Arrhenius-type and strain-compensated Arrhenius-type equations) and ANN model were quantified using a standard statistical parameter,

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namely R. It is expressed as: R=

̅ ̅ ∑𝑁 𝑖=1(𝐸𝑖 − 𝐸)(𝑃𝑖 −𝑃)

(12)

̅ )2 (𝑃 −𝑃 ̅ )2 √∑𝑁 (𝐸 −𝐸 𝑖 𝑖=1 𝑖

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Where N = No. of stress-strain samples, Pi= Predicted sample value, Ei= Sample of experimental

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̅andE ̅are the mean values of P and E, respectively. value,P

Fig. 9 Comparison of flow stress values from experiments with those predicted by Arrhenius-type

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equation for various strain rates [(a) 0.01 s-1 (b) 10s-1 (c) Overall full strain rate range (0.01, 0.1, 1 and 10 s-1)], Strain-compensated Arrhenius-type equation; [(d) 0.01 s-1 (e) 10s-1 (f) Overall full

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strain rate range (0.01, 0.1, 1 and 10 s-1)] and ANN model [(g) 0.01 s-1 (h) 10s-1 (i) Overall full strain rate range (0.01, 0.1, 1 and 10 s-1)]. As can be seen in Fig. 9(a & b), as per Arrhenius-type model, the flow stress values from experiments match well with those predicted, especially at a lower strain rate (0.01s-1) as compared to a higher strain rate (10 s-1). On the other hand, as per the strain-compensated Arrhenius-type equation, the flow stress predicted values match well with those obtained from experiments, especially at a higher strain rate (10 s-1) as compared to a lower strain rate (0.01 s-1)

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(Fig. 9(d & e)). But as per ANN model, the flow stress predicted values match well with those from the experiments at lower and higher strain rates (Fig.9 (g & h)). It is evident that as per Fig. 9(i) the ANN model predicts flow stress more accurately when compared to Arrhenius-type model (Fig. 9(c)) and strain-compensated Arrhenius-type equation (Fig. 9(f)) for values from experiments for the whole range. The preciseness of the models was again evaluated by the relative errors between the experimental and predicted values, which were determined by Eq. (13). Ei −Pi Ei

) × 100 %

(13)

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Relative percentage error =(

Where 𝐸𝑖 refers to the flow stress values from the experiments, while Pi values predicted by the

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model.

Fig. 10 Distribution of relative error in: a. Arrhenius-type equation; b. Strain-compensated

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Arrhenius-type equation; and c. ANN model.

It was observed that the relative error resulting from Arrhenius-type equation is in the range of -35.81 to 5.49%, while that for the strain-compensated Arrhenius-type equation -12 to 23%, and

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for the ANN model -11.53 to 3.9%. In Arrhenius-type model, approximately 40% of datasets have -20 to -25% of relative percentage error. In strain-compensated Arrhenius-type model,

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approximately 45% of datasets have -5 to -10% relative percentage error. In ANN model, 80% of datasets have -2 to 2% relative percentage error as shown in Fig.10. We can therefore conclude that, out of these models well-trained ANN model is more effective among the models and can therefore make the flow stress prediction more accurately. 4. Conclusion The flow curve data generated from isothermal high temperature compression experiments performed on a Gleeble-3800 thermomechanical equipment over a large strain rate range (0.01 – 16

10 s-1) and temperature range (1323-1473 K) were used to generate the Arrhenius-type equation, the strain-compensated Arrhenius-type and ANN models for the ferrous alloy. A comparison of the capability of the models to closely forecast the high temperature deformation characteristics of the alloy was made. The following are the conclusions obtained from the studies: 1. The values of correlation coefficient for different strain rate values (0.01, 0.1, 1 and 10 s-1) for the Arrhenius-type equation, the strain-compensated Arrhenius-type equation and the ANN model were found to be 0.9848, 0.9863 and 0.9965, respectively. 2. Referring to the Arrhenius-type equation, the flow stress predicted values match more closely with the experimental values at lower strain rates when compared to those at higher strain

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rates. The values of R for 𝜀̇ = 0.01-10s-1 were found to be 0.9557 and 0.9395, respectively. Approximately 40 % of dataset values have relative error percentage varying from 20 - 25% in comparison with the experimental values.

3. The flow stress predicted values by the strain-compensated Arrhenius-type equation match

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well with those from the experiments for higher rates of strain as compared to lower rates of strain. The values of R for 𝜀̇ = 0.01-10s-1 were found to be 0.9577 and 0.9748, respectively.

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Approximately 45% of dataset values have relative error percentage varying from 5 - 10 % when compared with the experimental values.

4. As per ANN model, the flow stress predicted values match closely with the experimental

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values at lower and higher strain rates. The values of R corresponding to 0.01 s-1 and 10 s-1 were found to be 0.9722 and 0.9771, respectively. Approximately 80% of dataset values have

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relative error percentage varying from ± 2% when compared with the experimental values. 5. The ANN model for the SMA accurately predicts the high temperature deformation characteristics over a large range of temperatures and strain rates. It has been found to be

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superior to other constitutive models in that it is more precise and consistent and faster. It can also be used as a tool to study and forecast the hot deformation behaviour of other alloys in the

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field of material science.

AUTHOR AGREEMENT FORM This statement is to certify that all Authors have seen and approved the manuscript being submitted. We warrant that the article is the Authors' original work. We warrant that the article has not received prior publication and is not under consideration for publication elsewhere. On

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behalf of all Co-Authors, the corresponding Author shall bear full responsibility for the submission. This research has not been submitted for publication nor has it been published in whole or in part elsewhere. We attest to the fact that all Authors listed on the title page have contributed significantly to the work, have read the manuscript, attest to the validity and legitimacy of the data and its interpretation, and agree to its submission to the Materials Today Communication Journal. All authors agree that author list is correct in its content and order and that no modification to the author list can be made without the formal approval of the Editor-in-Chief, and all authors accept that the Editor-in-Chief's decisions over acceptance or rejection or in the event of any breach of

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the Principles of Ethical Publishing in the Materials Today Communication Journal being discovered of retraction are final.

No additional authors will be added post submission, unless editors receive agreement from all

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authors and detailed information is supplied as to why the author list should be amended.

Acknowledgements

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We have no conflict of interest to declare.

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This work was supported by Department of Science and Technology, New Delhi, India (Project

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No. EEQ/2016/000500).

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