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Neural network modeling to evaluate the dynamic flow stress of high strength armor steels under high strain rate compression
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Available online at www.sciencedirect.com
ScienceDirect Defence Technology 10 (2014) 334e342 www.elsevier.com/locate/dt
Neural network modeling to evaluate the dynamic flow 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 Received 18 April 2014; revised 19 June 2014; accepted 24 June 2014 Available online 23 August 2014
Abstract An artificial 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 configuration consisting of both training and validation is effectively employed to predict flow stress. Tempering temperature, strain rate and strain are considered as inputs, whereas flow stress is taken as output of the neural network. A comparative study on JohnsoneCook (JeC) model and neural network model is performed. It was observed that the developed neural network model could predict flow stress under various strain rates and tempering temperatures. The experimental stressestrain data obtained from high strain rate compression tests using SHPB, over a range of tempering temperatures (500e650 C), strains (0.05e0.2) and strain rates (1000e5500/s) are employed to formulate JeC 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 coefficient (R) and average absolute relative error (AARE). R and AARE for the JeC 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. Copyright © 2014, China Ordnance Society. Production and hosting by Elsevier B.V. All rights reserved.
Keywords: Artificial neural network; High strength armor steel; JeC 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 [1e6]. The data obtained will be helpful for designing the products as well as developing the constitutive strength models of the materials. * Corresponding author. Tel.: þ40 24346332; fax: þ40 24342252. E-mail addresses: [email protected], ravindranadhb@gmail. com (R. BOBBILI). Peer review under responsibility of China Ordnance Society.
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 flow stress and parameters, such as strain, strain rate and deformation temperature. But, during the high strain rate deformation, many parameters influence the dynamic flow stress of materials. The effect of these parameters on the dynamic flow stress is extremely
http://dx.doi.org/10.1016/j.dt.2014.06.012 2214-9147/Copyright © 2014, China Ordnance Society. Production and hosting by Elsevier B.V. All rights reserved.
R. BOBBILI et al. / Defence Technology 10 (2014) 334e342
non-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. flow stress, strain rate sensitivity, etc., varies for each material. Lee et al. [7] observed an increase in flow stress with increase in strain rate in low, medium and high carbon steels. It has been found that increased carbon content enhances the dynamic flow 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 flow 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 significant variable and the strain has less influence on flow 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 0.01e50 s1. The Arrhenius constitutive model and a feed forward artificial 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-mechanical simulator to generate stress-strain data in a temperature range from 1000 C to 1150 C and at the strain rates of 0.01e10 s1. The back propagation neural network model was proved to be more accurate and efficient 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 0.001e10 s1. ANN model provided a simple and efficient 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 0.01e10 s1 and temperature range from 653 K to 743 K using Gleeble-1500 thermo-simulation machine. A modified constitutive model accommodating the effects of material behavior was proposed. Hou et al. [14] carried out high strain rate experiments using SHPB on MgGd-Y alloy over a range of temperatures. A modified JeC 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 (JohnsoneCook model, modified 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
335
(0.1e0.0001 s1) and temperatures (323e623 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 JeC 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 JohnsoneCook 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 flow 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 stressestrain data were obtained from high strain rate compression tests using split Hopkinson pressure bar (SHPB), over a wide range of strains (0.1e0.3) and strain rates (1000e5500/s). The whole SHPB setup consists of (a) pressure bars, (b) gas gun which propels a striker bar for producing the compressive 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 bars
Fig. 1. Schematic diagram of split Hopkinson pressure bar (SHPB).
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greatly reduced by applying a lubricant thin film. Hence a thin grease layer of CueMo 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 amplifiers 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 reflected wave (εr). Strain-rate generated in the specimen is Fig. 2. Experimental setup of Split Hopkinson pressure bar (SHPB) system.
ε_ ¼ 2cεr =l
ð1Þ
Hence the strain in the specimen is are 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 welldefined 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 reflected back to the incident bar. Fig. 3 represents incident wave, reflected 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. Length and diameter of the samples were changed to vary the strain rate. The sample ends were machined flat and parallel to the bar ends to provide good contact with them. Also the surfaces of the samples were polished to provide standard surface finish of 5 micron. Each sample was sandwiched between input and output bars. This frictional constraint can be
Fig. 3. Strain-time graph of high strength armor steel alloy tested at a strain rate of 1000/s in input and output bar.
Zt ε ¼ 2c=l
εr dt
ð2Þ
0
Stress in the specimen can be calculated as s ¼ Ab Eεt =AS
ð3Þ
where Ab and AS are the areas of the bar and specimens, respectively; l is the length of specimen; c is wave speed; εt is the strain in the transmitted bar; and E is the elastic modulus of the pressure bar. 2.2. JohnsoneCook 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 s ¼ f ðε; ε_ ; TÞ
ð4Þ
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 s ¼ so þ kεn， where so is the yield stress, n is work hardening exponent, and k is pre-exponential factor. The effect of strain rate on strength is generally expressed as s f ln_ε, but the above relationship breaks down at strain rate above 102 s1. The effects of temperature on the flow stress can be represented by m T Tr s ¼ sr 1 ð5Þ T m Tr where Tm is the melting temperature; and Tr is the reference temperature at which sr is the reference stress. The dynamic flow stress [16e21] depicting the effect of various parameters has been expressed by JohnsoneCook model as.
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Table 1 JohnsoneCook model constants for various steels. Material
A/MPa
B/MPa
n
C
High strength armor steel tempered at 500 C High strength armor steel tempered at 600 C High strength armor steel tempered at 650 C
1321
1124
0.506
0.0104
1227
1032
0.612
0.0127
1134
917
0.626
0.0114
s ¼ ðA þ Bεn Þð1 þ cln_ε* Þð1 T * m Þ
ð6Þ
where A is the yield stress; B and n represent the effect of strain hardening; C is 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_ =_ε0 for ε_ 0 ¼ 1 s1 ; T* is the homologous temperature referred as (TeTroom)/(TmelteTroom); and m is the thermal softening factor. Thus, the term present in first, second and third bracket in Eq. (4) represents strain, strain rate and temperature effect, respectively. The JeC model is independent of pressure. At reference strain rate and reference temperature, the functions of strain rate hardening and thermal softening are equal to unity. The JeC model is simplified as follows: s ¼ A þ Bεn
ð7Þ
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 (seA) at the reference strain rate and reference temperature gives B and n in Eq. (7). Strain rate sensitivity (C) is determined as the slope of linear fit of log (strain rate) vs dynamic flow stress/static stress using high strain rate data corresponding to a strain of 10%. The above constants are provided in Table 1. 2.3. Artificial neural network approach Neural networks are commonly employed in data prediction, categorization and data filtering 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 first 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
Fig. 4. The schematic of the ANN architecture.
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. 3. Results and discussion The experimental data were split into three sets, 70% for the training set, 15% for the verification set and 15% for the test set in ANN model. The input data (strain rate, strain and tempering temperature) and output data (flow stress) were standardized in the range (Fig. 4). The network model consists of ten hidden layers. To demonstrate the influence 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 the significance of these 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 influence 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. Influence of hidden neurons on the network performance.
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Fig. 6. Comparison between JeC Model, ANN Model and experimental flow stress of high strength armor steel tempered at 500 C by JeC 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 s1. The signals were processed by applying a dispersion correction to the signals and then time shifted to bring the three pulses (incident and reflected pulses in input bar and transmitted pulse from output bar) into coincidence at the
sampleebar 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. 6e8. The stress-strain curve is generated from test data received from specimens tested at strain rates of 1000 s1, 2000 s1, 3000 s1, 4000 s1 and
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339
Fig. 7. Comparison between JeC Model, ANN Model and experimental flow stress of high strength armor steel tempered at 600 C by JeC model at strain rates.
5000 s1. 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 s1 to 5000 s1. 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. The experimental data obtained from the high strain rate compression tests on split Hopkinson pressure bar, in a wide range of tempering temperatures (500 Ce650 C) and strain rates (1000 s1e5500 s1), were employed to develop JeC model and ANN model for high strength armor steels. Fig. 6 depicts the comparison of the experimental results with the predicted values at various strain rates and tempering
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Fig. 8. Comparison between JeC Model, ANN Model and experimental flow stress of high strength armor steel tempered at 650 C by JeC model at strain rates.
temperatures based on JeC strength model. It is observed that the predicted flow stress values obtained on JeC model are not consistent with the experimental values, particularly at high tempering temperatures; and the predicted values are smaller than the experimental results. So, JeC model is not so adequate in predicting the flow stress value in high tempering temperature region. The predicting performance of the JeC model was evaluated by comparing the experimental and
predicted data, as shown in Fig. 6. It was noticed that the JeC model could predict the experimental data only in the intermediate temperature range (500 C600 C). This variation may be attributed to the error introduced by the fitting of the material constants at some conditions and adiabatic temperature increment due to plastic deformation. The JeC model and the back-propagation ANN model were developed to predict the high strain rate deformation
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Fig. 9. Plot of predicted vs. experimental for modified JeC model.
behavior of high strength armor steels, and their predictability was evaluated in terms of correlation coefficient (R) and average absolute relative error (AARE). R and AARE for the JeC 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 flow stress by ANN model versus measured value for testing set. The predicted flow stresses of JeC 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%e10.6% for JeC model. Hence, the data obtained were better in the ANN model compared to the JeC model. It shows that the developed ANN model can offer an efficient prediction of flow stress at the tempering temperatures of 500 Ce650 C and the strain rates of 1000 s1 to 5500 s1.
behavior of high strength armor steels and their predictability was evaluated in terms of correlation coefficient (R) and average absolute relative error (AARE). R and AARE for the JeC 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.
4. Conclusions This paper has made an attempt to study the comparison of the results obtained from JeC model and ANN model with experimental values. The following conclusions are drawn: 1) The JeC model and the back-propagation ANN model were developed to predict the high strain rate deformation
Acknowledgments The authors would like to thank Defence Research and Development Organization, India for financial help in carrying out the experiments. References
Fig. 10. Plot of predicted vs. experimental for ANN model.
[1] Lee WS, Lin CF. Plastic deformation and fracture behaviour of Ti-6Al4V alloy loaded with high strain rate under various temperatures. Mat Sci Eng A 1998;241:48e59. [2] El-Magd E, Abouridouane A. Characterization, modeling and simulation of deformation and fracture behaviour of the light weight wrought alloys under high strain rate loading. Int J Impact Eng 2006;32:741e58. [3] Lee WS, Sue WC, Lin CF, Wu CJ. The strain rate and temperature dependence of the dynamic impact properties of 7075 aluminium alloy. J Mat Proc Tech 2000;100:116e22. [4] Meyers MA, Xu YB, Xue Q, Perez Prado MT, Mcnelley TR. Microstructural evolution in adiabatic shear localization in stainless steel. Acta mater 2003;51:1307e25. [5] Odeshi AG, Al-Ameeri S, Bassim MN. Effect of high strain rate on plastic deformation of a low alloy steel subjected to ballistic impact. J Mat Process Tech 2005:162e3. [6] Meyer LW, Seifert K, Malek AS. Behaviour of quenched and tempered steels under high strain rate compression loading. J Phys IV Fr 1997;7(C3):571e7. [7] Lee WS, Liu CY. The effects of temperature and strain rate on the dynamic flow behaviour of different steels. Mat Sci Eng A 2006;46:101e13.
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[8] Mohr D, Gary G, Lundberg B. Evaluation of stress-strain curve estimates in dynamic experiments. Int J Impact Eng 2010;37:161e9. [9] Mousavi ASH, Madaah-Hosseini HR, Bahrami A. Flow stress optimization for 304 stainless steel under cold and warm compression by artificial neural network and genetic algorithm. J Mater Design 2007;28:609e15. [10] Ji G, Li F, Li Q, Li H, Li Z. A comparative study on Arrhenius-type constitutive model and artificial neural network model to predict hightemperature deformation behavior in Aermet100 steel. J Mat Sci Eng A 2011;528:4774e82. [11] Han Y, Qiao G, Sun J, Zou D. A comparative study on constitutive relationship of as-cast 904L austenitic stainless steel during hot deformation based on Arrhenius-type and artificial neural network models. Comput Material Sci 2013;67:93e103. [12] Sun Y, Zeng WD, Zhao YQ, Qi YL, Ma X, Han YF. Development of constitutive relationship model of Ti600 alloy using artificial neural network. Comput Material Sci 2010;48:686e91. [13] Lin YC, Xia YC, Chen XM, Chen MS. Constitutive descriptions for hot pressed 2124-T851 aluminium alloy over a wide range of temperature and strain rate. Comput Material Sci 2010;50:227e33.
[14] Hou QY, Wang JT. A modified Johnson-Cook constitutive model for MgGd-Y alloy extended to a wide range of temperatures. Comput Material Sci 2010;50:147e52. [15] Gupta AK, Anirudh VK, Singh SK. Development of constitutive models for dynamic strain aging regime in Austenitic stainless steel 304. Mater Des 2013;43:410e8. [16] Lee WS, Liu CY. Dynamic compressive deformation behavior of S50C medium carbon steel. Mater Sci Technol 2004;20:919e24. [17] Lindholm US. Some experiments with the split Hopkinson pressure bar. J Mech Phys Solids 1964;12:317e35. [18] Jhonson GR, Cook WH. Fracture characteristics of three metals subjected to various strains, strain rates, temperatures, pressures. Eng Fract Mech 1985;21:31e48. [19] Remington BA, Allen P, Bringa EM, Aweliak J, Ho D, Lorenz K T, et al. Material dynamics under extreme conditions of pressure and strain rate. Mater Sci Tech 2006;22:474e88. [20] Hoge KG, Mukherjee AK. The temperature and strain rate dependence of the flow stress of tantalum. J Mater Sci 1977;12:1666e72. [21] Steinberg DJ, Chochran SG, Guinan MW. A constitutive model for metals applicable at high strain rates. J Appl Phys 1980;51:1498e504.
Available online at www.sciencedirect.com
ScienceDirect Defence Technology 10 (2014) 334e342 www.elsevier.com/locate/dt
Neural network modeling to evaluate the dynamic flow 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 Received 18 April 2014; revised 19 June 2014; accepted 24 June 2014 Available online 23 August 2014
Abstract An artificial 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 configuration consisting of both training and validation is effectively employed to predict flow stress. Tempering temperature, strain rate and strain are considered as inputs, whereas flow stress is taken as output of the neural network. A comparative study on JohnsoneCook (JeC) model and neural network model is performed. It was observed that the developed neural network model could predict flow stress under various strain rates and tempering temperatures. The experimental stressestrain data obtained from high strain rate compression tests using SHPB, over a range of tempering temperatures (500e650 C), strains (0.05e0.2) and strain rates (1000e5500/s) are employed to formulate JeC 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 coefficient (R) and average absolute relative error (AARE). R and AARE for the JeC 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. Copyright © 2014, China Ordnance Society. Production and hosting by Elsevier B.V. All rights reserved.
Keywords: Artificial neural network; High strength armor steel; JeC 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 [1e6]. The data obtained will be helpful for designing the products as well as developing the constitutive strength models of the materials. * Corresponding author. Tel.: þ40 24346332; fax: þ40 24342252. E-mail addresses: [email protected], ravindranadhb@gmail. com (R. BOBBILI). Peer review under responsibility of China Ordnance Society.
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 flow stress and parameters, such as strain, strain rate and deformation temperature. But, during the high strain rate deformation, many parameters influence the dynamic flow stress of materials. The effect of these parameters on the dynamic flow stress is extremely
http://dx.doi.org/10.1016/j.dt.2014.06.012 2214-9147/Copyright © 2014, China Ordnance Society. Production and hosting by Elsevier B.V. All rights reserved.
R. BOBBILI et al. / Defence Technology 10 (2014) 334e342
non-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. flow stress, strain rate sensitivity, etc., varies for each material. Lee et al. [7] observed an increase in flow stress with increase in strain rate in low, medium and high carbon steels. It has been found that increased carbon content enhances the dynamic flow 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 flow 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 significant variable and the strain has less influence on flow 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 0.01e50 s1. The Arrhenius constitutive model and a feed forward artificial 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-mechanical simulator to generate stress-strain data in a temperature range from 1000 C to 1150 C and at the strain rates of 0.01e10 s1. The back propagation neural network model was proved to be more accurate and efficient 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 0.001e10 s1. ANN model provided a simple and efficient 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 0.01e10 s1 and temperature range from 653 K to 743 K using Gleeble-1500 thermo-simulation machine. A modified constitutive model accommodating the effects of material behavior was proposed. Hou et al. [14] carried out high strain rate experiments using SHPB on MgGd-Y alloy over a range of temperatures. A modified JeC 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 (JohnsoneCook model, modified 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
335
(0.1e0.0001 s1) and temperatures (323e623 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 JeC 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 JohnsoneCook 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 flow 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 stressestrain data were obtained from high strain rate compression tests using split Hopkinson pressure bar (SHPB), over a wide range of strains (0.1e0.3) and strain rates (1000e5500/s). The whole SHPB setup consists of (a) pressure bars, (b) gas gun which propels a striker bar for producing the compressive 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 bars
Fig. 1. Schematic diagram of split Hopkinson pressure bar (SHPB).
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greatly reduced by applying a lubricant thin film. Hence a thin grease layer of CueMo 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 amplifiers 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 reflected wave (εr). Strain-rate generated in the specimen is Fig. 2. Experimental setup of Split Hopkinson pressure bar (SHPB) system.
ε_ ¼ 2cεr =l
ð1Þ
Hence the strain in the specimen is are 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 welldefined 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 reflected back to the incident bar. Fig. 3 represents incident wave, reflected 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. Length and diameter of the samples were changed to vary the strain rate. The sample ends were machined flat and parallel to the bar ends to provide good contact with them. Also the surfaces of the samples were polished to provide standard surface finish of 5 micron. Each sample was sandwiched between input and output bars. This frictional constraint can be
Fig. 3. Strain-time graph of high strength armor steel alloy tested at a strain rate of 1000/s in input and output bar.
Zt ε ¼ 2c=l
εr dt
ð2Þ
0
Stress in the specimen can be calculated as s ¼ Ab Eεt =AS
ð3Þ
where Ab and AS are the areas of the bar and specimens, respectively; l is the length of specimen; c is wave speed; εt is the strain in the transmitted bar; and E is the elastic modulus of the pressure bar. 2.2. JohnsoneCook 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 s ¼ f ðε; ε_ ; TÞ
ð4Þ
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 s ¼ so þ kεn， where so is the yield stress, n is work hardening exponent, and k is pre-exponential factor. The effect of strain rate on strength is generally expressed as s f ln_ε, but the above relationship breaks down at strain rate above 102 s1. The effects of temperature on the flow stress can be represented by m T Tr s ¼ sr 1 ð5Þ T m Tr where Tm is the melting temperature; and Tr is the reference temperature at which sr is the reference stress. The dynamic flow stress [16e21] depicting the effect of various parameters has been expressed by JohnsoneCook model as.
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Table 1 JohnsoneCook model constants for various steels. Material
A/MPa
B/MPa
n
C
High strength armor steel tempered at 500 C High strength armor steel tempered at 600 C High strength armor steel tempered at 650 C
1321
1124
0.506
0.0104
1227
1032
0.612
0.0127
1134
917
0.626
0.0114
s ¼ ðA þ Bεn Þð1 þ cln_ε* Þð1 T * m Þ
ð6Þ
where A is the yield stress; B and n represent the effect of strain hardening; C is 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_ =_ε0 for ε_ 0 ¼ 1 s1 ; T* is the homologous temperature referred as (TeTroom)/(TmelteTroom); and m is the thermal softening factor. Thus, the term present in first, second and third bracket in Eq. (4) represents strain, strain rate and temperature effect, respectively. The JeC model is independent of pressure. At reference strain rate and reference temperature, the functions of strain rate hardening and thermal softening are equal to unity. The JeC model is simplified as follows: s ¼ A þ Bεn
ð7Þ
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 (seA) at the reference strain rate and reference temperature gives B and n in Eq. (7). Strain rate sensitivity (C) is determined as the slope of linear fit of log (strain rate) vs dynamic flow stress/static stress using high strain rate data corresponding to a strain of 10%. The above constants are provided in Table 1. 2.3. Artificial neural network approach Neural networks are commonly employed in data prediction, categorization and data filtering 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 first 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
Fig. 4. The schematic of the ANN architecture.
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. 3. Results and discussion The experimental data were split into three sets, 70% for the training set, 15% for the verification set and 15% for the test set in ANN model. The input data (strain rate, strain and tempering temperature) and output data (flow stress) were standardized in the range (Fig. 4). The network model consists of ten hidden layers. To demonstrate the influence 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 the significance of these 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 influence 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. Influence of hidden neurons on the network performance.
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Fig. 6. Comparison between JeC Model, ANN Model and experimental flow stress of high strength armor steel tempered at 500 C by JeC 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 s1. The signals were processed by applying a dispersion correction to the signals and then time shifted to bring the three pulses (incident and reflected pulses in input bar and transmitted pulse from output bar) into coincidence at the
sampleebar 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. 6e8. The stress-strain curve is generated from test data received from specimens tested at strain rates of 1000 s1, 2000 s1, 3000 s1, 4000 s1 and
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Fig. 7. Comparison between JeC Model, ANN Model and experimental flow stress of high strength armor steel tempered at 600 C by JeC model at strain rates.
5000 s1. 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 s1 to 5000 s1. 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. The experimental data obtained from the high strain rate compression tests on split Hopkinson pressure bar, in a wide range of tempering temperatures (500 Ce650 C) and strain rates (1000 s1e5500 s1), were employed to develop JeC model and ANN model for high strength armor steels. Fig. 6 depicts the comparison of the experimental results with the predicted values at various strain rates and tempering
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Fig. 8. Comparison between JeC Model, ANN Model and experimental flow stress of high strength armor steel tempered at 650 C by JeC model at strain rates.
temperatures based on JeC strength model. It is observed that the predicted flow stress values obtained on JeC model are not consistent with the experimental values, particularly at high tempering temperatures; and the predicted values are smaller than the experimental results. So, JeC model is not so adequate in predicting the flow stress value in high tempering temperature region. The predicting performance of the JeC model was evaluated by comparing the experimental and
predicted data, as shown in Fig. 6. It was noticed that the JeC model could predict the experimental data only in the intermediate temperature range (500 C600 C). This variation may be attributed to the error introduced by the fitting of the material constants at some conditions and adiabatic temperature increment due to plastic deformation. The JeC model and the back-propagation ANN model were developed to predict the high strain rate deformation
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Fig. 9. Plot of predicted vs. experimental for modified JeC model.
behavior of high strength armor steels, and their predictability was evaluated in terms of correlation coefficient (R) and average absolute relative error (AARE). R and AARE for the JeC 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 flow stress by ANN model versus measured value for testing set. The predicted flow stresses of JeC 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%e10.6% for JeC model. Hence, the data obtained were better in the ANN model compared to the JeC model. It shows that the developed ANN model can offer an efficient prediction of flow stress at the tempering temperatures of 500 Ce650 C and the strain rates of 1000 s1 to 5500 s1.
behavior of high strength armor steels and their predictability was evaluated in terms of correlation coefficient (R) and average absolute relative error (AARE). R and AARE for the JeC 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.
4. Conclusions This paper has made an attempt to study the comparison of the results obtained from JeC model and ANN model with experimental values. The following conclusions are drawn: 1) The JeC model and the back-propagation ANN model were developed to predict the high strain rate deformation
Acknowledgments The authors would like to thank Defence Research and Development Organization, India for financial help in carrying out the experiments. References
Fig. 10. Plot of predicted vs. experimental for ANN model.
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