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Finite element analysis of AA1100 elasto-plastic behaviour using Johnson-Cook model
Available online at www.sciencedirect.com
ScienceDirect Materials Today: Proceedings 5 (2018) 5349–5353
www.materialstoday.com/proceedings
ICMPC 2017
Finite element analysis of AA1100 elasto-plastic behaviour using Johnson-Cook model Sonika Sahua, Dehi Pada Mondalb, Manmohan Dass Goelc, Mohd. Zahid Ansaria* a
PDPM-Indian Institute of Information Technology, Design and Manufacturing, Airport Road, Khamaria, Jabalpur 482 005, India b CSIR-Advanced Materials and Processes Research Institute, Bhopal 462 026, India c Department of Applied Mechanics, Visvesvaraya National Institute of Technology, South Ambazari Road, Nagpur 440 010, India
Abstract Johnson-Cook material model is generally used for computation analysis of impact and the penetration related problems involving ductile materials. This paper deals with determination of Johnson-Cook material parameters by considering the strainhardening effect for aluminum alloy AA1100. Uniaxial tensile tests, on AA1100, are performed at strain rates of 10-4 to 10-1 per second using a low capacity but high sensitivity testing machine to obtain the elasto-plastic deformation behaviour of the samples. This behaviour is then test simulated using finite element analysis software. Experimental results showed that the strain hardening behaviour of AA1100 is similar at all quasi-static strain rates considered in the present study at room temperature. Moreover, finite element results are in good agreement with the experimentally obtained results under the domain of the strain rates considered in the present study. © 2017 Elsevier Ltd. All rights reserved. Selection and/or Peer-review under responsibility of 7th International Conference of Materials Processing and Characterization. Keywords: Johnson-Cook model; tensile test; finite element analysis; true stress-strain; elasto-plastic behaviour.
1. Introduction Aluminum 1100 (AA1100) has many applications especially in aerospace, marines and sheet metal industries in the form of composites laminates [1-3]. The technical data of metals and its alloys provided by vendors generally determined by performing respective tests considering specific standards. However, in actual conditions, the material
* Corresponding author. E-mail address: [email protected]
2214-7853© 2017 Elsevier Ltd. All rights reserved. Selection and/or Peer-review under responsibility of 7th International Conference of Materials Processing and Characterization.
5350
Sahu et al. / Materials Today: Proceedings 5 (2018) 5349–5353
undergoes plastic deformation under the effects of dynamic loading, temperature, strain rate and pressure, and as a result of such combination of loadings, it becomes difficult to analyse their mechanical properties considering their end application. Moreover, when a material is subjected to impact or penetration, then it is difficult to analysis the complete behaviour of material in a short span of time. Performing experiments on material would be costly and time consuming. Therefore, researchers made their efforts in this field by developing mathematical models considering experiments data. These equation-based models are coded in the finite element analysis (FEA) software for analyzing the complete deformation behaviour of material. Simulated results can be truly reliable only when a proper constitutive equation is implanted. Many commercial FE software’s such as LSDYNA, ABAQUS, NASTRAN and ANSYS are embedded with user friendly material subroutine codes [4]. Different categories of constitutive models have been proposed such as phenomenological, physical and artificial neural network (ANN) based models [2,5]. These models do not include the data related to the actual physical behaviour of the material such as plastic anisotropic, elastic hysteresis and Bauschinger effect to avoid mathematical complexity. Therefore, these models give slight variation in the FE and experimental results. Phenomenological models are empirical and/or semiempirical based models containing less material parameters than physical and ANN based models. Cowper-Symonds [5], Johnson-Cook [6], Zener-Hollomon [7] and Zerilli-Armstrong [8] are some examples of phenomenological models. In this paper, a study has been done on the basic Johnson-Cook material model, which is an extensively used model for predicting dynamic response of material. This model is widely used in simulation impact and penetration related problems. This model is more popular due to containing simple form of equations and determination of the model constants. Five constants of this model can be determined by only conducting few tests under various loading conditions [9,10]. Therefore, in this study, Johnson-Cook model has been taken for predicting the flow stress behaviour of AA1100 under quasi-static strain rate at room temperature. Strain hardening material parameters of constitutive equation have been used in ABAQUS software. The predicted flow stress behaviour of AA1100 through FE analysis has been compared with the experimental data.
Fig.1. Schematic and actual tensile specimen (in mm) used in experiments as per ASTM E8.
2. Experimental and Simulation Analysis 2.1. Material and Testing AA 1100 is a pure aluminium alloy of chemical composition shown in Table 1. The tensile test specimens were prepared according to ASTM E8 standard as shown above in Fig. 1. The specimens were cut using abrasive water jet cutting machine. Samples were tested under different strain rates, which varies from 10-4 to 10-1 per second at
Sahu et al. / Materials Today: Proceedings 5 (2018) 5349–5353
5351
room temperature. Each test group contains three sets of test specimens. A low capacity but high sensitive tensile machine, Tinius Olsen H250K, was used for the test as shown in Fig. 1. This machine has a load cell capacity up to 25 kN and ram velocity varies from 0.001 mm/min to 500 mm/min. The experimental test results were used to evaluate constitutive Johnson-Cook equation parameters and elastic modulus. Yield point was determined at 0.2% of plastic strain. Table. 1. Chemical composition of AA1100 alloy Material
Si
Mg
Fe
Cu
Mn
Cr
Zn
Ti
Others
Al
(% wt.)
0.041
0.001
0.16
0.001
0.002
0.001
0.002
0.001
0.023
99.763
2.2. Johnson-Cook material model In 1983, Johnson and Cook developed a mathematical equation which involves an independent phenomenon of strain hardening, strain rate hardening and thermal softening. However, it is to be noted that their coupled effects were not involved. While combining effect of strain, strain rate and temperature on material, they proposed a phenomenological model as [6]:
[ A B n ][1 C ln * ][1 (T * )m ]
(1)
where, σ and ε is true stress and true strain; A, B, C are material parameters; and n is hardening exponent and m is softening exponent. * is the ratio of equivalent strain rate to reference strain rate and T * is homologous temperature. In this paper, uniaxial tensile tests were performed at room temperature, therefore only the work hardening part of the Johnson-Cook model will be used and the factor governing strain rate and temperature will be -1 set equal to unity. The reference strain rate is taken as 0.01 s . The flow stress of material does not show variation at room temperature as well as at quasi-strain rates. This indicates the effect of temperature and strain rate hardening could be neglected in Johnson-Cook model therefore the equation reduces to:
[ A B n ]
(2)
The experimental data are curve fitted in MATLAB software to obtain the strain hardening exponent, n, and the Johnson-Cook parameters A and B. 2.3. Simulation Finite element analysis of tensile test was performed by using ABAQUS software. Johnson-Cook material parameters were taken corresponding to 0.01 per second reference strain. The finite element model is shown in Fig. 2. The specimen was meshed with C3D8R element which is an 8 node linear brick element with reduced integration. The total number of element used was 10,900. Reduced integration element allows fast computation with reduced calculation due to lesser integration point. It also shows significant accuracy in the stress/displacement problems. Since one integration point is present at the middle of element, therefore strain is zero at the integration point which causes uncontrolled distortion in the mesh. This phenomena is known as hourglassing. ABAQUS software has element library which contains several techniques to control hourglass effect such as enhanced, relaxed stiffness, stiffness, viscous, combined. Another benefit of reduced integration is that it does not show volume locking effect. The symmetric positions at two sides of specimen were under clamped constraint and constant velocity loading, respectively. The tensile velocity applied as 0.36 mm/s.
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Sahu et al. / Materials Today: Proceedings 5 (2018) 5349–5353
Fig.2. Finite element model of quasi- static tensile specimen.
3. Results and Discussion Figure 3 shows the experimental results for the effect of strain rate on the elasto-plastic behaviour of AA1100 samples. It can be observed that the curve is typical of a ductile metal. The figure clearly indicates weak sensitivity of strain rate on deformation behaviour at room temperatures. The effect of strain rate is different in elastic and plastic regions. The material is showing a high strength in the plastic range as the strain rate is increased. Table 2 presents the results for Johnson-Cook parameters, A and B and exponent n for different strain rates considered in the present study. These were obtained by curve fitting the experimental curves shown in Fig. 3. The average R-square values for the elastic region and plastic region were about 0.995 and 0.906, respectively.
Fig. 3. Experimental results showing the elasto-plastic behaviour of AA1100 at different strain rates. Table 2. Johnson-Cook material model parameters for AA1100 alloy at strain rates.
A (MPa)
B (MPa)
n
0.0001
127.11
299.55
0.24
0.001
128.16
301.73
0.23
0.01
132.78
277.71
0.20
0.1
135.43
319.13
0.24
(/s)
Figure 4 shows comparison between experimental and simulated results for elasto-plastic deformation behaviour of the sample at a strain rate of 0.01 per second. The two results are in good agreement particularly, in the plastic region but same is not the case in elastic region. The reason for this may be attributed to the fact that the JohnsonCook model is strictly valid in the plastic region only, i.e., once the material has yielded; and is unsuitable to be used in elastic region. Thus, Fig. 4 is in fact showing a composite image obtained by joining the elastic region and the plastic region.
Sahu et al. / Materials Today: Proceedings 5 (2018) 5349–5353
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Fig. 4. Comparison between experimental and simulation results for elasto-plastic deformation behaviour at strain rate of 0.01 s-1.
4. Conclusion An experimentally validated method of determining Johnson-Cook constitutive equation parameters have been presented which included the strain-hardening effect. These parameters were obtained by curve fitting the true stress and true strain results obtained from testing AA1100 samples at different strain rates. In general, the behaviour did not show much variation at quasi-static rates and room temperature. However, it is found that strain hardening is pronounced in the plastic region where a high strain rate showed higher strength. The average values of JohnsonCook material parameters A and B are about 130 MPa and 299 MPa, respectively; and the value of n was 0.23 for the material and strain rates considered in the present investigation. Finally, the comparison of experimental and simulation results showed good conformity and thus FE simulation can be used to generate the exact stress-strain curve wherein limitation is on the availability of the experimental facilities. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13]
R.D. Sulamet-Ariobimo, J.W. Soedarsono, T. Sukarnoto, A. Rustandi, Y. Mujalis, D. Prayitno, Tensile properties analysis of AA1100 aluminium and SS400 steel using different JIS tensile standard specimen, J. Appl. Res. Technol. 14 (2016) 148–153. Y.C. Lin, X.M. Chen, A combined Johnson-Cook and Zerilli-Armstrong model for hot compressed typical high-strength alloy steel, Comput. Mater. Sci. 49 (2010) 628–633. E. Corona, G.E. Orient, An Evaluation of the Johnson-Cook Model to Simulate Puncture of 7075 aluminumPlates, Technical Report: Sandia National Lab. US (2014) 1-70. T. Wierzbicki, Y. Bao, Y.W. Lee, Y. Bai, Calibration and evaluation of seven fracture models, Int. J. Mech. Sci. 47 (2005) 719–743 D.N. Zhang, Q.Q. Shangguan, C.J. Xie, F. Liu, A modified Johnson-Cook model of dynamic tensile behaviors for 7075-T6 aluminum alloy, J. Alloys Compd. 619 (2015) 186–194. W.H. Cook, G.R. Johnson, A constitutive model and data for metals subjected to large strains, high strain rates and high temperatures. In: Proceedings of the seventh International symposium on Ballistic. Washington DC,USA: American Defence Preparedness Association.(1983) 541–547 L. Xue, T. Wierzbicki, Ductile fracture characterization of aluminum alloy 2024-T351 using damage plasticity theory, Int.J. Mech. (2009) 267–304. Y.C. Lin, X.M. Chen, G. Lin, A modified Johnson-Cook model for tensile behaviors of typical high-strength alloy steel, Mater. Sci. Eng. A. 527 (2010) 6980–6986. G.H. Majzoobi, F.R.Dehgolan, Determination of the constants of damage models, Procedia Eng. 10 (2011) 764–773. A. Banerjee, S. Dhar, S. Acharyya, D. Datta, N. Nayak, Determinationof Johnson Cook material and failure model constants andnumerical modelling of Charpy impact test of armour steel, Mater. Sci. Eng. A 640 (2015) 200-209 X. Teng, T. Wierzbicki, Evaluation of six fracture models in high velocity perforation, Eng. Fract. Mech. 73 (2006) 1653–1678 W.K. Rule, S.E. Jones, A revised form for the Johnson–Cook strength model, Int. J. Impact Eng. 21 (1998) 609–624 R. Bobbili, B. Ramakrishna, V. Madhu, A.K. Gogia, Prediction of flow stress of 7017 aluminium alloy under high strain rate compression at elevated temperatures, Def. Technol. 11 (2015) 93–98.
ScienceDirect Materials Today: Proceedings 5 (2018) 5349–5353
www.materialstoday.com/proceedings
ICMPC 2017
Finite element analysis of AA1100 elasto-plastic behaviour using Johnson-Cook model Sonika Sahua, Dehi Pada Mondalb, Manmohan Dass Goelc, Mohd. Zahid Ansaria* a
PDPM-Indian Institute of Information Technology, Design and Manufacturing, Airport Road, Khamaria, Jabalpur 482 005, India b CSIR-Advanced Materials and Processes Research Institute, Bhopal 462 026, India c Department of Applied Mechanics, Visvesvaraya National Institute of Technology, South Ambazari Road, Nagpur 440 010, India
Abstract Johnson-Cook material model is generally used for computation analysis of impact and the penetration related problems involving ductile materials. This paper deals with determination of Johnson-Cook material parameters by considering the strainhardening effect for aluminum alloy AA1100. Uniaxial tensile tests, on AA1100, are performed at strain rates of 10-4 to 10-1 per second using a low capacity but high sensitivity testing machine to obtain the elasto-plastic deformation behaviour of the samples. This behaviour is then test simulated using finite element analysis software. Experimental results showed that the strain hardening behaviour of AA1100 is similar at all quasi-static strain rates considered in the present study at room temperature. Moreover, finite element results are in good agreement with the experimentally obtained results under the domain of the strain rates considered in the present study. © 2017 Elsevier Ltd. All rights reserved. Selection and/or Peer-review under responsibility of 7th International Conference of Materials Processing and Characterization. Keywords: Johnson-Cook model; tensile test; finite element analysis; true stress-strain; elasto-plastic behaviour.
1. Introduction Aluminum 1100 (AA1100) has many applications especially in aerospace, marines and sheet metal industries in the form of composites laminates [1-3]. The technical data of metals and its alloys provided by vendors generally determined by performing respective tests considering specific standards. However, in actual conditions, the material
* Corresponding author. E-mail address: [email protected]
2214-7853© 2017 Elsevier Ltd. All rights reserved. Selection and/or Peer-review under responsibility of 7th International Conference of Materials Processing and Characterization.
5350
Sahu et al. / Materials Today: Proceedings 5 (2018) 5349–5353
undergoes plastic deformation under the effects of dynamic loading, temperature, strain rate and pressure, and as a result of such combination of loadings, it becomes difficult to analyse their mechanical properties considering their end application. Moreover, when a material is subjected to impact or penetration, then it is difficult to analysis the complete behaviour of material in a short span of time. Performing experiments on material would be costly and time consuming. Therefore, researchers made their efforts in this field by developing mathematical models considering experiments data. These equation-based models are coded in the finite element analysis (FEA) software for analyzing the complete deformation behaviour of material. Simulated results can be truly reliable only when a proper constitutive equation is implanted. Many commercial FE software’s such as LSDYNA, ABAQUS, NASTRAN and ANSYS are embedded with user friendly material subroutine codes [4]. Different categories of constitutive models have been proposed such as phenomenological, physical and artificial neural network (ANN) based models [2,5]. These models do not include the data related to the actual physical behaviour of the material such as plastic anisotropic, elastic hysteresis and Bauschinger effect to avoid mathematical complexity. Therefore, these models give slight variation in the FE and experimental results. Phenomenological models are empirical and/or semiempirical based models containing less material parameters than physical and ANN based models. Cowper-Symonds [5], Johnson-Cook [6], Zener-Hollomon [7] and Zerilli-Armstrong [8] are some examples of phenomenological models. In this paper, a study has been done on the basic Johnson-Cook material model, which is an extensively used model for predicting dynamic response of material. This model is widely used in simulation impact and penetration related problems. This model is more popular due to containing simple form of equations and determination of the model constants. Five constants of this model can be determined by only conducting few tests under various loading conditions [9,10]. Therefore, in this study, Johnson-Cook model has been taken for predicting the flow stress behaviour of AA1100 under quasi-static strain rate at room temperature. Strain hardening material parameters of constitutive equation have been used in ABAQUS software. The predicted flow stress behaviour of AA1100 through FE analysis has been compared with the experimental data.
Fig.1. Schematic and actual tensile specimen (in mm) used in experiments as per ASTM E8.
2. Experimental and Simulation Analysis 2.1. Material and Testing AA 1100 is a pure aluminium alloy of chemical composition shown in Table 1. The tensile test specimens were prepared according to ASTM E8 standard as shown above in Fig. 1. The specimens were cut using abrasive water jet cutting machine. Samples were tested under different strain rates, which varies from 10-4 to 10-1 per second at
Sahu et al. / Materials Today: Proceedings 5 (2018) 5349–5353
5351
room temperature. Each test group contains three sets of test specimens. A low capacity but high sensitive tensile machine, Tinius Olsen H250K, was used for the test as shown in Fig. 1. This machine has a load cell capacity up to 25 kN and ram velocity varies from 0.001 mm/min to 500 mm/min. The experimental test results were used to evaluate constitutive Johnson-Cook equation parameters and elastic modulus. Yield point was determined at 0.2% of plastic strain. Table. 1. Chemical composition of AA1100 alloy Material
Si
Mg
Fe
Cu
Mn
Cr
Zn
Ti
Others
Al
(% wt.)
0.041
0.001
0.16
0.001
0.002
0.001
0.002
0.001
0.023
99.763
2.2. Johnson-Cook material model In 1983, Johnson and Cook developed a mathematical equation which involves an independent phenomenon of strain hardening, strain rate hardening and thermal softening. However, it is to be noted that their coupled effects were not involved. While combining effect of strain, strain rate and temperature on material, they proposed a phenomenological model as [6]:
[ A B n ][1 C ln * ][1 (T * )m ]
(1)
where, σ and ε is true stress and true strain; A, B, C are material parameters; and n is hardening exponent and m is softening exponent. * is the ratio of equivalent strain rate to reference strain rate and T * is homologous temperature. In this paper, uniaxial tensile tests were performed at room temperature, therefore only the work hardening part of the Johnson-Cook model will be used and the factor governing strain rate and temperature will be -1 set equal to unity. The reference strain rate is taken as 0.01 s . The flow stress of material does not show variation at room temperature as well as at quasi-strain rates. This indicates the effect of temperature and strain rate hardening could be neglected in Johnson-Cook model therefore the equation reduces to:
[ A B n ]
(2)
The experimental data are curve fitted in MATLAB software to obtain the strain hardening exponent, n, and the Johnson-Cook parameters A and B. 2.3. Simulation Finite element analysis of tensile test was performed by using ABAQUS software. Johnson-Cook material parameters were taken corresponding to 0.01 per second reference strain. The finite element model is shown in Fig. 2. The specimen was meshed with C3D8R element which is an 8 node linear brick element with reduced integration. The total number of element used was 10,900. Reduced integration element allows fast computation with reduced calculation due to lesser integration point. It also shows significant accuracy in the stress/displacement problems. Since one integration point is present at the middle of element, therefore strain is zero at the integration point which causes uncontrolled distortion in the mesh. This phenomena is known as hourglassing. ABAQUS software has element library which contains several techniques to control hourglass effect such as enhanced, relaxed stiffness, stiffness, viscous, combined. Another benefit of reduced integration is that it does not show volume locking effect. The symmetric positions at two sides of specimen were under clamped constraint and constant velocity loading, respectively. The tensile velocity applied as 0.36 mm/s.
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Sahu et al. / Materials Today: Proceedings 5 (2018) 5349–5353
Fig.2. Finite element model of quasi- static tensile specimen.
3. Results and Discussion Figure 3 shows the experimental results for the effect of strain rate on the elasto-plastic behaviour of AA1100 samples. It can be observed that the curve is typical of a ductile metal. The figure clearly indicates weak sensitivity of strain rate on deformation behaviour at room temperatures. The effect of strain rate is different in elastic and plastic regions. The material is showing a high strength in the plastic range as the strain rate is increased. Table 2 presents the results for Johnson-Cook parameters, A and B and exponent n for different strain rates considered in the present study. These were obtained by curve fitting the experimental curves shown in Fig. 3. The average R-square values for the elastic region and plastic region were about 0.995 and 0.906, respectively.
Fig. 3. Experimental results showing the elasto-plastic behaviour of AA1100 at different strain rates. Table 2. Johnson-Cook material model parameters for AA1100 alloy at strain rates.
A (MPa)
B (MPa)
n
0.0001
127.11
299.55
0.24
0.001
128.16
301.73
0.23
0.01
132.78
277.71
0.20
0.1
135.43
319.13
0.24
(/s)
Figure 4 shows comparison between experimental and simulated results for elasto-plastic deformation behaviour of the sample at a strain rate of 0.01 per second. The two results are in good agreement particularly, in the plastic region but same is not the case in elastic region. The reason for this may be attributed to the fact that the JohnsonCook model is strictly valid in the plastic region only, i.e., once the material has yielded; and is unsuitable to be used in elastic region. Thus, Fig. 4 is in fact showing a composite image obtained by joining the elastic region and the plastic region.
Sahu et al. / Materials Today: Proceedings 5 (2018) 5349–5353
5353
Fig. 4. Comparison between experimental and simulation results for elasto-plastic deformation behaviour at strain rate of 0.01 s-1.
4. Conclusion An experimentally validated method of determining Johnson-Cook constitutive equation parameters have been presented which included the strain-hardening effect. These parameters were obtained by curve fitting the true stress and true strain results obtained from testing AA1100 samples at different strain rates. In general, the behaviour did not show much variation at quasi-static rates and room temperature. However, it is found that strain hardening is pronounced in the plastic region where a high strain rate showed higher strength. The average values of JohnsonCook material parameters A and B are about 130 MPa and 299 MPa, respectively; and the value of n was 0.23 for the material and strain rates considered in the present investigation. Finally, the comparison of experimental and simulation results showed good conformity and thus FE simulation can be used to generate the exact stress-strain curve wherein limitation is on the availability of the experimental facilities. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13]
R.D. Sulamet-Ariobimo, J.W. Soedarsono, T. Sukarnoto, A. Rustandi, Y. Mujalis, D. Prayitno, Tensile properties analysis of AA1100 aluminium and SS400 steel using different JIS tensile standard specimen, J. Appl. Res. Technol. 14 (2016) 148–153. Y.C. Lin, X.M. Chen, A combined Johnson-Cook and Zerilli-Armstrong model for hot compressed typical high-strength alloy steel, Comput. Mater. Sci. 49 (2010) 628–633. E. Corona, G.E. Orient, An Evaluation of the Johnson-Cook Model to Simulate Puncture of 7075 aluminumPlates, Technical Report: Sandia National Lab. US (2014) 1-70. T. Wierzbicki, Y. Bao, Y.W. Lee, Y. Bai, Calibration and evaluation of seven fracture models, Int. J. Mech. Sci. 47 (2005) 719–743 D.N. Zhang, Q.Q. Shangguan, C.J. Xie, F. Liu, A modified Johnson-Cook model of dynamic tensile behaviors for 7075-T6 aluminum alloy, J. Alloys Compd. 619 (2015) 186–194. W.H. Cook, G.R. Johnson, A constitutive model and data for metals subjected to large strains, high strain rates and high temperatures. In: Proceedings of the seventh International symposium on Ballistic. Washington DC,USA: American Defence Preparedness Association.(1983) 541–547 L. Xue, T. Wierzbicki, Ductile fracture characterization of aluminum alloy 2024-T351 using damage plasticity theory, Int.J. Mech. (2009) 267–304. Y.C. Lin, X.M. Chen, G. Lin, A modified Johnson-Cook model for tensile behaviors of typical high-strength alloy steel, Mater. Sci. Eng. A. 527 (2010) 6980–6986. G.H. Majzoobi, F.R.Dehgolan, Determination of the constants of damage models, Procedia Eng. 10 (2011) 764–773. A. Banerjee, S. Dhar, S. Acharyya, D. Datta, N. Nayak, Determinationof Johnson Cook material and failure model constants andnumerical modelling of Charpy impact test of armour steel, Mater. Sci. Eng. A 640 (2015) 200-209 X. Teng, T. Wierzbicki, Evaluation of six fracture models in high velocity perforation, Eng. Fract. Mech. 73 (2006) 1653–1678 W.K. Rule, S.E. Jones, A revised form for the Johnson–Cook strength model, Int. J. Impact Eng. 21 (1998) 609–624 R. Bobbili, B. Ramakrishna, V. Madhu, A.K. Gogia, Prediction of flow stress of 7017 aluminium alloy under high strain rate compression at elevated temperatures, Def. Technol. 11 (2015) 93–98.