Ravindranadh BOBBILI*,V.MADHU,A.K.GOGIA

Defence Metallurgical Research Laboratory,Hyderabad 500058,India

Neural network modeling to evaluate the dynamic fow stress of high strength armor steels under high strain rate compression

Ravindranadh BOBBILI*,V.MADHU,A.K.GOGIA

Defence Metallurgical Research Laboratory,Hyderabad 500058,India

An artifcial neural network(ANN)constitutive model is developed for high strength armor steel tempered at 500°C,600°C and 650°C based on high strain rate data generated from split Hopkinson pressure bar(SHPB)experiments.A new neural network confguration consisting of both training and validation is effectively employed to predict fow stress.Tempering temperature,strain rate and strain are considered as inputs,whereas fow stress is taken as output of the neural network.A comparative study on Johnson-Cook(J-C)model and neural network model is performed.It was observed that the developed neural network model could predict fow stress under various strain rates and tempering temperatures.The experimental stress-strain data obtained from high strain rate compression tests using SHPB,over a range of tempering temperatures(500-650°C),strains(0.05-0.2)and strain rates(1000-5500/s)are employed to formulate J-C model to predict the high strain rate deformation behavior of high strength armor steels.The J-C model and the back-propagation ANN model were developed to predict the high strain rate deformation behavior of high strength armor steel and their predictability is evaluated in terms of correlation coeffcient(R)and average absolute relative error(AARE).R and AARE for the J-C model are found to be 0.7461 and 27.624%,respectively,while R and AARE for the ANN model are 0.9995 and 2.58%,respectively.It was observed that the predictions by ANN model are in consistence with the experimental data for all tempering temperatures.

Artifcial neural network;High strength armor steel;J-C model;Tempering;SHPB

1.Introduction

High strength steels are of important candidate materials for a wide variety of engineering applications due to their superior mechanical properties.As these materials undergo severe plastic deformation conditions during their service period,it is essential to study the material deformation characteristics under high strain rate conditions for applications involving high strain rate deformations[1-6].The data obtained will be helpful for designing the products as well as developing the constitutive strength models of the materials. An iterative procedure involving dynamic material testing and computer modeling may reduce the time and expense required for the development of advanced materials for applications such as armor.Characterization of deformation,fracture and load carrying capability of material subjected to high strain rate is paramount for optimum material selection for design of armor materials which experience high strain rate dynamic deformation.

High strain rate deformation behavior of materials is always related with various mechanisms,such as strain hardening and thermal softening,etc.Constitutive relationship of materials is the basic function of fow stress and parameters, such as strain,strain rate and deformation temperature.But, during the high strain rate deformation,many parameters infuence the dynamic fow stress of materials.The effect of these parameters on the dynamic fow stress is extremelynon-linear.So,it is quite complex to develop the constitutive relationship model since it is highly non-linear and complex mapping.A large number of publications are available, explaining some of the above aspects along with high strain rate stress-strain data of various materials.

The effect of strain rate on properties,viz.fow stress,strain rate sensitivity,etc.,varies for each material.Lee et al.[7] observed an increase in fow stress with increase in strain rate in low,medium and high carbon steels.It has been found that increased carbon content enhances the dynamic fow stress of steel.Mohr[8]obtained the accurate stress-strain curves from SHPB testing by measuring input and output forces and velocities at the boundaries of specimen.Mousavi Anjidan[9]predicted fow stress of SS 304 under cold and warm compression tests by adopting neural network and genetic algorithm models.The results showed that temperature is a signifcant variable and the strain has less infuence on fow stress.Ji et al.[10]carried out the hot compression tests on Aermet 100 steel by using Gleeble-3800 thermo-mechanical simulator to generate stress-strain data,in a temperature range from 1073 K to 1473 K and at the strain rates of 0.01-50 s-1. The Arrhenius constitutive model and a feed forward artifcial neural network(ANN)model were developed to predict the high temperature deformation behavior of the abovementioned material.ANN was found to be superior for modeling the high temperature deformation behavior of materials.Han et al.[11]performed a comparative study on constitutive relationship of 904L austenitic steel during hot deformation based on Arrhenius and ANN models.Experimental data were gathered from hot compression tests on Gleeble-1500D thermo-mechanicalsimulatorto generate stress-strain data in a temperature range from 1000°C to 1150°C and at the strain rates of 0.01-10 s-1.The back propagation neural network model was proved to be more accurate and effcient in investigating the compressive deformation behavior at higher temperatures.Sun et al.[12] employed ANN model to develop a constitutive model for the hot compression of Ti600 alloy.These tests were performed on Gleeble-1500 thermo-mechanical simulator to generate stress-strain data in a temperature range from 800°C to 1100°C and at the strain rates of 0.001-10 s-1.ANN model provided a simple and effcient way to develop constitutive relationship for Ti600 alloy.Lin et al.[13]studied the compressive behavior of aluminium 2124-T851 alloy under the strain rates of 0.01-10 s-1and temperature range from 653 K to 743 K using Gleeble-1500 thermo-simulation machine.A modifed constitutive model accommodating the effects of material behavior was proposed.Hou et al.[14] carried out high strain rate experiments using SHPB on Mg-Gd-Y alloy over a range of temperatures.A modifed J-C model was proposed to predict the dynamic response of this material in a wide range of strain rates and temperatures. Gupta[15]developed various semi-empirical models(Johnson-Cook model,modifed Zerilli-Armstrong model and Arrhenius model)to study the effects of strain,strain rate and temperature.Tensile tests were performed on Austenitic stainless steel 316 using UTM machine at various strain rates (0.1-0.0001 s-1)and temperatures (323-623 K).A comparative study was undertaken among various constitutive models and ANN model.

The available literature has so far dealt with dynamic material characterization of various materials,their testing methodologies under different loading conditions and microstructural analysis of various steels.So far no attempt has been made to study the effect of tempering temperatures of high strength armor steel on dynamic properties using J-C and ANN models.Since there has been limited data available in literature regarding the constitutive behavior of this material, an effort has been made to evaluate the effect of tempering temperature on material parameters of the Johnson-Cook and ANN models.This generated high strain rate data will be useful to correlate the ballistic behaviors of these steels at different conditions.The objective of the present study is to develop ANN model for predicting the dynamic fow stress of tempered high strength armor steels during high strain rate deformation.

2.Experimental methods

2.1.Materials and test setup

The alloy under study contains 0.32%C,0.25%Si,0.6% Mn,1.5%Cr,1.7%Ni,0.4%Mo,and balance Fe,named as high strength armor steel.Armor steel plates were tempered at 500°C,600°C and 650°C for 2 h followed by cooling to room temperature in air.High strength armor steel samples with lengths of 3 mm,4 mm and 5 mm and diameters of 6 mm,8 mm and 10 mm were prepared with l/d ratio of 0.5 to carry out trials on SHPB system.The experimental stress-strain data were obtained from high strain rate compression tests using splitHopkinson pressure bar (SHPB),over a wide range of strains(0.1-0.3)and strain rates(1000-5500/s).The whole SHPB setup consists of(a) pressure bars,(b)gas gun which propels a striker bar for producing thecompressive wave,(c)strain gage for measuring the waves,(d)associated mounting and alignment hardwares,and(e)associated instrumentation and data acquisition system(Fig.1).SHPB apparatus(Fig.2)has two pressure bars,one called input or incident bar and another called output or transmitted bar[13,14].These pressure barsare made of material having yield strength higher than that of the material to be tested.The specimen to be tested is sandwiched between these 2 bars.The yield strength of the pressure bar determines the maximum stress attainable within the deforming specimen,because the cross section of the specimen approaches that of the pressure bar during deformation.A rectangular compression wave of welldefned amplitude and length is generated in the incident bar when the striker bar strikes it.When this wave reaches the specimen,part of the pressure pulse is transmitted into the specimen and into output bar,and part is refected back to the incident bar.Fig.3 represents incident wave,refected wave and transmitted wave(strain history).High strain rate stress-strain data can be generated by measuring the strains in the incident and transmitted bars with the help of strain gages by using one-dimensional wave propagation analysis.

Fig.2.Experimental setup of Split Hopkinson pressure bar(SHPB)system.

Fig.3.Strain-time graph of high strength armor steel alloy tested at a strain rate of 1000/s in input and output bar.

Length and diameter of the samples were changed to vary the strain rate.The sample ends were machined fat and parallel to the bar ends to provide good contact with them.Also the surfaces of the samples were polished to provide standard surface fnish of 5 micron.Each sample was sandwiched between input and output bars.This frictional constraint can be greatly reduced by applying a lubricant thin flm.Hence a thin grease layer of Cu-Mo paste was always applied to the sample ends to reduce friction between the bar and sample during compression test.The pressure wave was generated by using a striker bar(projectile)to impact the input(incident) pressure bar.The striker bar was propelled by a gas gun system attached at one end.The strain gages in conjunction with amplifers and associated instrumentation record these wave pulses.Since the specimen deforms uniformly,the strain rate in the specimen are directly proportional to the amplitude of the refected wave(εr).

Strain-rate generated in the specimen is

Hence the strain in the specimen is

Stress in the specimen can be calculated as whereAbandASare the areas of the bar and specimens, respectively;lis the length of specimen;cis wave speed;εtis the strain in the transmitted bar;andEis the elastic modulus of the pressure bar.

2.2.Johnson-Cook model

High-strain rate plastic deformation of materials can be described by various constitutive equations that basically attempt to address dependence of stress on strain,strain rate and temperature.In this regard,stress can be schematically presented as

There are a number of equations that have been proposed to describe the plastic behavior of materials as a function of strain rate and temperature.At low strain rates,metals are known to work harden along the well-known relationship known as parabolic hardening and expressed as σ=σo+kεn，where σois the yield stress,nis work hardening exponent,andkis pre-exponential factor.

The effect of strain rate on strength is generally expressed as σ∝ln˙ε,but the above relationship breaks down at strain rate above 102s-1.

The effects of temperature on the fow stress can be represented by

whereTmis the melting temperature;andTris the reference temperature at which σris the reference stress.

The dynamic fow stress[16-21]depicting the effect of various parameters has been expressed by Johnson-Cook model as.whereAis the yield stress;Bandnrepresent the effect of strain hardening;Cis the strain rate constant;ε is the equivalent plastic strain;˙ε is the strain rate;˙ε*is the dimensionless plastic strain rate represented as˙ε/˙ε0for˙ε0=1 s-1;T*is the homologous temperature referred as(T-Troom)/(Tmelt-Troom); andmis the thermal softening factor.Thus,the term present in frst,second and third bracket in Eq.(4)represents strain, strain rate and temperature effect,respectively.The J-C model is independent of pressure.

Table 1Johnson-Cook model constants for various steels.

At reference strain rate and reference temperature,the functions of strain rate hardening and thermal softening are equal to unity.The J-C model is simplifed as follows: where A is the yield stress which can be directly obtained from observing the strain-stress curve.

Plotting a line between ln ε and ln(σ-A)at the reference strain rate and reference temperature givesBandnin Eq.(7). Strain rate sensitivity(C)is determined as the slope of linear ft of log(strain rate)vs dynamic fow stress/static stress using high strain rate data corresponding to a strain of 10%.The above constants are provided in Table 1.

2.3.Artifcial neural network approach

Neural networks are commonly employed in data prediction,categorization and data fltering applications.Artificial neural networks(ANNs)imitate human brains to know the interaction between inputs and outputs through training.An ANN consists of neurons along with links of variable weights. Multi-layer ANN possesses input layers,hidden layers and output layers.The input layer frst receives data and conveys it to hidden layer for processing.The hidden layer locates between input and output layers.The processed data will be delivered as response to the output layer.The output layer receives the responses from the hidden layer and generates an output vector.Each layer has got a number of neurons connected by links with adaptable weights.These weights are adjusted during the training procedure.The inputs into a neuron are multiplied by their respective connection weights, summed together and a bias is added to the sum.This sum is converted through a transfer function to produce a single output.The nonlinear logarithmic sigmoid activation function was adopted in the hidden and output layers.The actual output obtained is compared to the required output to compute an error.The error for hidden layers is calculated by propagating back the error found out for the output layer;this technique is called back-propagation algorithm.

Fig.4.The schematic of the ANN architecture.

3.Results and discussion

The experimental data were split into three sets,70%for the training set,15%for the verifcation set and 15%for the test set in ANN model.The input data(strain rate,strain and tempering temperature)and output data(fow stress)were standardized in the range(Fig.4).The network model consists of ten hidden layers.To demonstrate the infuence of network variables,the number of hidden layer neurons was varied from 10 to 40.It has been noticed that the predicted results are reasonable and accurate,with 15 neurons in each hidden layer. The performance of the network also relies on learning parameters,such as the number of training epochs and the momentum,etc.To understand thesignifcance ofthese parameters,the number of epochs was varied from 1000 to 10,000,the learning rate was varied from 0.1 to 0.9,and the momentum rate was varied from 0.1 to 0.8.It shows that the momentum rate does not exhibit a substantial infuence on performance of the network.It is established that the optimum number of epochs is about 12,000,the number of neurons in each hidden layer is 15,the number of hidden layers is 10,and the learning rate is 0.8,with a momentum of 0.7 in all layers. Mean square error(MSE)of desired and predicted data was determined when MSE attained a minimum value of 0.001. Fig.5 shows MSE for various hidden neurons.It is found that ANN with 10 hidden layers and 15 hidden units has MSE.The results were obtained from a network with 15 neurons in a hidden layer and 1240 iterations.

Fig.5.Infuence of hidden neurons on the network performance.

Fig.6.Comparison between J-C Model,ANN Model and experimental fow stress of high strength armor steel tempered at 500°C by J-C model at strain rates.

Fig.3 gives strain-time graph received in incident and output bars for one of the samples tested at a strain rate of 1000 s-1.The signals were processed by applying a dispersion correction to the signals and then time shifted to bring the three pulses(incident and refected pulses in input bar and transmitted pulse from output bar)into coincidence at the sample-bar interfaces.The strain rate,stress and strain in the sample were calculated using equations described in Section 1. True stress-true strain curves for the specimen tested at varied strain rates are shown in Figs.6-8.The stress-strain curve is generated from test data received from specimens tested at strain rates of 1000 s-1,2000 s-1,3000 s-1,4000 s-1and5000 s-1.Based on the results,the following observations can be made:rapid increase in yield stress(the point where stressstrain curve deviates from linearity)has been observed to occur with increase in strain rate.This increase is nearly 100 MPa as strain rate increases from 1000 s-1to 5000 s-1. The increase in yield point with increasing strain rate occurs because the time available for the dislocations to jump over the barrier is shortened at higher strain rates.As a result,the dislocations start piling up at the barrier and consequently higher stress is required to overcome the barriers for continued motion of the dislocations.

Fig.7.Comparison between J-C Model,ANN Model and experimental fow stress of high strength armor steel tempered at 600°C by J-C model at strain rates.

Fig.8.Comparison between J-C Model,ANN Model and experimental fow stress of high strength armor steel tempered at 650°C by J-C model at strain rates.

The experimental data obtained from the high strain rate compression tests on split Hopkinson pressure bar,in a wide range of tempering temperatures(500°C-650°C)and strain rates(1000 s-1-5500 s-1),were employed to develop J-C model and ANN model for high strength armor steels.Fig.6 depicts the comparison of the experimental results with the predicted valuesat variousstrain ratesand tempering temperatures based on J-C strength model.It is observed that the predicted fow stress values obtained on J-C model are not consistent with the experimental values,particularly at high tempering temperatures;and the predicted values are smaller than the experimental results.So,J-C model is not so adequate in predicting the fow stress value in high tempering temperature region.The predicting performance of the J-C model was evaluated by comparing the experimental and predicted data,as shown in Fig.6.It was noticed that the J-C model could predict the experimental data only in the intermediate temperature range(500°C-600°C).This variation may be attributed to the error introduced by the ftting of the material constants at some conditions and adiabatic temperature increment due to plastic deformation.

Fig.9.Plot of predicted vs.experimental for modifed J-C model.

The J-C model and the back-propagation ANN model were developed to predict the high strain rate deformation behavior of high strength armor steels,and their predictability was evaluated in terms of correlation coeffcient(R)and average absolute relative error(AARE).R and AARE for the J-C model are found to be 0.8461 and 10.624%,respectively (Fig.9),while R and AARE for the ANN model are 0.9995 and 2.58%,respectively(Fig.10).Fig.10 illustrates the predicted fow stress by ANN model versus measured value for testing set.The predicted fow stresses of J-C model are given in Figs.6 and 7.It is found that that the relative error obtained from the ANN model was observed to vary from 1.2%to 4.5%,while it was in the range of 4.2%-10.6%for J-C model.Hence,the data obtained were better in the ANN model compared to the J-C model.It shows that the developed ANN model can offer an effcient prediction of fow stress at the tempering temperatures of 500°C-650°C and the strain rates of 1000 s-1to 5500 s-1.

4.Conclusions

This paper has made an attempt to study the comparison of the results obtained from J-C model and ANN model with experimental values.The following conclusions are drawn:

1)The J-C model and the back-propagation ANN model

Fig.10.Plot of predicted vs.experimental for ANN model.

were developed to predict the high strain rate deformation behavior of high strength armor steels and their predictability was evaluated in terms of correlation coeffcient (R)and average absolute relative error(AARE).R and AARE for the J-C model are found to be 0.8461 and 10.624%,respectively,while R and AARE for the ANN model are 0.9995 and 2.58%,respectively.

2)The established ANN model can effectively predict the experimental data over a wider range of tempering temperatures and strain rates.This represents that ANN model has superior capability to model the dynamic behavior of materials.This method circumvents the problems related to constitutive models that involve the determination of more number of constants.

3)The validation tests have also been conducted to verify the results obtained by ANN technique.The predictions of the ANN model were in good agreement with experimental data obtained from SHPB tests.

Acknowledgments

The authors would like to thank Defence Research and Development Organization,India for fnancial help in carrying out the experiments.

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Received 18 April 2014;revised 19 June 2014;accepted 24 June 2014 Available online 23 August 2014

*Corresponding author.Tel.:+40 24346332;fax:+40 24342252.

E-mail addresses:ravindranadhbobbili@gmail.com,ravindranadhb@gmail. com(R.BOBBILI).

Peer review under responsibility of China Ordnance Society.

http://dx.doi.org/10.1016/j.dt.2014.06.012

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Copyright?2014,China Ordnance Society.Production and hosting by Elsevier B.V.All rights reserved.