Research Article  Open Access
Peijie Liu, Yanming Quan, Guo Ding, "Dynamic Mechanical Characteristics and Constitutive Modeling of Rail Steel over a Wide Range of Temperatures and Strain Rates", Advances in Materials Science and Engineering, vol. 2019, Article ID 6862391, 15 pages, 2019. https://doi.org/10.1155/2019/6862391
Dynamic Mechanical Characteristics and Constitutive Modeling of Rail Steel over a Wide Range of Temperatures and Strain Rates
Abstract
Rail steel plays an indispensable role in the safety and stability of the railway system. Therefore, a suitable constitutive model is quite significant to understand the mechanical behavior of this material. Here, the compressive mechanical behavior of heattreated U71Mn rail steel over a wide range of strain rates (0.001 s^{−1}–10000 s^{−1}) and temperatures (20°C–800°C) was systematically investigated via uniaxial quasistatic and dynamic tests. The split Hopkinson pressure bar (SHPB) apparatus was utilized to perform dynamic mechanical tests. The effects of temperature, strain, and strain rate on the dynamic compressive characteristics of U71Mn were discussed, respectively. The results indicate that the flow response of U71Mn is both temperaturesensitive and strain ratesensitive. However, the influence of temperature on the flow response is more remarkable than that of strain rate. On the basis of the experimental data, the original and modified JohnsonCook (JC) models of the studied material were established, respectively. Using correlation coefficient and average absolute relative error parameters, it is revealed that better agreement between the experimental and predicted stress is reached by the modified JC model, which demonstrates that the modified one can characterize the mechanical behavior of the studied material preferably.
1. Introduction
Rail steel is one of the most significant infrastructures in the railway network, which plays an indispensable role in the security and stability of the railway system. With the fast expansion of railway transportation around the world, rail steel in service is subjected to complex and dynamic loading imposed by railway wheels, which easily generates rail defects involving corrugation, squats, spalling, cracks, and so on [1–4]. Grassie [5] conducted an indepth review on the development of rail corrugation and deepened its understanding. Zhong et al. [6] studied the rail crack growth with respect to rolling contact fatigue via the analysis of the chemical constituents and mechanical properties. Steenbergen and Dollevoet [7] studied the physical characteristics of rail squat initiation by a phenomenological analysis. Steenbergen [8] carried out an experimental investigation on the rail grindingrelated rolling contact fatigue and analyzed the characteristics of rail spalling defects. These aforementioned undesirable defects severely influence the safety of railway transportation if they are not effectively dealt with. Therefore, it has become increasingly significant to reprofile rails to remove defects in time. The dynamic flow response of materials is connected with temperature and strain rate, along with strain [9]; hence, an appropriate constitutive model to understand the flow behavior of rail steel is requested. For the sake of characterizing the stressstrain behavior under dynamic deformation conditions, the wellestablished split Hopkinson pressure bar (SHPB) technique which was initially designed by Kolsky is one of the most widely utilized testing means [10].
In recent years, some empirical (e.g., Johnson–Cook model), semiempirical (e.g., Zerilli–Armstrong model), and physicallybased (e.g., Mechanical Threshold Stress model) constitutive equations have been put forward to characterize the flow response of materials [11–13]. Zerilli and Armstrong [12] put forward the Zerilli–Armstrong (ZA) model on the basis of dislocation mechanics. The Mechanical Threshold Stress (MTS) model put forward by Follansbee and Kocks [13] is one of the physicallybased models, and these kinds of models are not always preferred for the reason that more data obtained from accurately manipulated experimental conditions are required. More importantly, there are more material constants encompassed in semiempirical and physicallybased models whose material constants are more difficult to derive than the empirical ones. The Johnson–Cook (JC) model [11] is an empiricallybased model and it is extensively utilized to express the mechanical behavior of materials at large strain rates and high temperatures. Thanks to the advantages of simple expression, low computational complexity, and fast calculation speed, the JC model has gained widespread implementation in a diversity of finite element analysis software [14]. Nevertheless, the JC constitutive model in its original form is often unable to guarantee enough accuracy on the flow stress prediction of materials. Therefore, it is a necessity to modify the conventional JC model to express the flow response with sufficient prediction precision over a wide range of temperatures and strain rates. Gambirasio and Rizzi [15] put forward the split JC model which demonstrated that the capacity of the model to characterize the mechanical response of the studied material was conspicuously enhanced. Wang et al. [16] conducted a research on the dynamic deformation behavior of Inconel 718 under large strain rates and elevated temperatures, and they established a modified JC model with adequate accuracy. Zhao et al. [17] performed a research on the original and modified JC models to describe the mechanical behavior of the FeCr alloy acquired by laser additive manufacturing under wide strain rates and temperature ranges and found that the modified JC model could generate precise prediction of the mechanical behavior of the studied material.
Currently, although comparatively more existing researches emphasizing the deformation mechanism under large strain rate situations [18, 19], hightemperature deformation [20], and dynamic fracture [21] of other steels and alloys are available, there are very few systematic researches available that address the dynamic mechanical response of rail materials over a wide range of strain rates and temperatures (especially under large strain rates and high temperatures). Nakkalil et al. [22] performed a research focused on the mechanical response of rail steels under large strain rates and elevated temperatures, but no constitutive model was established in the article. Jing et al. [23] investigated the mechanical performance of train wheel materials under large strain rates via the split Hopkinson tensile bar (SHTB) equipment, and their research demonstrated that the tensile properties of the materials relied on strain rate.
In this study, the dynamic mechanical characteristics of heattreated U71Mn rail steel are systematically investigated over a wide range of strain rates (0.001 s^{−1}–10000 s^{−1}) and temperatures (20°C–800°C) by virtue of the split Hopkinson pressure bar equipment. Based on the results of mechanical tests, the effects of strain rate and temperature along with strain on the flow stress of U71Mn rail steel are assessed. Furthermore, the strain rate sensitivity, temperature sensitivity, and strain hardening effect of the studied material are also discussed, respectively. Finally, the original and modified JC models are established on the basis of the experimental data, respectively. In contrast, it is verified that the modified JC model can represent the flow response of the studied material more accurately by means of correlation coefficient (R) and average absolute relative error (AARE).
2. Materials and Methods
2.1. Materials
The typical structure of a flat bottom rail comprises rail head, rail web, and rail foot, as shown in Figure 1. Compared with the rail web and rail foot, the rail head has a higher strength requirement to interact safely with railway wheels. The tested rail specimens were sampled from the heattreated Mnsteel rail (Chinese brand: U71Mn) which is made from high carbon steel to provide high fatigue toughness. After the hot rolling process of U71Mn rail steel by the universal mill, the surplus heat of the rail head was 850°C, and the afterheat generated after the hot rolling process was further used to online slackquench the rail via compressed air cooling technique performed in the rail manufacturer, during which the air cooling rate was set to 3°C/s. When the temperature of the rail head was 500°C, the rail was then air cooled to room temperature with the cooling rate of 0.1°C/s. The purpose of the rail heattreatment is to acquire the final expected mechanical properties to extend the rail service life. U71Mn rails (60 kg/m) are extensively employed in current Chinese highspeed railway network [24]. Tables 1 and 2 give the detailed chemical compositions and physical properties of the material, respectively. After the process of fine grinding, polishing, and etching, the typical microstructure of U71Mn rail steel is shown in Figure 2 under a scanning electron microscope (SEM, Quanta 200, FEI). The tested material displays a fully pearlitic microstructure with equiaxed pearlite colonies.


The tested rail material was in unused condition (never in service). All of the specimens for the purpose of mechanical testing were sampled from the rail head by electric discharge machining (EDM), and the sampling positions are shown in Figure 3. For the quasistatic compressive and dynamic compressive tests, all of the tested specimens were cylindrical in shape: the specimens used in the former test had a size of Φ8 mm × 12 mm, and the specimens used in the latter test had a size of Φ5 mm × 5 mm (for the test at the strain rate of 1500 s^{−1}) and Φ2 mm × 2 mm (for the test at 5000 s^{−1} and 10000 s^{−1}). Due to the same slenderness ratio of samples under dynamic loading conditions, the geometry effect of the specimens on the experimental results can be neglected. For every test condition, each group of the specimens was repeatedly conducted 6 times so as to ensure the reproducibility, and the averaged data under each test condition were adopted as the final experimental results.
2.2. Experimental Procedure
Uniaxial quasistatic compressive tests were performed via a Gleeble 3500 thermal simulation machine at 0.001 s^{−1} and room temperature 20°C. The Gleeble 3500 device is a thermal and mechanical testing machine with wholly integrated digital closedloop control. Each end face of the samples was properly lubricated to diminish the undesirable friction effect before the test.
The SHPB technique has been extensively accepted to measure the stressstrain behavior from 10^{2} s^{−1} to 10^{4} s^{−1} [25]. According to [26], the SHPB technique serves as an accurate method which makes the experimenter induce various stress histories within the cylindrical samples. A split Hopkinson pressure bar (SHPB) apparatus with a synchronously assembled heating system was utilized to perform dynamic compressive tests, as shown in Figure 4(a). This apparatus enables to keep the sample at the center of the furnace while keeping the incident and transmitted bars outside the heating range of the furnace whose temperature is controlled by a PID closedloop controller. In other words, when the sample with cylindrical shape is heated by the separate resistance furnace, the incident and transmitted bars are kept away from the heating zone of the furnace. Figure 4(b) displays the detailed configuration of the sample fixed during the hightemperature dynamic test, a Ktype thermocouple attached to the sample is used to measure the sample temperature. The thermocouple itself is supported by a sleeve (namely, a thin tube) which can move freely along the axial direction of the bars. The thermocouple also plays a role in maintaining the specimen at the appropriate position inside the furnace before dynamic loading. When the temperature of the sample initially reaches the required value, the sample is maintained at the constant temperature for 2 minutes to ensure the homogeneous temperature distribution within the specimen, and then the bars are brought into contact with the specimen through the bar mover immediately before the stress pulse arrives at the end of the incident bar. The bar mover is driven by the same gas gun which propels the striker bar towards the incident bar. To diminish the friction effect, the contact regions between the sample and the bars are properly lubricated with a grease containing molybdenum (Mo). When the incident bar is struck by the ultrahighstrength steel striker bar driven by a pressurized nitrogen gas gun, a compressively longitudinal wave is created and transmitted down the incident bar. As the incident pulse arrives at the samplebars interfaces, it partly returns to the incident bar whilst the remaining pulse is introduced into the transmitted bar on account of the mechanical impedance mismatch between the pressure bars and the sample. Strain gauges fixed onto the incident bar are employed to acquire the incident and reflected pulses, and strain gauges glued onto the transmitted bar are utilized to measure the transmitted pulse. These three strain pulses are recorded by an oscilloscope.
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On the basis of the theory of onedimensional wave propagation, the stress , strain , and strain rate of the specimen are expressed aswhere A_{0} represents the crosssectional area of the bars; E denotes Young’s modulus of the bars; (, where ρ is the density of the pressure bars) represents the propagation velocity of onedimensional elastic longitudinal wave; A_{s} means the initial crosssection area of the specimen; l_{s} refers to the specimen length; and and represent the strain signals of the transmitted and reflected wave, respectively.
3. Results and Discussion
Figure 5 shows the quasistatic compressive stressstrain curve of U71Mn rail steel at 0.001 s^{−1} and 20°C. It can be observed that the tested material undergoes remarkable strain hardening with the increasing plastic strain.
In order to verify the stressstate equilibrium at both end faces of the specimen during SHPB tests at the strain rate of 10000 s^{−1}, Figure 6 displays the typical set of waveforms recorded by strain gauges glued onto the pressure bars and the corresponding 1 and 2wave stress curves under high temperatures at the strain rate of 10000 s^{−1}. The 1wave stress analysis indicates the stress conditions at the specimentransmitted bar interface. The 2wave analysis denotes the stress conditions at the incident barspecimen interface, where the sum of incident wave and reflected waves is proportional to the sample front stress. As can been seen from Figures 6(b) and 6(d), a good agreement between the 1wave and the 2wave stressstrain response is reached, which represents the stressstate equilibrium within the specimen and therefore illustrates the validity of the SHPB test results.
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3.1. Strain Rate Hardening Effect on the Flow Stress
The dynamic compressive flow response under varying indicated temperatures is presented in Figure 7. Although deformed, none of the tested samples brought about fracture. Thus, the ending parts of the stressstrain curves indicate unloading instead of fracture. The influence of strain rate on the flow response of U71Mn as a function of temperature at = 0.1 is displayed in Figure 8. The impact of strain rates on flow stress at other strain values is also similar. From Figure 7, it is found that the plastic flow stress of the samples hinges upon strain rate and presents the strain rate hardening effect, which means that the flow stress increases as strain rate increases. The strain rate hardening effect is caused by dislocation pileup, since dislocation density is increased when the strain rate increases, and thus hardening occurs. Nevertheless, the impact of strain rates on the flow response is weakened as temperature increases. When the specimens were dynamically compressed at 800°C, as shown in Figure 7(e), all of the stressstrain curves under varied strain rates show tardily progressive tendency before unloading and the true stresses are quite low. The reason is that the tested specimens are softened severely at 800°C so that the flow stress is relatively low, which means that the specimens are easier to deform plastically.
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(d)
(e)
According to [27], strain rate dependence on the plastic flow stress can be determined quantitatively by introducing a strain rate sensitivity parameter β, and this parameter is expressed aswhere σ_{1} and σ_{2} are the experimentally obtained flow stresses at strain rates of and under a given plastic strain and temperature, respectively.
For U71Mn rail steel, the strain rate sensitivity β as a function of strain at 1500 s^{−1}–10000 s^{−1} under different temperatures is presented in Figure 9. It can be noticed that β decreases as the temperature rises when the temperature is under 600°C, particularly from 20°C to 200°C, whereas it enhances when the temperature is up to 800°C, which means that the material still has some certain strain rate strengthening effect at very high temperatures and the effect is not weaker than that at 600°C, indicating that the material still shows some certain strain rate strengthening effect. It can be also observed from Figure 9 that β decreases rapidly as strain increases under 20°C, 400°C, and 600°C, while it remains nearly invariable with regard to strain under 200°C and 800°C.
3.2. Thermal Softening Effect on the Flow Stress
Figure 10 presents the dynamic flow response at varying indicated strain rates. The impact of temperature on the flow behavior of U71Mn as a function of strain rate at = 0.1 is given in Figure 11. The temperature effect on the flow response under other strains is similar. Figures 10(a)–10(c) show that the temperature impact on the flow response of the studied material is far more pronounced than that of the strain rate; i.e., U71Mn rail steel shows strong thermal softening effect. The flow stress reduces rapidly as the temperature rises under varying prescribed strain rates at a constant strain. There is no much difference in flow stress when the temperatures are 400°C and 600°C, respectively, which means that the flow response is insensitive to temperatures at this temperature range. However, the flow stress decreases more evidently as the temperature rises from 600°C to 800°C, which indicates the enhanced temperature sensitivity of the material within this temperature range.
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According to [21], temperature dependence on flow stress can be quantitatively evaluated by introducing a temperature sensitivity parameter , which is expressed aswhere σ_{1} and σ_{2} are the flow stresses acquired from tests performed at temperatures of T_{1} and T_{2}, respectively; T_{1} is the reference temperature (taken as 20°C), and T_{2} (T_{2} > T_{1}) is the experimental temperature.
For U71Mn rail steel, Figure 12 shows the temperature sensitivity as a function of true strain at 5000 s^{−1} under varying temperature ranges. It shows that the increases as temperature rises. A significant observation is that the undergoes a drastic increment when the temperature is 800°C and there is not much difference in the temperature sensitivity when the temperatures are 400°C and 600°C, respectively. As per Figure 12, the temperature sensitivity slightly decreases with the increase of strain except that there is a more remarkable decrease of the temperature sensitivity with the rise of strain at 800°C.
3.3. Strain Hardening Effect on the Flow Stress
The strain hardening rate hinges upon the competition among the impacts of strain rate and strain hardening, together with thermal softening. In Figure 5, it displays that the strain hardening rate is evident under quasistatic compressive condition, which can be explicated by the fact that the deformationgenerated heat at 0.001 s^{−1} can dissipate into surroundings quickly so that the thermal softening effect can be ignored. For dynamic tests conducted under dynamic loading conditions, the heat caused by plastic deformation usually occurs in few milliseconds so that it is very hard for the heat to disperse into the ambient environment because of the very short time, which will eventually bring about the temperature increment of the specimen. The large adiabatic heat can reduce the strength of the tested sample, bringing about the reduction of strain hardening rate. Therefore, temperature softening effect serves as a striking part in the specimen deformation under large strain rates, which causes the insensitivity of the flow stress to strains. The high adiabatic temperature rise generated within the specimen during plastic deformation needs to be taken into account because adiabatic temperature rise can cause the weakening of the strain hardening rate. The temperature increment value is normally calculated by [28]where ρ_{s} is the density of the studied material (adopted as 7.8 g/cm^{3}), C_{v} represents the heat capacity (i.e., 0.5 J/g°C), and ε and σ denote the plastic strain and flow stress, respectively. ζ is the Taylor–Quinney coefficient, which means the proportion of the plastic deformation induced heat. In this study, ζ is taken as approximately 1.0 for the dynamic loading cases [28]. The integral part in equation (4) signifies the plastic work, namely, the area beneath the stressstrain curve.
Table 3 lists the temperature increment values induced within the deformed specimens under dynamic compressive conditions. It can been seen from Table 3 that adiabatic temperature increment decreases as the experimental temperature rises and that adiabatic temperature increment increases with the increase of strain and strain rate. Therefore, the flow softening phenomenon of U71Mn rail steel at 20°C–600°C and strain rates of 5000 s^{−1} and 10000 s^{−1} in Figures 7(a)–7(d) is probably induced by the temperature softening effect that overshadows that of strain rate and strain. However, the corresponding flow response in Figure 7(e) displays continuous progressive tendency at 800°C, which implies that the flow response of U71Mn rail steel is dominated by the impacts of the strain rate and strain. In the process of plastic deformation, the slope of the curves in Figure 7 is dominated by the combination result of the competition process between the strain hardening effect and the temperature softening effect.

3.4. Establishment of the Johnson–Cook Model
As an empiricallybased model, the JC model takes into account the effects of strain rate and strain along with temperature on the plastic flow response. These three total effects on the plastic deformation of materials are considered by incorporating them into the model as a multiplicative form. The JC model is written aswhere ε represents the plastic strain, σ denotes the Von Mises flow stress, represents the dimensionless strain rate ( means the true strain rate, while denotes the reference strain rate, i.e., 0.001 s^{−1} in the study), A is the yield stress in MPa under the reference strain rate and temperature, B means the strain hardening coefficient (MPa), C is the strain rate hardening coefficient, m denotes the thermal softening exponent, n means the strain hardening exponent, and represents the homologous temperature where T_{m} denotes the melting point (taken as 1469°C for the studied steel) and T_{r} denotes the reference temperature (taken as 20°C, T ≥ T_{r}).
The adiabatic temperature rise induced by dynamic loading exerts a profound influence on the constitutive behavior of the studied material because of the thermal softening effect. Therefore, the conventional JC model needs modifying to involve the temperature softening phenomenon, which is realized by incorporating equation (4) into the temperature term of the original JC model. The temperature term in equation (5) is transformed to
On the basis of the experimentally obtained stressstrain curves given in Figures 5 and 10, the material constants in the original JC model are calculated by regression analysis technique as the preconditioning values. The value of A (namely, the stress under the strain of 0.002) at 20°C and 0.001 s^{−1} is calculated as 1262 MPa from Figure 5.
The strain rate and temperature impacts are ignored during the uniaxial quasistatic compressive condition. Under 20°C and 0.001 s^{−1}, equation (5) will reduce to
Taking the logarithmic transformation on the both sides of equation (7), then equation (7) will transform to
Based on the results of quasistatic experiment from Figure 5, the curve of vs. can be plotted by the substitution of A into equation (8) and by utilizing the values of the stressstrain curve under varying strains from the hardening part of the stressstrain curve. B can be derived from the intercept of the plot, and meanwhile, n can be acquired from the slope of the plot.
At 20°C, there is no temperature term. Therefore, equation (5) changes to
Employing the results of stressstrain curves under a constant strain under 1500 s^{−1}, 5000 s^{−1}, and 10000 s^{−1} from the dynamic compressive tests, the value of C is derived from the slope of vs. plot.
At 0.001 s^{−1}, equation (5) can be transformed as
Using the flow stress values under a fixed strain at varying temperatures (200°C, 400°C, 600°C, and 800°C) and the strain rate of 5000 s^{−1}, the value of m is acquired from the slope of vs. plot.
Based on the initial fitting results obtained from the preconditioning method above, the least square method is further employed to acquire the final material constants. Table 4 lists the original JC model parameters. Therefore, the expression of the JC constitutive equation of U71Mn rail steel is given as follows:

Figure 13 presents the comparison result between the measured and predicted stress derived from the JC model in its original form at 200°C and 10000 s^{−1}. Obviously, appreciable deviation between the experimental and predicted values is observed. Thus, it is a necessity to modify the original JC model. Considering the impact of temperature on the flow response of the studied material, the temperature term in the original JC constitutive model is modified as follows:where a, b, and are the constants which describe the relationship between temperature and flow stress. The modified constitutive model of U71Mn rail steel can be acquired as follows:
The comparison results between the measured and predicted flow stress employing the modified JC model are presented in Figure 14. Aiming at assessing the predictability of the original and modified JC models, two standard statistical indexes, i.e., correlation coefficient (R) and average absolute relative error (AARE), are employed. These two statistical parameters are defined aswhere is the experimentallyobtained flow stress, represents the predicted flow stress, denotes the mean values of , and means the mean values of . N means the amount of data points. The correlation coefficient R represents the precision and degree of linear relationship between the measured and estimated data. As an unbiased statistical index, AARE can be employed to evaluate the predictability of a numerical model or equation. Figure 15 presents the correlation between the predicted and measured stress data using the original and modified JC models. In Figures 15(a) and 15(b), the correlation coefficients for the original and modified JC models are 0.84 and 0.98, respectively. The corresponding average absolute relative errors are 21.1% and 8.9%, respectively. By contrast, it can be corroborated that the predictability of the modified JC model is much better than the original one, which indicates that equation (13) can be utilized to express the mechanical behavior of U71Mn rail steel.
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4. Conclusions
In this paper, the true stressstrain data of U71Mn rail steel have been obtained from the uniaxial compressive experiments performed at a wide range of strain rates (0.001–10000 s^{−1}) and temperatures (20°C–800°C). The dynamic response of U71Mn rail steel has been investigated by means of the aforementioned experimental results. The following conclusions can be drawn based on the study:(1)The strain rate sensitivity of U71Mn decreases with the increase of temperature. Nevertheless, the strain rate sensitivity increases when the temperature is 800°C. Besides, the strain also has a remarkable impact on the strain rate sensitivity of the studied material.(2)U71Mn rail steel displays remarkable temperature sensitivity. The influence of temperature on the flow behavior is evident with respect to the increasing experimental temperature. However, the temperature effect of the studied material is not conspicuous when the temperatures are 400°C and 600°C, respectively. The temperature effect on the flow response is more remarkable than that of strain rate.(3)The strain hardening effect of the studied material slightly decreases at temperatures from 20°C to 600°C and strain rates of 5000 s^{−1} and 10000 s^{−1}. However, when the temperature is 800°C, the strain hardening effect is obvious.(4)On the basis of the experimental results, the modified JC model of U71Mn was established by taking into account the effects of high temperature and high strain rate. Compared with the experimental data, the correlation coefficient (R) and average absolute relative error (AARE) of the modified JC model are 0.98 and 8.9%, respectively, which shows that the proposed JC model can better express the dynamic mechanical behavior of U71Mn rail steel.
Data Availability
The data used to support the findings of this study are available from the corresponding author upon request.
Conflicts of Interest
The authors declare no conflicts of interest.
Acknowledgments
The authors gratefully acknowledge the National Natural Science Foundation of China (Grant no. 51675184) and the Provincial Science and Technology Project of Guangdong Province (Grant no. 2016A010102006).
References
 P. T. Torstensson and J. C. O. Nielsen, “Monitoring of rail corrugation growth due to irregular wear on a railway metro curve,” Wear, vol. 267, no. 1, pp. 556–561, 2009. View at: Publisher Site  Google Scholar
 S. Kaewunruen, M. Ishida, and S. Marich, “Dynamic wheelrail interaction over rail squat defects,” Acoustics Australia, vol. 43, no. 1, pp. 97–107, 2015. View at: Publisher Site  Google Scholar
 D. F. Cannon, K.O. Edel, S. L. Grassie, and K. Sawley, “Rail defects: an overview,” Fatigue, vol. 26, no. 10, pp. 865–886, 2003. View at: Publisher Site  Google Scholar
 S. MayaJohnson, A. J. Ramirez, and A. Toro, “Fatigue crack growth rate of two pearlitic rail steels,” Engineering Fracture Mechanics, vol. 138, pp. 63–72, 2015. View at: Publisher Site  Google Scholar
 S. L. Grassie, “Rail corrugation: advances in measurement, understanding and treatment,” Wear, vol. 258, no. 78, pp. 1224–1234, 2005. View at: Publisher Site  Google Scholar
 W. Zhong, J. J. Hu, Z. B. Li, Q. Y. Liu, and Z. R. Zhou, “A study of rolling contact fatigue crack growth in U75V and U71Mn rails,” Wear, vol. 271, no. 12, pp. 388–392, 2011. View at: Publisher Site  Google Scholar
 M. Steenbergen and R. Dollevoet, “On the mechanism of squat formation on train rails—Part I: origination,” International Journal of Fatigue, vol. 47, pp. 361–372, 2013. View at: Publisher Site  Google Scholar
 M. Steenbergen, “Rolling contact fatigue in relation to rail grinding,” Wear, vol. 356357, pp. 110–121, 2016. View at: Publisher Site  Google Scholar
 S. Seo, O. Min, and H. Yang, “Constitutive equation for Ti–6Al–4V at high temperatures measured using the SHPB technique,” International Journal of Impact Engineering, vol. 31, no. 6, pp. 735–754, 2005. View at: Publisher Site  Google Scholar
 G. I. Mylonas and G. N. Labeas, “Mechanical characterisation of aluminium alloy 7449‐T7651 at high strain rates and elevated temperatures using split Hopkinson bar testing,” Experimental Techniques, vol. 38, no. 2, pp. 26–34, 2014. View at: Publisher Site  Google Scholar
 G. R. Johnson and W. H. Cook, “A constitutive model and data for metals subjected to large strains, high strain rates and high temperatures,” in Proceedings of the 7th International Symposium on Ballistics, pp. 541–547, Hague, Netherlands, April 1983. View at: Google Scholar
 F. J. Zerilli and R. W. Armstrong, “Dislocation‐mechanics‐based constitutive relations for material dynamics calculations,” Journal of Applied Physics, vol. 61, no. 5, pp. 1816–1825, 1987. View at: Publisher Site  Google Scholar
 P. S. Follansbee and U. F. Kocks, “A constitutive description of the deformation of copper based on the use of the mechanical threshold stress as an internal state variable,” Acta Metallurgica, vol. 36, no. 1, pp. 81–93, 1988. View at: Publisher Site  Google Scholar
 Y. Karpat, “Temperature dependent flow softening of titanium alloy Ti6Al4V: an investigation using finite element simulation of machining,” Journal of Materials Processing Technology, vol. 211, no. 4, pp. 737–749, 2011. View at: Publisher Site  Google Scholar
 L. Gambirasio and E. Rizzi, “An enhanced Johnson–Cook strength model for splitting strain rate and temperature effects on lower yield stress and plastic flow,” Computational Materials Science, vol. 113, pp. 231–265, 2016. View at: Publisher Site  Google Scholar
 X. Wang, C. Huang, B. Zou, H. Liu, H. Zhu, and J. Wang, “Dynamic behavior and a modified Johnson–Cook constitutive model of Inconel 718 at high strain rate and elevated temperature,” Materials Science and Engineering: A, vol. 580, no. 37, pp. 385–390, 2013. View at: Publisher Site  Google Scholar
 Y. Zhao, J. Sun, J. Li, Y. Yan, and P. Wang, “A comparative study on JohnsonCook and modified JohnsonCook constitutive material model to predict the dynamic behavior laser additive manufacturing FeCr alloy,” Journal of Alloys and Compounds, vol. 723, 2017. View at: Google Scholar
 M. Hokka, T. Leemet, A. Shrot, M. Baeker, and V. T. Kuokkala, “Characterization and numerical modeling of high strain rate mechanical behavior of Ti153 alloy for machining simulations,” Materials Science and Engineering: A, vol. 550, no. 6, pp. 350–357, 2012. View at: Publisher Site  Google Scholar
 F. Abed, A. AbdulLatif, and A. Yehia, “Experimental study on the mechanical behavior of EN08 steel at different temperatures and strain rates,” Metals, vol. 8, no. 9, p. 736, 2018. View at: Publisher Site  Google Scholar
 W. S. Lee and C. F. Lin, “Hightemperature deformation behaviour of Ti6Al4V alloy evaluated by high strainrate compression tests,” Journal of Materials Processing Technology, vol. 75, no. 13, pp. 127–136, 1998. View at: Publisher Site  Google Scholar
 W. S. Lee, C. F. Lin, and T. J. Liu, “Impact and fracture response of sintered 316L stainless steel subjected to high strain rate loading,” Materials Characterization, vol. 58, no. 4, pp. 363–370, 2007. View at: Publisher Site  Google Scholar
 R. Nakkalil, J. R. H. Jr, and M. N. Bassim, “Characterization of the compression properties of rail steels at high temperatures and strain rates,” Materials Science and Engineering: A, vol. 141, no. 2, pp. 247–260, 1991. View at: Publisher Site  Google Scholar
 L. Jing, L. Han, L. Zhao, and Y. Zhang, “The dynamic tensile behavior of railway wheel steel at high strain rates,” Journal of Materials Engineering and Performance, vol. 25, no. 11, pp. 4959–4966, 2016. View at: Publisher Site  Google Scholar
 K. K. Gu, Q. Lin, W. J. Wang et al., “Analysis on the effects of rotational speed of grinding stone on removal behavior of rail material,” Wear, vol. 342343, no. 3, pp. 52–59, 2015. View at: Publisher Site  Google Scholar
 S NematNasser, ASM Handbook: Mechanical Testing and Evaluation, vol. 8, Materials Park, OH, USA, 2000.
 R. Zwiessler, T. Kenkmann, M. H. Poelchau, S. Nau, and S. Hess, “On the use of a split Hopkinson pressure bar in structural geology: high strain rate deformation of Seeberger sandstone and Carrara marble under uniaxial compression,” Journal of Structural Geology, vol. 97, pp. 225–236, 2017. View at: Publisher Site  Google Scholar
 W. S. Lee and C. F. Lin, “Plastic deformation and fracture behaviour of Ti–6Al–4V alloy loaded with high strain rate under various temperatures,” Materials Science and Engineering: A, vol. 241, no. 12, pp. 48–59, 1998. View at: Publisher Site  Google Scholar
 S. NematNasser, W. G. Guo, and D. P. Kihl, “Thermomechanical response of AL6XN stainless steel over a wide range of strain rates and temperatures,” Journal of the Mechanics and Physics of Solids, vol. 49, no. 8, pp. 1823–1846, 2001. View at: Publisher Site  Google Scholar
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Copyright © 2019 Peijie Liu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.