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Development of Johnson Cook Model for Zircaloy-4 with Low Oxygen Content
Available online at www.sciencedirect.com
ScienceDirect Materials Today: Proceedings 4 (2017) 966–974
www.materialstoday.com/proceedings
5th International Conference of Materials Processing and Characterization (ICMPC 2016)
Development of Johnson Cook Model for Zircaloy-4 with Low Oxygen Content K Limbadria, Hansoge Nitin Krishnamurthyb, A. Maruthi Ramc, N Saibabac, V.V. Kutumba Raod, J N Murthya, Amit Kumar Guptab and Swadesh Kumar Singha* a
Gokaraju Rangaraju Institute of Engineering and Technology, Hyderabad, 500090, India. b Birla Institute of Technology and Science, Hyderabad, 500090, India. c Nuclear Fuel Complex, Hyderabad, 500062, India. d Former Pprofessor, Iindian Institute of Technology, Bhubaneswar, 751013,India.
Abstract Strain, temperature, strain rate and the material properties such as strength coefficient, strain hardening exponent and coefficient of strain rate hardening affect the flow stress of a material which in turn affects the formability. In this present study, tensile tests are conducted on Zircaloy-4 at different strain rates(0.01s-1, 0.005s-1, 0.001s-1)andin the temperature range of 298K to 423K. The flow stress behaviour is analysed and the material constants are determined by modelling the flow behaviour using the JohnsonCook constitutive equation. The predictability of this constitutive model is determined based on the statistical parameters such as correlation coefficient(R), average absolute error (∆) and standard deviation (s). ©2017 Elsevier Ltd. All rights reserved. Selection and peer-review under responsibility of Conference Committee Members of 5th International Conference of Materials Processing and Characterization (ICMPC 2016). Keywords:Zircaloy-4; strain rate; flow stress; Johnson Cook; coefficient of correlation.
1.
Introduction:
In recent years, Zircaloy-4 has proved to be a good structural material for pressurized water, boiling water reactor,CANDUreactor applications such as uranium fuel cladding tubes[1], [2], grids and channels[3] in BWRs and pressure,calandria tubes in PHWRs[3] and other internal components in fuel-bundleassembly. This is mainly due to its less absorption of neutrons and high corrosion resistance in water at high temperatures. Formability is
2214-7853©2017 Elsevier Ltd. All rights reserved. Selection and peer-review under responsibility of Conference Committee Members of 5th International Conference of Materials Processing and Characterization (ICMPC 2016).
K Limbadri et al / Materials Today: Proceedings 4 (2017) 966–974
967
animportant characteristic of the material to understand thedeformationof different kinds of tubes. Hence,a comprehensive knowledge of the material flow behaviorin various realistic conditions is to be analyzed. Finite element methods (FE) have become important in almost all the research sectors, and especially in mechanical research organizations for reducing the processing times and gaining the economic benefit in avoiding very expensive tools. In establishing effective manufacturing process and optimizing the process parameters, the FE methods are benedictions in the industrial world. The FE codes are inexhaustible tools for simulating the ambient parameters such as loading conditions, temperature, etc.[4]. These finite element simulations require mathematical expressions to predict exact material behaviour under different conditions. The mathematical expressions are generally called as constitutive equations which require material constants to be evaluated. There are many constitutive models and their use and accuracy of predictions are rapidly increasing in their respective fields.These can be differentiated in broadly two groups, phenomenological and physical based constitutive models. The phenomenological constitutive models are developed based on the empirical observations by statistical mathematical tools. The physical based constitutive models are built on the physical phenomena that the materials undergo during the deformation of the material. Generally these are developed by observing the dislocation movement at various temperatures, theory of thermodynamics and kinetics of slips. The physical based constitutive modes are more precise in predicting the flow stress values as compared to the phenomenological, but these modes require a large number of constants to accurately predict the deformation behaviour of the material. The empirical models are very simple and involve less number of constants.Theseequations have less complicated functions, but are not as accurate as the physical based in predicting the results. The Johnson-Cook model (JC)[5], Arrhenius equations and Khan-Huang-Liang (KHL) are some of the phenomenological models. Zerilli-Armstrong Model (ZA)[6], Rusinek–Klepaczkomodel (RK)[7] and Bonder-Partom (BP) model are physical based models. Various researchers have studied different aspects of characterization of the material and modelled them. Schaffler et al.[8]have studied the effect of neutron-irradiation on thermo mechanical properties of Zircaloy-4 in temperature range between 350oC and 400oC, fluency range 0-85× 1023nm-2 and also proposed a visco-elastic model to illustrate irradiation induced hardening. Grange et al.[2]developed ductile fracture mechanism of the material for centre-crack panel specimens by using Gurson-Tvergaar Needleman model and it was validated by simulating crack initiation and propagation. Shearing edges and burr formation of the material has been analysed and GursonTvergaar Needleman model and also Johnson-Cook model was developed and validated by comparing characteristics of sheared edges of grid and punch-force displacement[9]. JinyuanZhai et al.[10]have developed continuum plasticity model using intervals of stress deviators to display the tension-compression asymmetry of beta treated Zircaloy4. Even though research insights have the displayed characterization of Zircaloy-4 in the case of different manufacturing processes, the flow stress behaviour which mainly depends on strain, strain rate, temperature and other material constants has not yet been modelled. Lee et al.[11], observed DSA phenomena between 250oC and 400oC and have developed a model which predicts the temperature dependency of ductility and strain. As the Zircaloy-4 tubes are manufactured from the rolled sheets or from pilger process, the understanding of deformation behaviour of this material is important and therefore in the present study, the flow stress behaviour of this material is modelled by Johnson Cook constitutive equation.This mathematical expression will be used in the Finite element codes to predict the flow stress behaviour at different conditions. The JC model, being a phenomenological method, lacks physical background behind the equation. However, it is one of the widely used amongst the phenomenological constitutive models due its simplicity and less number of material constants. The complete JC model, developed for the Zircaloy-4, has been explained in detail in the following discussion. 2.
Experimentation:
The Zircaloy-4, with less oxygen content, of 1mm thick sheet was milled into tensile test samples in rolling direction. The samples were manufactured according to standards [12]at Nuclear Fuel Complex (NFC) and is shown in Fig.1.The chemical composition of the material is listed in the Table 1.
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Table 1: Chemical composition of Zircaloy-4 with low oxygen content (wt. %) Element
Sn
Fe
Cr
Oxygen
Wt%
1.33
0.22
0.1
834 ppm
The tensile tests were conducted on a computer controlled UTM, shown inFig.2, which has a maximum load capacity of 100kN. The machine is equipped with a controlsystem to impose an exponential increase of the actuator speed to obtain constant true strain rates. A contact type extensometer is used to measure the strains. The resistance heating split furnace is used to heat the tensile test specimen up to 423K. The pull rods for the high temperature testing at UTM are designed to be made of Nickel base super-alloy CM-247. The experiments were conducted at temperatures of 298K–423 K at intervals of 50 K and at strain rates of 0.01, 0.005 and 0.001s-1. Standard equations were used to convert the load–displacement data to true stress–true strain data. The elastic region was subtracted from the true stress–true strain curve to get true stress–true plastic strain data. The data range of strain rate, temperature and true plastic strain is shown in Table 2.
Figure 1: Dimensions of the UTM specimen for tensile testing (in mm)
3.
Johnson Cook model:
The development of stress in materials, according to Johnson-Cook model[13], is dependent on three independent phenomena: isotropic hardening, strain rate hardening and thermal hardening, and all these are independentofeach other. The mathematical expression or constitutive equation representing all the three terms is a multiplication of all the stresses for [12]. The resultant equation is 𝜎 = (𝐴 + 𝐵𝜀 𝑛 )(1 + 𝐶 ln 𝜀̇∗ )(1 − 𝑇 ∗𝑚 )
(1)
Table 2: Tensile test data range for JC model development Strain rate (s-1)
Temperature(K)
True plastic strain
0.01
298
0.05
0.005
348
0.1
0.001
423
0.15 0.2 0.25
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Fig.2: Computerized UTM of 100kN capacity with resistance heating 3-zone split furnace
where 𝜎 is the Von-mises flow stress and (𝐴 + 𝐵𝜀 𝑛 ) , (1 + 𝐶 ln 𝜀̇∗ ), (1 − 𝑇 ∗𝑚 ) are the stresses resulting from isotropic strain hardening, strain rate hardening and thermal softening respectively. A indicates the yield stress at reference temperature and reference strain rate, B indicates the coefficient of strain hardening, 𝜀 indicates the true plastic strain, 𝑛 is strain hardening exponent, 𝐶 is the coefficient of strain rate hardening, 𝜀̇∗ = 𝜀̇/𝜀̇0, where 𝜀̇ is true strain rate, 𝜀̇0 is the reference strain rate, 𝑇 ∗ is the homologous temperature and the 𝑚 is the thermal softening exponent. The homologous temperature is expressed as 𝑇∗ =
𝑇−𝑇𝑟𝑒𝑓
𝑇𝑚 −𝑇𝑟𝑒𝑓
(2)
where, 𝑇 is absolute temperature, 𝑇𝑟𝑒𝑓 is the reference temperature (in general the minimum temperature of the experimental range is considered as reference temperature) in such a way that 𝑇 ≥ 𝑇𝑟𝑒𝑓 , 𝑇𝑚 is the melting temperature. 4. Determination of material constants in JC model: The material constants for the JC model is derived in three steps 4.1. Step 1: At reference strain rate and reference temperature Under this condition, ln 𝜀̇∗ = 0 and in the same way at reference temperature the homologous temperature value is also zero. Since the stresses resulting from strain rate hardening and thermal softening become unity, the resultant stress, from the equation (1) becomes
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𝜎 = (𝐴 + 𝐵𝜀 𝑛 )
(3)
the value of A has been taken at the reference strain rate 0.01s-1 and the reference temperature 298 K. The coefficient of strain hardening,𝐵, and the strain hardening exponent,𝑛, are determined by plotting a straight line between ln(𝜎 − 𝐴)and ln(𝜀). 4.2. Step 2: At a fixed strain and reference temperature
At reference temperature the homologous temperature becomes zero therefore the stress resulted from thermal softening is unity. So only the first two terms in the Eq(1) remain 𝜎 = (𝐴 + 𝐵𝜀 𝑛 )(1 + 𝐶 ln 𝜀̇∗ )
(4)
The strain rate hardening coefficient,𝐶, is determined by solving the equations of lines formed by variables between 𝜎 �(𝐴+𝐵𝜀𝑛 ) − 1� and ln(𝜀̇∗ ). Therefore there are as many values of 𝐶 as the number of true plastic strain 𝜀 values. 4.3. Step 3: At a fixed strain and reference strain rate
Under this condition, 𝜀̇∗ = 1 and therefore the ln 𝜀̇∗ = 0 which accounts for the stresses caused by strain hardening.The equation is modified to 𝜎 = (𝐴 + 𝐵𝜀 𝑛 )(1 − 𝑇 ∗𝑚 )
(5) 𝜎
The thermal softening exponent (𝑚) can be obtained by plotting the line between �1 − (𝐴+𝐵𝜀𝑛 )� and ln(𝑇 ∗ ). Hence, for each value of true plastic strain (𝜀) there is a value of𝑚.
4.4. Optimization of the C and m values As there are 5 values of C and m corresponding to 5 different values of strain, there is a need for optimization to obtain the best fit. This is done using the statistical method of least squares which involve the minimization of average absolute error (∆) between the experimental flow stress (𝜎𝑒𝑥𝑝 ) and predicted flow stress (𝜎𝑝 ) values. The mathematical expression of the average absolute error (∆) is given below Eq (6). 1
∆= ∑𝑖=𝑁 𝑖=1 � 𝑁
𝑖 −𝜎 𝑖 𝜎𝑒𝑥𝑝 𝑝 𝑖 𝜎𝑒𝑥𝑝
(6)
� × 100
where𝜎𝑒𝑥𝑝 is the experimental flow stress, 𝜎𝑝 is the predicted flow stress and the N is the number of data points taken in to account. 4.5. Predictability of the constitutive equation All the constants, A, B, C, n and m have been determined, therefore, the complete constitutive equation for predicting the flow stress has been expressed by substituting these values in Eq. (1). Moreover, the predictability of the constitutive equation can be computed based on the statistical parameters of correlation coefficient (R), average absolute error (∆) and standard deviation (s). The correlation coefficient is the most frequently used statistical parameter for expressing the information on strength of linear relationship between the experimental and predicted values. It is illustrated in Eq. (7).
𝑅=
𝑖 ∑𝑖=𝑁 �𝑒𝑥𝑝 ��𝜎𝑝𝑖 −𝜎 �𝑝 � 𝑖=1 �𝜎𝑒𝑥𝑝 −𝜎 2
𝑖 −𝜎 𝑖 � � �∑𝑖=𝑁�𝜎𝑒𝑥𝑝 �𝑒𝑥𝑝 � ∑𝑖=𝑁 𝑝 𝑖=1 𝑖=1 �𝜎𝑝 −𝜎
2
(7)
K Limbadri et al / Materials Today: Proceedings 4 (2017) 966–974
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where𝜎�𝑒𝑥𝑝 , 𝜎�𝑝 are the average experimental flow stress and average predicted flow stress respectively. Here, R and ∆ are biased and unbiased statistical values, in other words, if the R value is high, it does not necessarily imply the performance of the model as high. The average error∆is unbiased, as it calculates the values term by term for determining the predictability of the model. All the above equations are solved by using MATLAB. All the constants thus obtained for JC model are listed in Table 3. Table 3: Material constants for JC model
5.
Material Constant
A
B
C
n
m
Value
283
774.2
0.0152
0.8499
0.8012
Results and Conclusion:
The equation for JC model is given by 𝜎 = (283 + 774.2 × 𝜀 0.8499 )(1 + 0.0152 × ln 𝜀̇∗ )(1 − 𝑇 ∗0.8012)
600
(8)
(a) 0.01s-1
1
2
True Stress(MPa)
500 400 300 200 100 0 0
0.05
0.1
ExperimentalPredicted-
298K 1: 298K
0.15
0.2
True Plastic Strain 348K
2: 348K
423K 3: 423K
0.25
0.3
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600
(b) 0.005s-1
1
2
True Stress(MPa)
500 400 300 200 100 0 0
0.05 ExperimentalPredicted-
600
0.1
0.15
0.2
298K 1: 298K
348K
2: 348K
0.3
423K 3: 423K
(c) 0.001s-1 1
500
True Stress(MPa)
0.25
True Plastic Strain
2
400 300 200 100 0 0
0.05 Experimental-
0.1
0.15
0.2
0.25
0.3
True Plastic Strain
298K 1: 298K
348K
2: 348K
423K 3: 423K
Fig.3 : Graphs showing experimental and predicted stress for JC Model at various strains for strain rates of (a) 0.01s-1, (b) 0.005s-1, (c) 0.001s-1.
K Limbadri et al / Materials Today: Proceedings 4 (2017) 966–974
Predicted True Stress (Mpa)
600
973
R=0.7616
500 400 300 200 100 0 0
100
200 300 400 Experimental True Stress (Mpa) JC Model
500
600
Linear (JC Model)
Fig. 4: Correlation between experimental and predicted stress.
Figures 3 (a), (b) and (c) show the predicted values of flow stress for JC model at strain rates of 0.01, 0.005 and 0.001s-1. However, the R value for the model came out to be 0.7616(as shown in Fig. 4) and value of average absolute error was 11.95% and standard deviation of 12.13. The predictive capability of the JC model is low mainly due to the influence of the coupled effect of strain and temperature, and of strain rate and temperature. This effect is not captured in the model. Hence, it can be concluded that the flow behaviour of Zircaloy-4 with low oxygen content at elevated temperature is not governed by three independent phenomena, viz., thermal softening, strain rate hardening and isotropic strain hardening. In order to correctly model the deformation behaviour of Zircaloy-4 with low oxygen content, there is a need to consider other constitutive models which consider the coupled effect of strain and temperature, and of strain rate and temperature. The future work will involve the development of physical based models such as the Zerilli-Armstrong Model (ZA) andRusinek–Klepaczkomodel (RK) along with the empirical Arrhenius model. Acknowledgment: The financial support received for this research work from Department of Atomic Energy, Government of India, 36(2)/14/56/2014-BRNS/2699 is gratefully acknowledged. References: [1]
P. Geyer and P. Bouffioux, “Thermomechanical Behavior and Modeling Between 350 ° C and 400 ° C of Zircaloy-4 Cladding Tubes From an Unirradiated State to High Fluence,” vol. 122, no. April 2000, pp. 168– 176, 2016.
[2]
M. Grange, J. Besson, and E. Andrieu, “An anisotropic Gurson type model to represent the ductile rupture of hydrided Zircaloy-4 sheets,” Int. J. Fract., vol. 105, pp. 273–293, 2000.
[3]
K. L. Murty and I. Charit, “Texture development and anisotropic deformation of zircaloys,” Prog. Nucl. Energy, 2006.
[4]
A. K. Gupta, H. N. Krishnamurthy, P. Puranik, S. K. Singh, and A. Balu, “An exponential strain dependent Rusinek-Klepaczko model for flow stress prediction in austenitic stainless steel 304 at elevated temperatures,” J. Mater. Res. Technol., vol. 3, no. 4, pp. 370–377, 2014.
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[5]
A. K. Gupta, H. N. Krishnamurthy, Y. Singh, K. M. Prasad, and S. K. Singh, “Development of constitutive models for dynamic strain aging regime in Austenitic stainless steel 304,” Mater. Des., vol. 45, pp. 616–627, 2013.
[6]
N. Kotkunde, H. N. Krishnamurthy, P. Puranik, A. K. Gupta, and S. K. Singh, “Microstructure study and constitutive modeling of Ti-6Al-4V alloy at elevated temperatures,” Mater. Des., vol. 54, pp. 96–103, 2014.
[7]
Y. Tian, L. Huang, H. Ma, and J. Li, “Establishment and comparison of four constitutive models of 5A02 aluminium alloy in high-velocity forming process,” Mater. Des., vol. 54, pp. 587–597, 2014.
[8]
I. Schaffler, “Thermomechanical Behavior and Modeliing Between 350 C and 400 C of Zircaloy-4 Cladding Tubes From an Unirradiated State to Highh Fluence,” J. Eng. Mater. Technol., vol. 122, pp. 168–176, 2000.
[9]
H. Jeong, J. Ha, J. Hwang, H. Lee, D. Kim, and N. Kim, “A study on the shearing process and the burr formation of zircaloy-4 sheet by using GTN model,” Int. J. Precis. Eng. Manuf., 2014.
[10]
J. Zhai, X. Gao, J. C. Sobotka, B. A. Webler, and B. V. Cockeram, “Modeling the tension-compression asymmetric yield behavior of ??-treated Zircaloy-4,” J. Nucl. Mater., 2014.
[11]
K. W. Lee, S. K. Kim, K. T. Kim, and S. I. Hong, “Ductility and strain rate sensitivity of Zircaloy-4 nuclear fuel claddings,” J. Nucl. Mater., 2001.
[12]
A. K. Gupta, V. K. Anirudh, and S. K. Singh, “Constitutive models to predict flow stress in Austenitic Stainless Steel 316 at elevated temperatures,” J. Mater., vol. 43, pp. 410–418, 2013.
[13]
D. Samantaray, S. Mandal, and A. K. Bhaduri, “A comparative study on Johnson Cook, modified ZerilliArmstrong and Arrhenius-type constitutive models to predict elevated temperature flow behaviour in modified 9Cr-1Mo steel,” Comput. Mater. Sci., vol. 47, no. 2, pp. 568–576, 2009.
ScienceDirect Materials Today: Proceedings 4 (2017) 966–974
www.materialstoday.com/proceedings
5th International Conference of Materials Processing and Characterization (ICMPC 2016)
Development of Johnson Cook Model for Zircaloy-4 with Low Oxygen Content K Limbadria, Hansoge Nitin Krishnamurthyb, A. Maruthi Ramc, N Saibabac, V.V. Kutumba Raod, J N Murthya, Amit Kumar Guptab and Swadesh Kumar Singha* a
Gokaraju Rangaraju Institute of Engineering and Technology, Hyderabad, 500090, India. b Birla Institute of Technology and Science, Hyderabad, 500090, India. c Nuclear Fuel Complex, Hyderabad, 500062, India. d Former Pprofessor, Iindian Institute of Technology, Bhubaneswar, 751013,India.
Abstract Strain, temperature, strain rate and the material properties such as strength coefficient, strain hardening exponent and coefficient of strain rate hardening affect the flow stress of a material which in turn affects the formability. In this present study, tensile tests are conducted on Zircaloy-4 at different strain rates(0.01s-1, 0.005s-1, 0.001s-1)andin the temperature range of 298K to 423K. The flow stress behaviour is analysed and the material constants are determined by modelling the flow behaviour using the JohnsonCook constitutive equation. The predictability of this constitutive model is determined based on the statistical parameters such as correlation coefficient(R), average absolute error (∆) and standard deviation (s). ©2017 Elsevier Ltd. All rights reserved. Selection and peer-review under responsibility of Conference Committee Members of 5th International Conference of Materials Processing and Characterization (ICMPC 2016). Keywords:Zircaloy-4; strain rate; flow stress; Johnson Cook; coefficient of correlation.
1.
Introduction:
In recent years, Zircaloy-4 has proved to be a good structural material for pressurized water, boiling water reactor,CANDUreactor applications such as uranium fuel cladding tubes[1], [2], grids and channels[3] in BWRs and pressure,calandria tubes in PHWRs[3] and other internal components in fuel-bundleassembly. This is mainly due to its less absorption of neutrons and high corrosion resistance in water at high temperatures. Formability is
2214-7853©2017 Elsevier Ltd. All rights reserved. Selection and peer-review under responsibility of Conference Committee Members of 5th International Conference of Materials Processing and Characterization (ICMPC 2016).
K Limbadri et al / Materials Today: Proceedings 4 (2017) 966–974
967
animportant characteristic of the material to understand thedeformationof different kinds of tubes. Hence,a comprehensive knowledge of the material flow behaviorin various realistic conditions is to be analyzed. Finite element methods (FE) have become important in almost all the research sectors, and especially in mechanical research organizations for reducing the processing times and gaining the economic benefit in avoiding very expensive tools. In establishing effective manufacturing process and optimizing the process parameters, the FE methods are benedictions in the industrial world. The FE codes are inexhaustible tools for simulating the ambient parameters such as loading conditions, temperature, etc.[4]. These finite element simulations require mathematical expressions to predict exact material behaviour under different conditions. The mathematical expressions are generally called as constitutive equations which require material constants to be evaluated. There are many constitutive models and their use and accuracy of predictions are rapidly increasing in their respective fields.These can be differentiated in broadly two groups, phenomenological and physical based constitutive models. The phenomenological constitutive models are developed based on the empirical observations by statistical mathematical tools. The physical based constitutive models are built on the physical phenomena that the materials undergo during the deformation of the material. Generally these are developed by observing the dislocation movement at various temperatures, theory of thermodynamics and kinetics of slips. The physical based constitutive modes are more precise in predicting the flow stress values as compared to the phenomenological, but these modes require a large number of constants to accurately predict the deformation behaviour of the material. The empirical models are very simple and involve less number of constants.Theseequations have less complicated functions, but are not as accurate as the physical based in predicting the results. The Johnson-Cook model (JC)[5], Arrhenius equations and Khan-Huang-Liang (KHL) are some of the phenomenological models. Zerilli-Armstrong Model (ZA)[6], Rusinek–Klepaczkomodel (RK)[7] and Bonder-Partom (BP) model are physical based models. Various researchers have studied different aspects of characterization of the material and modelled them. Schaffler et al.[8]have studied the effect of neutron-irradiation on thermo mechanical properties of Zircaloy-4 in temperature range between 350oC and 400oC, fluency range 0-85× 1023nm-2 and also proposed a visco-elastic model to illustrate irradiation induced hardening. Grange et al.[2]developed ductile fracture mechanism of the material for centre-crack panel specimens by using Gurson-Tvergaar Needleman model and it was validated by simulating crack initiation and propagation. Shearing edges and burr formation of the material has been analysed and GursonTvergaar Needleman model and also Johnson-Cook model was developed and validated by comparing characteristics of sheared edges of grid and punch-force displacement[9]. JinyuanZhai et al.[10]have developed continuum plasticity model using intervals of stress deviators to display the tension-compression asymmetry of beta treated Zircaloy4. Even though research insights have the displayed characterization of Zircaloy-4 in the case of different manufacturing processes, the flow stress behaviour which mainly depends on strain, strain rate, temperature and other material constants has not yet been modelled. Lee et al.[11], observed DSA phenomena between 250oC and 400oC and have developed a model which predicts the temperature dependency of ductility and strain. As the Zircaloy-4 tubes are manufactured from the rolled sheets or from pilger process, the understanding of deformation behaviour of this material is important and therefore in the present study, the flow stress behaviour of this material is modelled by Johnson Cook constitutive equation.This mathematical expression will be used in the Finite element codes to predict the flow stress behaviour at different conditions. The JC model, being a phenomenological method, lacks physical background behind the equation. However, it is one of the widely used amongst the phenomenological constitutive models due its simplicity and less number of material constants. The complete JC model, developed for the Zircaloy-4, has been explained in detail in the following discussion. 2.
Experimentation:
The Zircaloy-4, with less oxygen content, of 1mm thick sheet was milled into tensile test samples in rolling direction. The samples were manufactured according to standards [12]at Nuclear Fuel Complex (NFC) and is shown in Fig.1.The chemical composition of the material is listed in the Table 1.
K Limbadri et al / Materials Today: Proceedings 4 (2017) 966–974
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Table 1: Chemical composition of Zircaloy-4 with low oxygen content (wt. %) Element
Sn
Fe
Cr
Oxygen
Wt%
1.33
0.22
0.1
834 ppm
The tensile tests were conducted on a computer controlled UTM, shown inFig.2, which has a maximum load capacity of 100kN. The machine is equipped with a controlsystem to impose an exponential increase of the actuator speed to obtain constant true strain rates. A contact type extensometer is used to measure the strains. The resistance heating split furnace is used to heat the tensile test specimen up to 423K. The pull rods for the high temperature testing at UTM are designed to be made of Nickel base super-alloy CM-247. The experiments were conducted at temperatures of 298K–423 K at intervals of 50 K and at strain rates of 0.01, 0.005 and 0.001s-1. Standard equations were used to convert the load–displacement data to true stress–true strain data. The elastic region was subtracted from the true stress–true strain curve to get true stress–true plastic strain data. The data range of strain rate, temperature and true plastic strain is shown in Table 2.
Figure 1: Dimensions of the UTM specimen for tensile testing (in mm)
3.
Johnson Cook model:
The development of stress in materials, according to Johnson-Cook model[13], is dependent on three independent phenomena: isotropic hardening, strain rate hardening and thermal hardening, and all these are independentofeach other. The mathematical expression or constitutive equation representing all the three terms is a multiplication of all the stresses for [12]. The resultant equation is 𝜎 = (𝐴 + 𝐵𝜀 𝑛 )(1 + 𝐶 ln 𝜀̇∗ )(1 − 𝑇 ∗𝑚 )
(1)
Table 2: Tensile test data range for JC model development Strain rate (s-1)
Temperature(K)
True plastic strain
0.01
298
0.05
0.005
348
0.1
0.001
423
0.15 0.2 0.25
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969
Fig.2: Computerized UTM of 100kN capacity with resistance heating 3-zone split furnace
where 𝜎 is the Von-mises flow stress and (𝐴 + 𝐵𝜀 𝑛 ) , (1 + 𝐶 ln 𝜀̇∗ ), (1 − 𝑇 ∗𝑚 ) are the stresses resulting from isotropic strain hardening, strain rate hardening and thermal softening respectively. A indicates the yield stress at reference temperature and reference strain rate, B indicates the coefficient of strain hardening, 𝜀 indicates the true plastic strain, 𝑛 is strain hardening exponent, 𝐶 is the coefficient of strain rate hardening, 𝜀̇∗ = 𝜀̇/𝜀̇0, where 𝜀̇ is true strain rate, 𝜀̇0 is the reference strain rate, 𝑇 ∗ is the homologous temperature and the 𝑚 is the thermal softening exponent. The homologous temperature is expressed as 𝑇∗ =
𝑇−𝑇𝑟𝑒𝑓
𝑇𝑚 −𝑇𝑟𝑒𝑓
(2)
where, 𝑇 is absolute temperature, 𝑇𝑟𝑒𝑓 is the reference temperature (in general the minimum temperature of the experimental range is considered as reference temperature) in such a way that 𝑇 ≥ 𝑇𝑟𝑒𝑓 , 𝑇𝑚 is the melting temperature. 4. Determination of material constants in JC model: The material constants for the JC model is derived in three steps 4.1. Step 1: At reference strain rate and reference temperature Under this condition, ln 𝜀̇∗ = 0 and in the same way at reference temperature the homologous temperature value is also zero. Since the stresses resulting from strain rate hardening and thermal softening become unity, the resultant stress, from the equation (1) becomes
K Limbadri et al / Materials Today: Proceedings 4 (2017) 966–974
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𝜎 = (𝐴 + 𝐵𝜀 𝑛 )
(3)
the value of A has been taken at the reference strain rate 0.01s-1 and the reference temperature 298 K. The coefficient of strain hardening,𝐵, and the strain hardening exponent,𝑛, are determined by plotting a straight line between ln(𝜎 − 𝐴)and ln(𝜀). 4.2. Step 2: At a fixed strain and reference temperature
At reference temperature the homologous temperature becomes zero therefore the stress resulted from thermal softening is unity. So only the first two terms in the Eq(1) remain 𝜎 = (𝐴 + 𝐵𝜀 𝑛 )(1 + 𝐶 ln 𝜀̇∗ )
(4)
The strain rate hardening coefficient,𝐶, is determined by solving the equations of lines formed by variables between 𝜎 �(𝐴+𝐵𝜀𝑛 ) − 1� and ln(𝜀̇∗ ). Therefore there are as many values of 𝐶 as the number of true plastic strain 𝜀 values. 4.3. Step 3: At a fixed strain and reference strain rate
Under this condition, 𝜀̇∗ = 1 and therefore the ln 𝜀̇∗ = 0 which accounts for the stresses caused by strain hardening.The equation is modified to 𝜎 = (𝐴 + 𝐵𝜀 𝑛 )(1 − 𝑇 ∗𝑚 )
(5) 𝜎
The thermal softening exponent (𝑚) can be obtained by plotting the line between �1 − (𝐴+𝐵𝜀𝑛 )� and ln(𝑇 ∗ ). Hence, for each value of true plastic strain (𝜀) there is a value of𝑚.
4.4. Optimization of the C and m values As there are 5 values of C and m corresponding to 5 different values of strain, there is a need for optimization to obtain the best fit. This is done using the statistical method of least squares which involve the minimization of average absolute error (∆) between the experimental flow stress (𝜎𝑒𝑥𝑝 ) and predicted flow stress (𝜎𝑝 ) values. The mathematical expression of the average absolute error (∆) is given below Eq (6). 1
∆= ∑𝑖=𝑁 𝑖=1 � 𝑁
𝑖 −𝜎 𝑖 𝜎𝑒𝑥𝑝 𝑝 𝑖 𝜎𝑒𝑥𝑝
(6)
� × 100
where𝜎𝑒𝑥𝑝 is the experimental flow stress, 𝜎𝑝 is the predicted flow stress and the N is the number of data points taken in to account. 4.5. Predictability of the constitutive equation All the constants, A, B, C, n and m have been determined, therefore, the complete constitutive equation for predicting the flow stress has been expressed by substituting these values in Eq. (1). Moreover, the predictability of the constitutive equation can be computed based on the statistical parameters of correlation coefficient (R), average absolute error (∆) and standard deviation (s). The correlation coefficient is the most frequently used statistical parameter for expressing the information on strength of linear relationship between the experimental and predicted values. It is illustrated in Eq. (7).
𝑅=
𝑖 ∑𝑖=𝑁 �𝑒𝑥𝑝 ��𝜎𝑝𝑖 −𝜎 �𝑝 � 𝑖=1 �𝜎𝑒𝑥𝑝 −𝜎 2
𝑖 −𝜎 𝑖 � � �∑𝑖=𝑁�𝜎𝑒𝑥𝑝 �𝑒𝑥𝑝 � ∑𝑖=𝑁 𝑝 𝑖=1 𝑖=1 �𝜎𝑝 −𝜎
2
(7)
K Limbadri et al / Materials Today: Proceedings 4 (2017) 966–974
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where𝜎�𝑒𝑥𝑝 , 𝜎�𝑝 are the average experimental flow stress and average predicted flow stress respectively. Here, R and ∆ are biased and unbiased statistical values, in other words, if the R value is high, it does not necessarily imply the performance of the model as high. The average error∆is unbiased, as it calculates the values term by term for determining the predictability of the model. All the above equations are solved by using MATLAB. All the constants thus obtained for JC model are listed in Table 3. Table 3: Material constants for JC model
5.
Material Constant
A
B
C
n
m
Value
283
774.2
0.0152
0.8499
0.8012
Results and Conclusion:
The equation for JC model is given by 𝜎 = (283 + 774.2 × 𝜀 0.8499 )(1 + 0.0152 × ln 𝜀̇∗ )(1 − 𝑇 ∗0.8012)
600
(8)
(a) 0.01s-1
1
2
True Stress(MPa)
500 400 300 200 100 0 0
0.05
0.1
ExperimentalPredicted-
298K 1: 298K
0.15
0.2
True Plastic Strain 348K
2: 348K
423K 3: 423K
0.25
0.3
K Limbadri et al / Materials Today: Proceedings 4 (2017) 966–974
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600
(b) 0.005s-1
1
2
True Stress(MPa)
500 400 300 200 100 0 0
0.05 ExperimentalPredicted-
600
0.1
0.15
0.2
298K 1: 298K
348K
2: 348K
0.3
423K 3: 423K
(c) 0.001s-1 1
500
True Stress(MPa)
0.25
True Plastic Strain
2
400 300 200 100 0 0
0.05 Experimental-
0.1
0.15
0.2
0.25
0.3
True Plastic Strain
298K 1: 298K
348K
2: 348K
423K 3: 423K
Fig.3 : Graphs showing experimental and predicted stress for JC Model at various strains for strain rates of (a) 0.01s-1, (b) 0.005s-1, (c) 0.001s-1.
K Limbadri et al / Materials Today: Proceedings 4 (2017) 966–974
Predicted True Stress (Mpa)
600
973
R=0.7616
500 400 300 200 100 0 0
100
200 300 400 Experimental True Stress (Mpa) JC Model
500
600
Linear (JC Model)
Fig. 4: Correlation between experimental and predicted stress.
Figures 3 (a), (b) and (c) show the predicted values of flow stress for JC model at strain rates of 0.01, 0.005 and 0.001s-1. However, the R value for the model came out to be 0.7616(as shown in Fig. 4) and value of average absolute error was 11.95% and standard deviation of 12.13. The predictive capability of the JC model is low mainly due to the influence of the coupled effect of strain and temperature, and of strain rate and temperature. This effect is not captured in the model. Hence, it can be concluded that the flow behaviour of Zircaloy-4 with low oxygen content at elevated temperature is not governed by three independent phenomena, viz., thermal softening, strain rate hardening and isotropic strain hardening. In order to correctly model the deformation behaviour of Zircaloy-4 with low oxygen content, there is a need to consider other constitutive models which consider the coupled effect of strain and temperature, and of strain rate and temperature. The future work will involve the development of physical based models such as the Zerilli-Armstrong Model (ZA) andRusinek–Klepaczkomodel (RK) along with the empirical Arrhenius model. Acknowledgment: The financial support received for this research work from Department of Atomic Energy, Government of India, 36(2)/14/56/2014-BRNS/2699 is gratefully acknowledged. References: [1]
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