- Email: [email protected]
Determination of Johnson Cook Material Model Constants for 93% WHA and Optimization using Genetic Algorithm
Available online at www.sciencedirect.com
ScienceDirect Materials Today: Proceedings 5 (2018) 18911–18919
www.materialstoday.com/proceedings
ICMPC_2018
Determination of Johnson Cook Material Model Constants for 93% WHA and Optimization using Genetic Algorithm Chithajalu Kiran Sagara, Abhiram Chilukurib and Amrita Priyadrashinic* a
PhD Scholar, Mechanical Engineering Department, BITS-Pilani, Hyderabad campus, Telangana-500078, INDIA. b Student, Mechanical Engineering Department, BITS-Pilani, Hyderabad campus, Telangana-500078, INDIA c Assitant Professor, Mechanical Engineering Department, BITS-Pilani, Hyderabad campus, Telangana-500078, INDIA
Abstract An accurate material constitutive model is a key concern to develop a realistic FE models both for machining as well as penetration mechanisms. A commonly used material model in both the cases is Johnson–Cook (JC) model which is generally available in most of the commercial FE codes. The constants of the JC constitutive model are generally determined by Split Hopkinson Pressure Bar (SPHB). In this paper, experimental SHPB test data taken from literature has been utilized to determine J-C model constants under wide range of strain rates and temperatures for 93% W Tungsten Heavy alloy (WHA). The calculated values of material model constants are then optimized using Evolutionary Algorithm based technique i.e. Genetic Algorithm (GA). The predicted results would help compare the efficiency of the optimization technique used in the present work for finetuning the JC model constants in terms of absolute error and coefficient of correlation. Furthermore, the predicted values of JC model constants can be appropriately used for FE simulations of both machining as well as penetration mechanisms for WHA alloys. The material taken into consideration for the present study finds application typically in armour piercing ammunition. © 2018 Elsevier Ltd. All rights reserved. Selection and/or Peer-review under responsibility of Materials Processing and characterization. Keywords: Tungsten heavy alloy (WHA), Genetic Algorithm, Johnson-Cook Model
1. Introduction Tungsten Heavy Alloys provide a unique combination of density, mechanical strength, machinability, corrosion resistance, and economy. Consequently, WHAs are widely used for counterweights, inertial masses, radiation shielding, sporting goods, and ordnance products. * Corresponding author. Tel.:040 66303645. E-mail address: [email protected] 2214-7853 © 2018 Elsevier Ltd. All rights reserved. Selection and/or Peer-review under responsibility of Materials Processing and characterization.
18912
Amrita Priyadrashini et al./ Materials Today: Proceedings 5 (2018) 18911–18919
These versatile materials provide distinct advantages when compared to alternate high-density materials such as lead and Depleted Uranium (DU), especially, in terms of toxicity [1]. WHA is truly the material of choice for high density applications such as aircraft counterbalances, radiation shielding, casing material for down hole logging of oil wells, low chatter boring bars and long extension tool holders and armour piercing (AP) ammunition. Most of these components require machining as the finishing operation to get the final shape and surface finish, thus, making machining indispensable. But WHAs are generally considered to have lower machinability index because of tungsten abrasiveness, high elastic stiffness and high temperatures attained due to the heat generated in high speed and heavy feed machining. To carry the machining operations efficiently, we need to find the right combination of cutting parameters as well as cutting tool materials and geometry. This certainly requires proper understanding of the deformation behavior of work material as well as mechanism of chip formation during machining. The trial and error experimental tests have, no doubt, laid the foundation stone in the area of metal cutting studies; but they are very time consuming and expensive. The focus, nowadays, is mainly to simulate the real machining operation by Finite Element models which may substitute the expensive cutting experiments to a great extent. Furthermore, WHAs are currently one of the basic materials for Kinetic Energy (KE) penetrators not only for their high density and strength but also for environment friendly attributes. KE penetrator is a type of ammunition that uses kinetic energy to penetrate the target. Consequently, the ballistic penetration modeling is very much required for development of armor solutions. Much of the emphasis is given to simulate the penetration mechanisms by FE models as compared to experimentally derived empirical formulations and analytical model derivations. Penetration mechanism during ballistic impact, however, is equally a challenging task to represent numerically since it involves large deformations, high strain-rates, temperatures, localization of failure of target material. This essentially needs complicated modeling of material behaviour as observed in the case of machining. An accurate material constitutive model is a key concern to develop a realistic FE models both for machining as well as penetration mechanism. A commonly used material model in both the cases is Johnson–Cook model which is generally available in most of the commercial FE codes. Typical strain rates in machining are usually from 104 /s to 107 /s [2]. These values are so high that, so far, no experimental mean of test has been developed to reach these extreme conditions of strains and strain rates, what is more at sufficiently high temperatures. Empirical constitutive models are widely adopted although they only represent accurately the material for lower strains, strain rates and temperatures than what is observed during machining. The values used during the computation of the model are therefore extrapolated ones. The Johnson-Cook flow stress dissociates plastic, viscous and thermal aspects without taking the strain softening into account as proposed by more recent models [3]. The parameters of the Johnson-Cook constitutive model are generally determined by Split Hopkinson Pressure Bar (SPHB) [4]. This experimental configuration allows to reach strains up to 0.5 and strain rates lower than 104 /s. These values are lower than what is generally observed during machining. The flow stress for higher values is then extrapolated. Tusit and John [5] had conducted high strain rate test at 750 s-1 using SHPB test for failure analysis of 93% WHA at room temperature. Tusit [6] determined the J-C parameters using SHPB tests at 103 and 104 /s strain rates and from room temperature to 732 K for 93% WHA. Lee et al [7] had conducted high strain rate of 103 s-1 using SHPB test at wide range of temperatures ranging from room temperature to 1100 0C for metallographic failure examination. Rohr and Nahme et al [4] had determined J-C parameters by logarithmic curve fitting using Taylor impact test at 103 and 104 /s strain rates and from room temperature to 827 K for 93% WHA. The JC model parameters were identified with a combination of analytical modelling and orthogonal cutting experiments suggesting good correlation with experimental flow stress data [8]. Inverse identification algorithms have been used to characterize material flow stress laws to define material behavior [9]. An analysis of inverse algorithms suggested that material parameters derived from analytical methods are error prone and suggests the use of secondary shear zone parameters in conjunction with FE analysis for better results [10]. The identification of material parameters through deterministic methods suggests that different optimum parameters are possible depending on the starting point and hence not conclusive [10]. An improvement over conventional optimization methods is the use of evolutionary computational algorithms which identified material parameters through a fine grain search technique and claimed precedence over classical data fitting techniques [11]. The concepts of artificial neural networks and genetic algorithms have been explored with reasonable success in the prediction and optimization of flow stress [3]. Dusunceli et al [12] applied the genetic algorithm (GA) optimization technique to determine the parameters for the visco-plasticity model based
Amrita Priyadrashini et al./ Materials Today: Proceedings 5 (2018) 18911–18919
18913
on overstress by minimizing the errors between experimental and simulated stress. Similarly, Anaraki et al [10] presented an inverse method to determine the material constants for AZ61 magnesium alloy by minimizing the errors between calculated and experimental results. Very few studies have focused on the optimization of constitutive model parameters under high strain rate conditions and for wide range of temperatures. chen et al [13] applied GA-based multi-objective optimization technique to determine J-C parameters for Ti-6Al-4V alloy with strain rates (2000 and 2500 s-1) at temperature ranging 25 to 900 0C. In this paper, experimental SHPB test data taken from Lee et al [7] has been utilized to determine J-C model constants under wide range of strain rates and temperatures for 93% WHA. The calculated values of material model constants are then optimized using Evolutionary Algorithm based technique, namely, Genetic Algorithm. The predicted results would help in fine-tuning the JC model constants in terms of absolute error and coefficient of correlation. Furthermore, the calculated as well as optimized values of JC model constants can be appropriately used for FE simulations of machining simulations as well as penetration mechanisms for WHA alloys. 2. Methodology and JC model optimization In this paper, experimental data are taken from Lee et al [7] and utilized to determine J-C model constants under wide range of strain rates and temperatures for 93% WHA. Experimental stress-strain curves obtained from the mentioned literature are converted to true stress- strain curves and data points are taken in plasticity region till Ultimate Tensile Strength (UTS). True stress-strain data has been plotted for given strain rates (4000, 2500, 1600 /s) and temperatures (298, 573, 773, 973, 1173, 1373 K) in Fig. 1 (a-c). These data points are then used for determining the JC Model constants and this approach is considered as M1. Next step is to fine tune the calculated JC model constants obtained from M1 using GA based optimization technique. The new constants are determined and this approach is considered as M2. In addition, another set of JC constant are considered namely, M3, which have been directly taken from the literature where in Rohr and Nahme [4] had derived JC model constants for 93% WHA at strain rates of 2x10-1, 1x10-2, 8x10-4, 6x10-5, 1000, 10000 /s and temperatures of 300 and 827 K and are listed in Table 1. The values obtained from M1, M2 and M3 are compared by calculating the absolute error and coefficient of correlation in each of the cases. This would help in proposing a better methodology to determine the JC model constants with least absolute error. Table 1. Details of M1, M2 and M3. Model Woei and Guo [3]
M1& M2
Rohr and Nahme [4]
M3
A, B, n, c, m
Temperatures K
Strain rates /s
298, 573, 773, 973, 1173, 1373
800,1600,2500,4000
300, 827
2x10 , 1x10 , 8x10 , 6x10 , 1000, 10000
-1
(a)
1197, 580, 0.05, 0.025, 1.9
(b)
-2
-4
-5
18914
Amrita Priyadrashini et al./ Materials Today: Proceedings 5 (2018) 18911–18919
(c)
Fig. 1. Literature [7] True stress-strain curves at different temperatures (a) for Strain rate 4000 /s; (b) for Strain rate 2500 /s, (c) for Strain rate 1600 /s
2.1. Determination of JC model constants Johnson Cook constitutive equation is expressed as: ∗ )(1 )(1 + (1) =( + − ∗ ) where σ is the flow stress, A is the yield stress at reference temperature and strain rate, B is the coefficient of strain hardening, n is the strain hardening exponent, C is the Co-efficient of strain rate dependency, m is the thermal softening exponent, ε is the plastic strain, ∗ = is the dimensionless strain rate and is strain rate and is reference strain rate and T ∗ is homologous temperature and expressed as: ∗ =
(2)
where T is current temperature, is melting temperature (2173 K for 93%WHA) and is reference temperature ). Minimum temperature of the test is considered as reference temperature and maximum strain rate in test ( as reference strain rate. The JC model considers strain hardening, strain rate hardening and thermal softening as three independent phenomena and these can be isolated from each other and calculated individually under certain conditions. Hence, the flow stress can be determined by multiplying these three parameters in Eq. (1). Reference temperature is taken as 298 K while reference strain rate is taken as 4000 /s. Step 1: At reference strain rate and reference temperature. The strain hardening and thermal softening effect will be nil, so second and third term in equation will be equal to unity thereby only first term is left and Eq. (1) reduces to (3) =( + ) yield stress of material at room temperature is 1103 MPa (i.e. the stress at 0.002 strain) as A, Equating the value of A in Eq. (3) and plotting graph between ln (σ – A) vs ln . B is substituted from the intercept of the plot, while n is the slope. Step 2: At fixed plastic strain and reference temperature. The strain hardening effect will be observed and thermal softening effect will be nil, so T* is equal to zero Eq. (1) reduces to. ∗ )(1 + (4) =( + ) Hence, from flow stress data for every fixed plastic strain at various strain rates a value of C is calculated from the slope of − 1 vs ln( ∗ )plot. (
)
Amrita Priyadrashini et al./ Materials Today: Proceedings 5 (2018) 18911–18919
18915
Step 3: At fixed plastic strain and reference strain rate. The strain hardening effect will be nil and thermal softening effect is observed, so ln ( *) is equal to zero Eq. (1) reduces to )(1 − ∗ ) (5) =( + Hence, from flow stress data for every fixed plastic strain at different temperatures a value of m is calculated from the slope of ln 1 − vs ln ( ∗ ) plot. The material constants C and m are determined using the least-square (
)
method. In order to determine optimal C and m values. It involves minimizing average absolute error (Δ) between the experimental and predicted flow stress values. (6) − 1 = × 100 is the experimental flow stress, is the calculated flow stress and N is the number of data points Where considered. Correlation coefficient is commonly used statistical tool which provides linear relationship between experimental and predicted values. it is expressed as. (7) ∑ ( − )( − ) = ∑ ( − ) ∑ ( − ) and are the average values of and respectively. Usually it is considered as higher the value of Where R closer is the correlation with experimental values. but always it not true as equation has tendency to be biased towards higher or lower values [14]. 2.2. Genetic Algorithm (GA) based optimization The genetic algorithm is a method for solving both constrained and unconstrained optimization problems that is based on natural selection, the process that drives biological evolution. The genetic algorithm repeatedly modifies a population of individual solutions. At each step, the genetic algorithm selects individuals at random from the current population to be parents and uses them to produce the children for the next generation. Over successive generations, the population evolves toward an optimal solution [15]. Following are the steps for optimizing the JC constant: Step 1: Create initial population Initial population is created by taking a range of B, C, n & m values and randomly twenty set of initial population has been formed within this range, where function = , , , are considered as design variables. Step 2: Fitness function Fitness has been defined using three objective functions which is the difference between experimental and calculated data of stress- strain curve. , (8) − 1 ( )= × 100 , ,
Δ (x) =
−
1 ,
Δ (x) =
,
× 100
,
−
1
(9)
(10) × 100
where ( ), ( ) and ( ) are absolute error functions under three strain rates 4000, 2500, 1600 /s respectively. i, g, l are strain level at strain rate 4000, 2500, 1600 /s and j, h, r, are number of temperature values considered at
18916
Amrita Priyadrashini et al./ Materials Today: Proceedings 5 (2018) 18911–18919
strain rate 4000, 2500, 1600 /s, N is total no of data points considered,
,
,
are experimental flow
, , are calculated flow stresses respectively. stresses and The main objective of this fitness function is to minimize absolute error function ( ) , ( ) and ( ) simultaneously. Since it is a multi-objective optimization problem the stress-strain responses at different temperature and strain rates are considered as a set of curves. Penalty function method has been induced for objective function as. ( )= (11) ( )+ ( )+ ( ) (12) ( , , , ) = { ( )} are weight coefficients for ( ), ( ) and Δ (x). where , , As fitness function Δ(x) derived from the objective function and is utilized in successive genetic operations. Individuals with high fitness will survive to the next generation. Since optimization is a minimization problem, fitness function is redefined as (13) – ( ) ( )= Where is a predefined constant weight function to make sure fitness function has a positive value. Step 3: Parent Selection From initial population randomly three members are selected and the one with the least Δ is selected as parent 1. The same procedure is repeated for parent 2. Similarly, a set of 10 parents are selected. Step4: Encode The selected 10 parent’s values of B, C, n and m are converted into binary string and concatenated. This long string of 0’s and 1’s is now our chromosome. Step 5: Recombination Selected parents are then crossed over randomly among them and the chromosomes in between these are interchanged. Step 6: Mutation During cross over between two parents a probability of 0.1 is considered as mutation factor and one of its bit is flipped to form new children. Step 7: Decode The new children chromosomes are then decoded to form the constants B, C, n and m. Step 8: Generations The above-mentioned procedure has been repeated for 200 times, i.e. for 200 generations. The whole algorithm is coded in python using libraries numpy, scipy and pandas. The program takes an input of the value A, temperature set and strain rate set at which experimental flow stress data has been recorded and outputs as the optimal B, C, n and m values along with absolute error associate with it. 3. Result and Discussion JC model constants have been calculated using Eq.1-5 based on the experimental data from the literature [7]. The optimal values of C and m value have been determined using least square method (see Eq. 6-7). The obtained values of JC constants have been listed as M1 in Table 2. Genetic algorithm based optimization has been carried out for constants B, C, n, m using Eq. 8-13. Figure 2 shows the evolution process of genetic algorithm. It is observed that from 50th generation absolute error reduced considerably which further got reduced and converged at 180th generation values and there after remained constant. Binary string values of minimum error chromosome has been decoded for B, n, C and m values and are listed as M2 in Table 2. Material constants obtained by Rohr and Nahme [4] have also been given in Table 2 as M3. Furthermore, absolute error percentage (∆) and coefficient of correlation (R) are calculated for M1, M2 and M3 and presented in Table 2 for comparison. Table 2. Details of M1, M2 and M3. Model
A MPa
B
n
c
m
∆
(R)
MPa
M1
1103
707.5
0.220
0.112
1.407
8.72
0.963
M2
1103
875.50
0.353
0.091
1.903
2.93
0.973
M3 [4]
1197
580
0.05
0.025
1.9
16.54
0.922
Amrita Priyadrashini et al./ Materials Today: Proceedings 5 (2018) 18911–18919
18917
Fig. 2 The objection function versus Generation for GA based optimization
It is observed that there is a considerable improvement in predicting the values of JC constants using GA i.e. M2 in terms of error percentage (lowest) and R (highest). In order to have better insight, flow stress curves were plotted obtained from predicted results (M1, M2 and M3) under different strain rates and temperatures compared with that of experimentally obtained flow stresses (see Fig. 3 (a), (b) and (c)). It is clearly observed that flow stress curves obtained from M2 closely matches with that of experimental flow stress curves as compared to that of M1 and M3 material constants.
(a)
(b)
(c) Fig. 3. Comparison of experimental [7] and calculated flow stress obtained from M1, M2 and M3 for different temperatures at strain rate (a) 4000 /s; (b) 2500 /s; (c) 1600 /s
18918
Amrita Priyadrashini et al./ Materials Today: Proceedings 5 (2018) 18911–18919
Fig. 4 (a-c) presents the percentage error of flow stress for M1, M2 and M3 at 298 K for different strain rates. It is observed that percentage error of flow stress at 298 K for M1 is in the range of 1 – 4% for 4000 /s strain rate, 1 – 4% for 2500 /s strain rate and 4 – 8 % for 1600 /s strain rate. Similarly, for M2 percentage error of flow stress is in the range of 1 – 3 % for 4000 /s strain rate, 1 – 2 % for 2500 /s strain rate and 1 – 5 % for 1600 /s strain rate; whereas for M3 percentage error of flow stress is in the range of 7 – 18 %for 4000 /s, 7 – 23 % for 2500 /s strain rate and 6 – 28 % for 1600 /s strain rate. From this observation, it can be stated that JC constants obtained using GA i.e. M2 is capable of predicting better results consistently as compared to M1 and M3.
(a)
(b)
(c)
Fig. 4. Error between experimental [9] and calculated flow stress obtained by JC model constants for M1, M2 and M3 at 298 K: (a) strain rate 4000 /s; (b) strain rate 2500 /s; (c) strain rate 1600 /s.
4. Conclusion JC model constants have been determined based on the experimental data from the literature [7].The optimal values of C and m have been determined using least square method. Genetic algorithm based optimization has been used for fine tuning the constants B, C, n, m. In addition, JC constants obtained by Rohr and Nahme [4] have also been taken into consideration for comparison with the obtained results in terms of absolute percentage error and coefficient of correlation. It was found that JC constants obtained using GA yields the lowest absolute error percentage and highest R. The flow stress curves obtained from GA closely matches with that of experimental flow stress curves. The JC model constants predicted in the present work can be used efficiently for FE simulations of machining process as well penetration mechanisms during ballistic impact.
Amrita Priyadrashini et al./ Materials Today: Proceedings 5 (2018) 18911–18919
18919
References [1]
A. Arora, V.G. Rao “Tungsten Heavy Alloy For Defence Applications,” Materials Technology, Vols. 19-4, no. 26-August-2016, pp. 201263, 2004.
[2]
P.J. Arrazola, O. zel, D. Umbrello, M. Davies, I.S. Jawahir “Recent advances in modelling of metal machining processes,” CIRP Annals Manufacturing Technology, vol. 62, pp. 695-718, 2013.
[3]
M.Sasso*, G.Newaz, D.Amodio “Material characterization at high strain rate by Hopkinson bar tests and Finite element optimization,” Materials Science and Engineering A, vol. 487, pp. 289-300, 2008.
[4]
I. Rohr, H. Nahme, K. Thoma “Material characterisation and constitutive modelling of a tungsten sintered alloy for a wide range of strain rates,” International Journal of Impact Engineering, vol. 35, no. 8, pp. 811-819, 2008.
[5]
Clayton, T. Weerasooriya, John “FAILURE BEHAVIOR OF A TUNGSTEN HEAVY ALLOY,” in International Conference on Tungsten, Refractory & Hardmetals VI, Florida, 2006.
[6] [7]
T. Weerasooriya, “Deformation Behavior of 93W-5Ni-2Fe at Different Rate of cCompression Loading and Temperatures,” USA, 1998. W. Lee, G. Xiea, C. Lin “The strain rate and temperature dependence of the dynamic impact response of tungsten composite,” Materials Science and Engineering A , vol. 257, pp. 256-267, 1998.
[8]
N. Tounsi, J. Vincenti, A. Otho, M.A. Elbestawi “From the basic mechanics of orthogonal metal cutting toward the identification of the constitutive equation,” International Journal of Machine Tools & Manufacture, vol. 42, p. 1373–1383, 2002.
[9]
T. Özel, E. Zeren “Determination of work material flow stress and friction for FEA of machining using orthogonal cutting tests,” Journal of Materials Processing Technology, vol. 153–154, p. 1019–1025, 2004.
[10]
Anaraki MT, Sanjari .M and Akbarzadeh .A “Modeling of high temperature rheological behavior of AZ61 Mg-alloy using inverse method and ANN,” Materials and Design, vol. 29, p. 1701–1706, 2008.
[11]
T. Özel, Y. Karpat “Multi-objective optimization for turning processes using neural network modeling and dynamic-neighborhood particle swarm optimization,” The International Journal of Advanced Manufacturing Technology, vol. 35, no. 3-4, p. 234–247, 2007.
[12]
Dusunceli N, Colak .O.U, Filiz .C “Determination of material parameters of a viscoplastic model by genetic algorithm,” Materials and Design, vol. 31, p. 1250–1255, 2010.
[13]
G. Chen, C. Ren, W. Yu, X. Yang, L. Zhang “Aplication of genetic algorithms for optimizing the johnson-cook constitutive model parameters when simulating the titanium alloy Ti-6Al-4V machining process,” Journal of Engineering Manufacture, pp. 1-11, 2012.
[14]
A. K. Gupta, H.N. Krishnamurthy, Y. Singh, K.M. Prasad “Development of constitutive models for dynamic strain aging regime in Austenitic stainless steel 304,” Materials and design, vol. 45, pp. 616-627, 2013.
[15] [16]
D.M.D. Addona*, R. Teti “Genetic algorithm-based optimization of cutting parameters,” in Procedia CIRP 7, 2013. E. Pink, S. Kumar “Deformation mechanisms operating in a tungsten heavy alloy,” Materials science & Engineering A, Vols. 234-236, pp. 102-105, 1997.
[17]
J. Das, G.A. Rao, S.K. Pabi “Microstructure and mechanical properties of tungsten heavy alloys,” Materials Science and Engineering A, vol. 527, pp. 7841-7847, 2010.
[18]
N.F. Johari, A.M. Zain, N.H. Mustaffa, A. Udin “Machining Parameters Optimization using Hybrid Firefly,” in international conference on computer science and computational mathematics, Langkawi, 2017.
[19]
R. Gero, L. Borukhin, I. Pikus “Some structural effects of plastic deformation on tungsten heavy metal alloys,” Materials Science and Engineering A, vol. 302, pp. 162-167, 2001.
[20]
R.D. Koyee, S. Schmauder, U.Heisel, R. Eisseler “Numerical modeling and optimization of machining duplex stainless,” Production & Manufacturing Research, vol. 3, pp. 36-83, 2015.
[21]
X. Gong, J. Fan, F. Ding “Tensile mechanical properties and fracture behavior of tungsten heavy alloys at 25 to 1100 deg C,” Materials Science and Engineering A, vol. 646, no. 1, pp. 316-321, 2015.
ScienceDirect Materials Today: Proceedings 5 (2018) 18911–18919
www.materialstoday.com/proceedings
ICMPC_2018
Determination of Johnson Cook Material Model Constants for 93% WHA and Optimization using Genetic Algorithm Chithajalu Kiran Sagara, Abhiram Chilukurib and Amrita Priyadrashinic* a
PhD Scholar, Mechanical Engineering Department, BITS-Pilani, Hyderabad campus, Telangana-500078, INDIA. b Student, Mechanical Engineering Department, BITS-Pilani, Hyderabad campus, Telangana-500078, INDIA c Assitant Professor, Mechanical Engineering Department, BITS-Pilani, Hyderabad campus, Telangana-500078, INDIA
Abstract An accurate material constitutive model is a key concern to develop a realistic FE models both for machining as well as penetration mechanisms. A commonly used material model in both the cases is Johnson–Cook (JC) model which is generally available in most of the commercial FE codes. The constants of the JC constitutive model are generally determined by Split Hopkinson Pressure Bar (SPHB). In this paper, experimental SHPB test data taken from literature has been utilized to determine J-C model constants under wide range of strain rates and temperatures for 93% W Tungsten Heavy alloy (WHA). The calculated values of material model constants are then optimized using Evolutionary Algorithm based technique i.e. Genetic Algorithm (GA). The predicted results would help compare the efficiency of the optimization technique used in the present work for finetuning the JC model constants in terms of absolute error and coefficient of correlation. Furthermore, the predicted values of JC model constants can be appropriately used for FE simulations of both machining as well as penetration mechanisms for WHA alloys. The material taken into consideration for the present study finds application typically in armour piercing ammunition. © 2018 Elsevier Ltd. All rights reserved. Selection and/or Peer-review under responsibility of Materials Processing and characterization. Keywords: Tungsten heavy alloy (WHA), Genetic Algorithm, Johnson-Cook Model
1. Introduction Tungsten Heavy Alloys provide a unique combination of density, mechanical strength, machinability, corrosion resistance, and economy. Consequently, WHAs are widely used for counterweights, inertial masses, radiation shielding, sporting goods, and ordnance products. * Corresponding author. Tel.:040 66303645. E-mail address: [email protected] 2214-7853 © 2018 Elsevier Ltd. All rights reserved. Selection and/or Peer-review under responsibility of Materials Processing and characterization.
18912
Amrita Priyadrashini et al./ Materials Today: Proceedings 5 (2018) 18911–18919
These versatile materials provide distinct advantages when compared to alternate high-density materials such as lead and Depleted Uranium (DU), especially, in terms of toxicity [1]. WHA is truly the material of choice for high density applications such as aircraft counterbalances, radiation shielding, casing material for down hole logging of oil wells, low chatter boring bars and long extension tool holders and armour piercing (AP) ammunition. Most of these components require machining as the finishing operation to get the final shape and surface finish, thus, making machining indispensable. But WHAs are generally considered to have lower machinability index because of tungsten abrasiveness, high elastic stiffness and high temperatures attained due to the heat generated in high speed and heavy feed machining. To carry the machining operations efficiently, we need to find the right combination of cutting parameters as well as cutting tool materials and geometry. This certainly requires proper understanding of the deformation behavior of work material as well as mechanism of chip formation during machining. The trial and error experimental tests have, no doubt, laid the foundation stone in the area of metal cutting studies; but they are very time consuming and expensive. The focus, nowadays, is mainly to simulate the real machining operation by Finite Element models which may substitute the expensive cutting experiments to a great extent. Furthermore, WHAs are currently one of the basic materials for Kinetic Energy (KE) penetrators not only for their high density and strength but also for environment friendly attributes. KE penetrator is a type of ammunition that uses kinetic energy to penetrate the target. Consequently, the ballistic penetration modeling is very much required for development of armor solutions. Much of the emphasis is given to simulate the penetration mechanisms by FE models as compared to experimentally derived empirical formulations and analytical model derivations. Penetration mechanism during ballistic impact, however, is equally a challenging task to represent numerically since it involves large deformations, high strain-rates, temperatures, localization of failure of target material. This essentially needs complicated modeling of material behaviour as observed in the case of machining. An accurate material constitutive model is a key concern to develop a realistic FE models both for machining as well as penetration mechanism. A commonly used material model in both the cases is Johnson–Cook model which is generally available in most of the commercial FE codes. Typical strain rates in machining are usually from 104 /s to 107 /s [2]. These values are so high that, so far, no experimental mean of test has been developed to reach these extreme conditions of strains and strain rates, what is more at sufficiently high temperatures. Empirical constitutive models are widely adopted although they only represent accurately the material for lower strains, strain rates and temperatures than what is observed during machining. The values used during the computation of the model are therefore extrapolated ones. The Johnson-Cook flow stress dissociates plastic, viscous and thermal aspects without taking the strain softening into account as proposed by more recent models [3]. The parameters of the Johnson-Cook constitutive model are generally determined by Split Hopkinson Pressure Bar (SPHB) [4]. This experimental configuration allows to reach strains up to 0.5 and strain rates lower than 104 /s. These values are lower than what is generally observed during machining. The flow stress for higher values is then extrapolated. Tusit and John [5] had conducted high strain rate test at 750 s-1 using SHPB test for failure analysis of 93% WHA at room temperature. Tusit [6] determined the J-C parameters using SHPB tests at 103 and 104 /s strain rates and from room temperature to 732 K for 93% WHA. Lee et al [7] had conducted high strain rate of 103 s-1 using SHPB test at wide range of temperatures ranging from room temperature to 1100 0C for metallographic failure examination. Rohr and Nahme et al [4] had determined J-C parameters by logarithmic curve fitting using Taylor impact test at 103 and 104 /s strain rates and from room temperature to 827 K for 93% WHA. The JC model parameters were identified with a combination of analytical modelling and orthogonal cutting experiments suggesting good correlation with experimental flow stress data [8]. Inverse identification algorithms have been used to characterize material flow stress laws to define material behavior [9]. An analysis of inverse algorithms suggested that material parameters derived from analytical methods are error prone and suggests the use of secondary shear zone parameters in conjunction with FE analysis for better results [10]. The identification of material parameters through deterministic methods suggests that different optimum parameters are possible depending on the starting point and hence not conclusive [10]. An improvement over conventional optimization methods is the use of evolutionary computational algorithms which identified material parameters through a fine grain search technique and claimed precedence over classical data fitting techniques [11]. The concepts of artificial neural networks and genetic algorithms have been explored with reasonable success in the prediction and optimization of flow stress [3]. Dusunceli et al [12] applied the genetic algorithm (GA) optimization technique to determine the parameters for the visco-plasticity model based
Amrita Priyadrashini et al./ Materials Today: Proceedings 5 (2018) 18911–18919
18913
on overstress by minimizing the errors between experimental and simulated stress. Similarly, Anaraki et al [10] presented an inverse method to determine the material constants for AZ61 magnesium alloy by minimizing the errors between calculated and experimental results. Very few studies have focused on the optimization of constitutive model parameters under high strain rate conditions and for wide range of temperatures. chen et al [13] applied GA-based multi-objective optimization technique to determine J-C parameters for Ti-6Al-4V alloy with strain rates (2000 and 2500 s-1) at temperature ranging 25 to 900 0C. In this paper, experimental SHPB test data taken from Lee et al [7] has been utilized to determine J-C model constants under wide range of strain rates and temperatures for 93% WHA. The calculated values of material model constants are then optimized using Evolutionary Algorithm based technique, namely, Genetic Algorithm. The predicted results would help in fine-tuning the JC model constants in terms of absolute error and coefficient of correlation. Furthermore, the calculated as well as optimized values of JC model constants can be appropriately used for FE simulations of machining simulations as well as penetration mechanisms for WHA alloys. 2. Methodology and JC model optimization In this paper, experimental data are taken from Lee et al [7] and utilized to determine J-C model constants under wide range of strain rates and temperatures for 93% WHA. Experimental stress-strain curves obtained from the mentioned literature are converted to true stress- strain curves and data points are taken in plasticity region till Ultimate Tensile Strength (UTS). True stress-strain data has been plotted for given strain rates (4000, 2500, 1600 /s) and temperatures (298, 573, 773, 973, 1173, 1373 K) in Fig. 1 (a-c). These data points are then used for determining the JC Model constants and this approach is considered as M1. Next step is to fine tune the calculated JC model constants obtained from M1 using GA based optimization technique. The new constants are determined and this approach is considered as M2. In addition, another set of JC constant are considered namely, M3, which have been directly taken from the literature where in Rohr and Nahme [4] had derived JC model constants for 93% WHA at strain rates of 2x10-1, 1x10-2, 8x10-4, 6x10-5, 1000, 10000 /s and temperatures of 300 and 827 K and are listed in Table 1. The values obtained from M1, M2 and M3 are compared by calculating the absolute error and coefficient of correlation in each of the cases. This would help in proposing a better methodology to determine the JC model constants with least absolute error. Table 1. Details of M1, M2 and M3. Model Woei and Guo [3]
M1& M2
Rohr and Nahme [4]
M3
A, B, n, c, m
Temperatures K
Strain rates /s
298, 573, 773, 973, 1173, 1373
800,1600,2500,4000
300, 827
2x10 , 1x10 , 8x10 , 6x10 , 1000, 10000
-1
(a)
1197, 580, 0.05, 0.025, 1.9
(b)
-2
-4
-5
18914
Amrita Priyadrashini et al./ Materials Today: Proceedings 5 (2018) 18911–18919
(c)
Fig. 1. Literature [7] True stress-strain curves at different temperatures (a) for Strain rate 4000 /s; (b) for Strain rate 2500 /s, (c) for Strain rate 1600 /s
2.1. Determination of JC model constants Johnson Cook constitutive equation is expressed as: ∗ )(1 )(1 + (1) =( + − ∗ ) where σ is the flow stress, A is the yield stress at reference temperature and strain rate, B is the coefficient of strain hardening, n is the strain hardening exponent, C is the Co-efficient of strain rate dependency, m is the thermal softening exponent, ε is the plastic strain, ∗ = is the dimensionless strain rate and is strain rate and is reference strain rate and T ∗ is homologous temperature and expressed as: ∗ =
(2)
where T is current temperature, is melting temperature (2173 K for 93%WHA) and is reference temperature ). Minimum temperature of the test is considered as reference temperature and maximum strain rate in test ( as reference strain rate. The JC model considers strain hardening, strain rate hardening and thermal softening as three independent phenomena and these can be isolated from each other and calculated individually under certain conditions. Hence, the flow stress can be determined by multiplying these three parameters in Eq. (1). Reference temperature is taken as 298 K while reference strain rate is taken as 4000 /s. Step 1: At reference strain rate and reference temperature. The strain hardening and thermal softening effect will be nil, so second and third term in equation will be equal to unity thereby only first term is left and Eq. (1) reduces to (3) =( + ) yield stress of material at room temperature is 1103 MPa (i.e. the stress at 0.002 strain) as A, Equating the value of A in Eq. (3) and plotting graph between ln (σ – A) vs ln . B is substituted from the intercept of the plot, while n is the slope. Step 2: At fixed plastic strain and reference temperature. The strain hardening effect will be observed and thermal softening effect will be nil, so T* is equal to zero Eq. (1) reduces to. ∗ )(1 + (4) =( + ) Hence, from flow stress data for every fixed plastic strain at various strain rates a value of C is calculated from the slope of − 1 vs ln( ∗ )plot. (
)
Amrita Priyadrashini et al./ Materials Today: Proceedings 5 (2018) 18911–18919
18915
Step 3: At fixed plastic strain and reference strain rate. The strain hardening effect will be nil and thermal softening effect is observed, so ln ( *) is equal to zero Eq. (1) reduces to )(1 − ∗ ) (5) =( + Hence, from flow stress data for every fixed plastic strain at different temperatures a value of m is calculated from the slope of ln 1 − vs ln ( ∗ ) plot. The material constants C and m are determined using the least-square (
)
method. In order to determine optimal C and m values. It involves minimizing average absolute error (Δ) between the experimental and predicted flow stress values. (6) − 1 = × 100 is the experimental flow stress, is the calculated flow stress and N is the number of data points Where considered. Correlation coefficient is commonly used statistical tool which provides linear relationship between experimental and predicted values. it is expressed as. (7) ∑ ( − )( − ) = ∑ ( − ) ∑ ( − ) and are the average values of and respectively. Usually it is considered as higher the value of Where R closer is the correlation with experimental values. but always it not true as equation has tendency to be biased towards higher or lower values [14]. 2.2. Genetic Algorithm (GA) based optimization The genetic algorithm is a method for solving both constrained and unconstrained optimization problems that is based on natural selection, the process that drives biological evolution. The genetic algorithm repeatedly modifies a population of individual solutions. At each step, the genetic algorithm selects individuals at random from the current population to be parents and uses them to produce the children for the next generation. Over successive generations, the population evolves toward an optimal solution [15]. Following are the steps for optimizing the JC constant: Step 1: Create initial population Initial population is created by taking a range of B, C, n & m values and randomly twenty set of initial population has been formed within this range, where function = , , , are considered as design variables. Step 2: Fitness function Fitness has been defined using three objective functions which is the difference between experimental and calculated data of stress- strain curve. , (8) − 1 ( )= × 100 , ,
Δ (x) =
−
1 ,
Δ (x) =
,
× 100
,
−
1
(9)
(10) × 100
where ( ), ( ) and ( ) are absolute error functions under three strain rates 4000, 2500, 1600 /s respectively. i, g, l are strain level at strain rate 4000, 2500, 1600 /s and j, h, r, are number of temperature values considered at
18916
Amrita Priyadrashini et al./ Materials Today: Proceedings 5 (2018) 18911–18919
strain rate 4000, 2500, 1600 /s, N is total no of data points considered,
,
,
are experimental flow
, , are calculated flow stresses respectively. stresses and The main objective of this fitness function is to minimize absolute error function ( ) , ( ) and ( ) simultaneously. Since it is a multi-objective optimization problem the stress-strain responses at different temperature and strain rates are considered as a set of curves. Penalty function method has been induced for objective function as. ( )= (11) ( )+ ( )+ ( ) (12) ( , , , ) = { ( )} are weight coefficients for ( ), ( ) and Δ (x). where , , As fitness function Δ(x) derived from the objective function and is utilized in successive genetic operations. Individuals with high fitness will survive to the next generation. Since optimization is a minimization problem, fitness function is redefined as (13) – ( ) ( )= Where is a predefined constant weight function to make sure fitness function has a positive value. Step 3: Parent Selection From initial population randomly three members are selected and the one with the least Δ is selected as parent 1. The same procedure is repeated for parent 2. Similarly, a set of 10 parents are selected. Step4: Encode The selected 10 parent’s values of B, C, n and m are converted into binary string and concatenated. This long string of 0’s and 1’s is now our chromosome. Step 5: Recombination Selected parents are then crossed over randomly among them and the chromosomes in between these are interchanged. Step 6: Mutation During cross over between two parents a probability of 0.1 is considered as mutation factor and one of its bit is flipped to form new children. Step 7: Decode The new children chromosomes are then decoded to form the constants B, C, n and m. Step 8: Generations The above-mentioned procedure has been repeated for 200 times, i.e. for 200 generations. The whole algorithm is coded in python using libraries numpy, scipy and pandas. The program takes an input of the value A, temperature set and strain rate set at which experimental flow stress data has been recorded and outputs as the optimal B, C, n and m values along with absolute error associate with it. 3. Result and Discussion JC model constants have been calculated using Eq.1-5 based on the experimental data from the literature [7]. The optimal values of C and m value have been determined using least square method (see Eq. 6-7). The obtained values of JC constants have been listed as M1 in Table 2. Genetic algorithm based optimization has been carried out for constants B, C, n, m using Eq. 8-13. Figure 2 shows the evolution process of genetic algorithm. It is observed that from 50th generation absolute error reduced considerably which further got reduced and converged at 180th generation values and there after remained constant. Binary string values of minimum error chromosome has been decoded for B, n, C and m values and are listed as M2 in Table 2. Material constants obtained by Rohr and Nahme [4] have also been given in Table 2 as M3. Furthermore, absolute error percentage (∆) and coefficient of correlation (R) are calculated for M1, M2 and M3 and presented in Table 2 for comparison. Table 2. Details of M1, M2 and M3. Model
A MPa
B
n
c
m
∆
(R)
MPa
M1
1103
707.5
0.220
0.112
1.407
8.72
0.963
M2
1103
875.50
0.353
0.091
1.903
2.93
0.973
M3 [4]
1197
580
0.05
0.025
1.9
16.54
0.922
Amrita Priyadrashini et al./ Materials Today: Proceedings 5 (2018) 18911–18919
18917
Fig. 2 The objection function versus Generation for GA based optimization
It is observed that there is a considerable improvement in predicting the values of JC constants using GA i.e. M2 in terms of error percentage (lowest) and R (highest). In order to have better insight, flow stress curves were plotted obtained from predicted results (M1, M2 and M3) under different strain rates and temperatures compared with that of experimentally obtained flow stresses (see Fig. 3 (a), (b) and (c)). It is clearly observed that flow stress curves obtained from M2 closely matches with that of experimental flow stress curves as compared to that of M1 and M3 material constants.
(a)
(b)
(c) Fig. 3. Comparison of experimental [7] and calculated flow stress obtained from M1, M2 and M3 for different temperatures at strain rate (a) 4000 /s; (b) 2500 /s; (c) 1600 /s
18918
Amrita Priyadrashini et al./ Materials Today: Proceedings 5 (2018) 18911–18919
Fig. 4 (a-c) presents the percentage error of flow stress for M1, M2 and M3 at 298 K for different strain rates. It is observed that percentage error of flow stress at 298 K for M1 is in the range of 1 – 4% for 4000 /s strain rate, 1 – 4% for 2500 /s strain rate and 4 – 8 % for 1600 /s strain rate. Similarly, for M2 percentage error of flow stress is in the range of 1 – 3 % for 4000 /s strain rate, 1 – 2 % for 2500 /s strain rate and 1 – 5 % for 1600 /s strain rate; whereas for M3 percentage error of flow stress is in the range of 7 – 18 %for 4000 /s, 7 – 23 % for 2500 /s strain rate and 6 – 28 % for 1600 /s strain rate. From this observation, it can be stated that JC constants obtained using GA i.e. M2 is capable of predicting better results consistently as compared to M1 and M3.
(a)
(b)
(c)
Fig. 4. Error between experimental [9] and calculated flow stress obtained by JC model constants for M1, M2 and M3 at 298 K: (a) strain rate 4000 /s; (b) strain rate 2500 /s; (c) strain rate 1600 /s.
4. Conclusion JC model constants have been determined based on the experimental data from the literature [7].The optimal values of C and m have been determined using least square method. Genetic algorithm based optimization has been used for fine tuning the constants B, C, n, m. In addition, JC constants obtained by Rohr and Nahme [4] have also been taken into consideration for comparison with the obtained results in terms of absolute percentage error and coefficient of correlation. It was found that JC constants obtained using GA yields the lowest absolute error percentage and highest R. The flow stress curves obtained from GA closely matches with that of experimental flow stress curves. The JC model constants predicted in the present work can be used efficiently for FE simulations of machining process as well penetration mechanisms during ballistic impact.
Amrita Priyadrashini et al./ Materials Today: Proceedings 5 (2018) 18911–18919
18919
References [1]
A. Arora, V.G. Rao “Tungsten Heavy Alloy For Defence Applications,” Materials Technology, Vols. 19-4, no. 26-August-2016, pp. 201263, 2004.
[2]
P.J. Arrazola, O. zel, D. Umbrello, M. Davies, I.S. Jawahir “Recent advances in modelling of metal machining processes,” CIRP Annals Manufacturing Technology, vol. 62, pp. 695-718, 2013.
[3]
M.Sasso*, G.Newaz, D.Amodio “Material characterization at high strain rate by Hopkinson bar tests and Finite element optimization,” Materials Science and Engineering A, vol. 487, pp. 289-300, 2008.
[4]
I. Rohr, H. Nahme, K. Thoma “Material characterisation and constitutive modelling of a tungsten sintered alloy for a wide range of strain rates,” International Journal of Impact Engineering, vol. 35, no. 8, pp. 811-819, 2008.
[5]
Clayton, T. Weerasooriya, John “FAILURE BEHAVIOR OF A TUNGSTEN HEAVY ALLOY,” in International Conference on Tungsten, Refractory & Hardmetals VI, Florida, 2006.
[6] [7]
T. Weerasooriya, “Deformation Behavior of 93W-5Ni-2Fe at Different Rate of cCompression Loading and Temperatures,” USA, 1998. W. Lee, G. Xiea, C. Lin “The strain rate and temperature dependence of the dynamic impact response of tungsten composite,” Materials Science and Engineering A , vol. 257, pp. 256-267, 1998.
[8]
N. Tounsi, J. Vincenti, A. Otho, M.A. Elbestawi “From the basic mechanics of orthogonal metal cutting toward the identification of the constitutive equation,” International Journal of Machine Tools & Manufacture, vol. 42, p. 1373–1383, 2002.
[9]
T. Özel, E. Zeren “Determination of work material flow stress and friction for FEA of machining using orthogonal cutting tests,” Journal of Materials Processing Technology, vol. 153–154, p. 1019–1025, 2004.
[10]
Anaraki MT, Sanjari .M and Akbarzadeh .A “Modeling of high temperature rheological behavior of AZ61 Mg-alloy using inverse method and ANN,” Materials and Design, vol. 29, p. 1701–1706, 2008.
[11]
T. Özel, Y. Karpat “Multi-objective optimization for turning processes using neural network modeling and dynamic-neighborhood particle swarm optimization,” The International Journal of Advanced Manufacturing Technology, vol. 35, no. 3-4, p. 234–247, 2007.
[12]
Dusunceli N, Colak .O.U, Filiz .C “Determination of material parameters of a viscoplastic model by genetic algorithm,” Materials and Design, vol. 31, p. 1250–1255, 2010.
[13]
G. Chen, C. Ren, W. Yu, X. Yang, L. Zhang “Aplication of genetic algorithms for optimizing the johnson-cook constitutive model parameters when simulating the titanium alloy Ti-6Al-4V machining process,” Journal of Engineering Manufacture, pp. 1-11, 2012.
[14]
A. K. Gupta, H.N. Krishnamurthy, Y. Singh, K.M. Prasad “Development of constitutive models for dynamic strain aging regime in Austenitic stainless steel 304,” Materials and design, vol. 45, pp. 616-627, 2013.
[15] [16]
D.M.D. Addona*, R. Teti “Genetic algorithm-based optimization of cutting parameters,” in Procedia CIRP 7, 2013. E. Pink, S. Kumar “Deformation mechanisms operating in a tungsten heavy alloy,” Materials science & Engineering A, Vols. 234-236, pp. 102-105, 1997.
[17]
J. Das, G.A. Rao, S.K. Pabi “Microstructure and mechanical properties of tungsten heavy alloys,” Materials Science and Engineering A, vol. 527, pp. 7841-7847, 2010.
[18]
N.F. Johari, A.M. Zain, N.H. Mustaffa, A. Udin “Machining Parameters Optimization using Hybrid Firefly,” in international conference on computer science and computational mathematics, Langkawi, 2017.
[19]
R. Gero, L. Borukhin, I. Pikus “Some structural effects of plastic deformation on tungsten heavy metal alloys,” Materials Science and Engineering A, vol. 302, pp. 162-167, 2001.
[20]
R.D. Koyee, S. Schmauder, U.Heisel, R. Eisseler “Numerical modeling and optimization of machining duplex stainless,” Production & Manufacturing Research, vol. 3, pp. 36-83, 2015.
[21]
X. Gong, J. Fan, F. Ding “Tensile mechanical properties and fracture behavior of tungsten heavy alloys at 25 to 1100 deg C,” Materials Science and Engineering A, vol. 646, no. 1, pp. 316-321, 2015.