# Dynamic Mechanical Characteristics and Constitutive Modeling of Rail Steel over a Wide Range of Temperatures and Strain Rates

Authors: Peijie Liu; Yanming Quan; Guo Ding
Journal: Advances in Materials Science and Engineering
 (2019)

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## 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 heat-treated 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 temperature-sensitive and strain rate-sensitive. 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 Johnson-Cook (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.


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## Body

## 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 in-depth 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 grinding-related 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 stress-strain behavior under dynamic deformation conditions, the well-established 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 physically-based (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 physically-based 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 physically-based models whose material constants are more difficult to derive than the empirical ones. The Johnson–Cook (JC) model [11] is an empirically-based 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], high-temperature 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 heat-treated 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 Figure1. 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 heat-treated Mn-steel 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 slack-quench 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 heat-treatment 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 high-speed 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.Figure 1
Typical structure of a flat bottom rail track.Table 1
Chemical constituents (wt.%) of U71Mn.
Material
C
Mn
Si
S
P
Fe
U71Mn
0.65–0.76
0.70–1.20
0.15–0.58
≤0.025
≤0.03
BalanceTable 2
Physical properties of U71Mn.
Mechanical properties
Values
Thermal properties
Values
Density (kg/m3)
7800
Thermal conductivity (W/(m·°C))
51
Young’s modulus (GPa)
210
Specific heat (J/(kg·°C))
500
Poisson’s ratio
0.3
Melting point (°C)
1469Figure 2
SEM graph of heat-treated U71Mn rail steel.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 Figure3. 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.Figure 3
Sampling positions of the tested specimens.


### 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 closed-loop 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 stress-strain behavior from 102 s−1 to 104 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 closed-loop 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 high-temperature dynamic test, a K-type 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 ultrahigh-strength 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 sample-bars 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.Figure 4
Experimental setup for dynamic tests. (a) Schematic of the SHPB equipment with a synchronously assembled heating system (εI, εR, and εT refer to strain signals from incident waves, reflected waves, and transmitted waves, respectively). 1, nitrogen gas gun; 2, inlet valve; 3, air chamber; 4, barrel; 5, striker bar; 6, incident bar; 7, strain gauge; 8, specimen; 9, heating furnace; 10, transmitted bar; 11, energy absorption bar; 12, bar mover; 13, support; 14, air pipe; 15, magnetic outlet valve. (b) Detailed configuration of the sample fixed during the high-temperature dynamic test.
(a)
(b)On the basis of the theory of one-dimensional wave propagation, the stressσt, strain εt, and strain rate ε˙t of the specimen are expressed as(1)σt=A0ASEεT,εt=−2c0lS∫0tεRdt,ε˙t=−2c0lSεR,where A0 represents the cross-sectional area of the bars; E denotes Young’s modulus of the bars; c0 (c0=E/ρ, where ρ is the density of the pressure bars) represents the propagation velocity of one-dimensional elastic longitudinal wave; As means the initial cross-section area of the specimen; ls refers to the specimen length; and εT and εR represent the strain signals of the transmitted and reflected wave, respectively.


## 2.1. Materials

The typical structure of a flat bottom rail comprises rail head, rail web, and rail foot, as shown in Figure1. 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 heat-treated Mn-steel 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 slack-quench 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 heat-treatment 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 high-speed 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.Figure 1
Typical structure of a flat bottom rail track.Table 1
Chemical constituents (wt.%) of U71Mn.
Material
C
Mn
Si
S
P
Fe
U71Mn
0.65–0.76
0.70–1.20
0.15–0.58
≤0.025
≤0.03
BalanceTable 2
Physical properties of U71Mn.
Mechanical properties
Values
Thermal properties
Values
Density (kg/m3)
7800
Thermal conductivity (W/(m·°C))
51
Young’s modulus (GPa)
210
Specific heat (J/(kg·°C))
500
Poisson’s ratio
0.3
Melting point (°C)
1469Figure 2
SEM graph of heat-treated U71Mn rail steel.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 Figure3. 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.Figure 3
Sampling positions of the tested specimens.


## 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 closed-loop 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 stress-strain behavior from 102 s−1 to 104 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 closed-loop 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 high-temperature dynamic test, a K-type 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 ultrahigh-strength 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 sample-bars 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.Figure 4
Experimental setup for dynamic tests. (a) Schematic of the SHPB equipment with a synchronously assembled heating system (εI, εR, and εT refer to strain signals from incident waves, reflected waves, and transmitted waves, respectively). 1, nitrogen gas gun; 2, inlet valve; 3, air chamber; 4, barrel; 5, striker bar; 6, incident bar; 7, strain gauge; 8, specimen; 9, heating furnace; 10, transmitted bar; 11, energy absorption bar; 12, bar mover; 13, support; 14, air pipe; 15, magnetic outlet valve. (b) Detailed configuration of the sample fixed during the high-temperature dynamic test.
(a)
(b)On the basis of the theory of one-dimensional wave propagation, the stressσt, strain εt, and strain rate ε˙t of the specimen are expressed as(1)σt=A0ASEεT,εt=−2c0lS∫0tεRdt,ε˙t=−2c0lSεR,where A0 represents the cross-sectional area of the bars; E denotes Young’s modulus of the bars; c0 (c0=E/ρ, where ρ is the density of the pressure bars) represents the propagation velocity of one-dimensional elastic longitudinal wave; As means the initial cross-section area of the specimen; ls refers to the specimen length; and εT and εR represent the strain signals of the transmitted and reflected wave, respectively.


## 3. Results and Discussion

Figure5 shows the quasistatic compressive stress-strain 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.Figure 5
True flow strain-stress curve of U71Mn rail steel at 0.001 s−1 and 20°C.In order to verify the stress-state 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 2-wave stress curves under high temperatures at the strain rate of 10000 s−1. The 1-wave stress analysis indicates the stress conditions at the specimen-transmitted bar interface. The 2-wave analysis denotes the stress conditions at the incident bar-specimen 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 1-wave and the 2-wave stress-strain response is reached, which represents the stress-state equilibrium within the specimen and therefore illustrates the validity of the SHPB test results.Figure 6
Typical strain gauge signals and the corresponding 1- and 2-wave stress curves obtained during the SHPB test with U71Mn samples: (a) strain gauge signals at 10000 s−1 and 600°C; (b) 1- and 2-wave stress curves at 10000 s−1 and 600°C; (c) strain gauge signals at 10000 s−1 and 800°C; (d) 1- and 2-wave stress curves at 10000 s−1 and 800°C.
(a)
(b)
(c)
(d)


### 3.1. Strain Rate Hardening Effect on the Flow Stress

The dynamic compressive flow response under varying indicated temperatures is presented in Figure7. Although deformed, none of the tested samples brought about fracture. Thus, the ending parts of the stress-strain curves indicate unloading instead of fracture. The influence of strain rate on the flow response of U71Mn as a function of temperature at ε = 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 stress-strain 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.Figure 7
Flow stress-strain curves of U71Mn rail steel in SHPB tests under varying prescribed temperatures: (a) 20°C; (b) 200°C; (c) 400°C; (d) 600°C; (e) 800°C.
(a)
(b)
(c)
(d)
(e)Figure 8
Effect of strain rate on the flow stress of U71Mn rail steel atε = 0.1, as a function of temperature.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 as(2)β=σ2−σ1lnε˙2/ε˙1,where σ1 and σ2 are the experimentally obtained flow stresses at strain rates of ε˙1 and ε˙2 under a given plastic strain and temperature, respectively.For U71Mn rail steel, the strain rate sensitivityβ 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.Figure 9
Strain rate sensitivity at 1500 s−1–10000 s−1 under varying temperatures.


### 3.2. Thermal Softening Effect on the Flow Stress

Figure10 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 ε = 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.Figure 10
Flow stress-strain curves of U71Mn steel in SHPB tests under different strain rates: (a) 1500 s−1; (b) 5000 s−1; (c) 10000 s−1.
(a)
(b)
(c)Figure 11
Effect of temperature on the flow stress of U71Mn steel at a true strain of 0.1, as a function of strain rate.According to [21], temperature dependence on flow stress can be quantitatively evaluated by introducing a temperature sensitivity parameter na, which is expressed as(3)na=−lnσ2/σ1lnT2/T1,where σ1 and σ2 are the flow stresses acquired from tests performed at temperatures of T1 and T2, respectively; T1 is the reference temperature (taken as 20°C), and T2 (T2 > T1) is the experimental temperature.For U71Mn rail steel, Figure12 shows the temperature sensitivity na as a function of true strain at 5000 s−1 under varying temperature ranges. It shows that the na increases as temperature rises. A significant observation is that the na 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.Figure 12
Temperature sensitivity at 5000 s−1 under various temperature ranges.


### 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 Figure5, it displays that the strain hardening rate is evident under quasistatic compressive condition, which can be explicated by the fact that the deformation-generated 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](4)ΔT=ζρsCv∫0εσdε,where ρs is the density of the studied material (adopted as 7.8 g/cm3), Cv 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 stress-strain curve.Table3 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.Table 3
Variation of adiabatic temperature increment under various compressive conditions.
Temperature (°C)
Strain rate (s−1)
Temperature increment ΔT (°C)
ε = 0.05
ε = 0.1
ε = 0.15
ε = 0.2
ε = 0.25
20
1500
14.02
32.74
51.66
—
—
5000
15.59
35.68
55.76
—
—
10000
16.47
37.86
59.22
80.02
100.36
200
1500
12.11
29.18
46.46
—
—
5000
12.60
30.28
47.94
—
—
10000
13.31
31.63
50.20
68.33
86.12
400
1500
10.15
24.23
—
—
—
5000
10.63
25.33
39.96
—
—
10000
11.58
26.81
41.99
56.89
71.60
600
1500
9.44
22.47
—
—
—
5000
10.30
24.12
39.09
—
—
10000
9.89
23.72
37.94
52.14
66.33
800
1500
2.66
6.74
—
—
—
5000
3.03
7.48
12.85
—
—
10000
3.12
8.15
14.14
20.92
28.22


### 3.4. Establishment of the Johnson–Cook Model

As an empirically-based 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 as(5)σ=A+Bεn1+Clnε˙∗1−T∗m,where ε represents the plastic strain, σ denotes the Von Mises flow stress, ε˙∗=ε˙/ε˙0 represents the dimensionless strain rate (ε˙ means the true strain rate, while ε˙0 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 T∗=T−Tr/Tm−Tr represents the homologous temperature where Tm denotes the melting point (taken as 1469°C for the studied steel) and Tr denotes the reference temperature (taken as 20°C, T ≥ Tr).The adiabatic temperature riseΔT 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 T∗ in equation (5) is transformed to(6)T∗=T+ζ/ρsCv∫0εσdε−TrTm−Tr.On the basis of the experimentally obtained stress-strain curves given in Figures5 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(7)σ=A+Bεn.Taking the logarithmic transformation on the both sides of equation (7), then equation (7) will transform to(8)lnσ−A=lnB+nlnε.Based on the results of quasistatic experiment from Figure5, the curve of lnσ−A vs. lnε can be plotted by the substitution of A into equation (8) and by utilizing the values of the stress-strain curve under varying strains from the hardening part of the stress-strain 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(9)σ=A+Bεn1+Clnε˙∗.Employing the results of stress-strain 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 σ/A+Bεn−1 vs. lnε˙∗ plot.At 0.001 s−1, equation (5) can be transformed as(10)σA+Bεn1+Clnε˙∗=1−T∗m.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 ln1−σ/A+Bεn1+Clnε˙∗ vs. lnT∗ 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. Table4 lists the original JC model parameters. Therefore, the expression of the JC constitutive equation of U71Mn rail steel is given as follows:(11)σ=1231.54+2501.96ε1.5571+0.01284lnε˙0.0011−T−2014490.85624.Table 4
Material constants in the original JC model for U71Mn.
Parameter
A (MPa)
B (MPa)
n
C
m
Values
1231.54
2501.96
1.557
0.01284
0.85624Figure13 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:(12)σ=A+Bεn1+Clnε˙∗1−m0T∗maT+b,where a, b, and m0 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:(13)σ=1231.54+2501.96ε1.5571+0.01284lnε˙0.0011−1.27T−2014490.85624−0.48T+361.8.Figure 13
Comparison between the experimental and predicted stress using the original JC model at 200°C and 10000 s−1.The comparison results between the measured and predicted flow stress employing the modified JC model are presented in Figure14. 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 as(14)R=∑i=1i=Nσexpi−σ¯expσpi−σ¯p∑i=1i=Nσexpi−σ¯exp2∑i=1i=Nσpi−σ¯p2,AARE%=1N∑i=1i=Nσexpi−σpiσexpi×100,where σexp is the experimentally-obtained flow stress, σp represents the predicted flow stress, σ¯exp denotes the mean values of σexp, and σ¯p means the mean values of σp. 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.Figure 14
Comparison between the experimental and predicted stress using the modified JC model predictions: (a) 1500 s−1; (b) 5000 s−1; (c) 10000 s−1.
(a)
(b)
(c)Figure 15
Correlation between the measured and predicted flow stresses under varying deformation conditions. (a) Original JC model; (b) modified JC model.
(a)
(b)


## 3.1. Strain Rate Hardening Effect on the Flow Stress

The dynamic compressive flow response under varying indicated temperatures is presented in Figure7. Although deformed, none of the tested samples brought about fracture. Thus, the ending parts of the stress-strain curves indicate unloading instead of fracture. The influence of strain rate on the flow response of U71Mn as a function of temperature at ε = 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 stress-strain 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.Figure 7
Flow stress-strain curves of U71Mn rail steel in SHPB tests under varying prescribed temperatures: (a) 20°C; (b) 200°C; (c) 400°C; (d) 600°C; (e) 800°C.
(a)
(b)
(c)
(d)
(e)Figure 8
Effect of strain rate on the flow stress of U71Mn rail steel atε = 0.1, as a function of temperature.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 as(2)β=σ2−σ1lnε˙2/ε˙1,where σ1 and σ2 are the experimentally obtained flow stresses at strain rates of ε˙1 and ε˙2 under a given plastic strain and temperature, respectively.For U71Mn rail steel, the strain rate sensitivityβ 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.Figure 9
Strain rate sensitivity at 1500 s−1–10000 s−1 under varying temperatures.


## 3.2. Thermal Softening Effect on the Flow Stress

Figure10 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 ε = 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.Figure 10
Flow stress-strain curves of U71Mn steel in SHPB tests under different strain rates: (a) 1500 s−1; (b) 5000 s−1; (c) 10000 s−1.
(a)
(b)
(c)Figure 11
Effect of temperature on the flow stress of U71Mn steel at a true strain of 0.1, as a function of strain rate.According to [21], temperature dependence on flow stress can be quantitatively evaluated by introducing a temperature sensitivity parameter na, which is expressed as(3)na=−lnσ2/σ1lnT2/T1,where σ1 and σ2 are the flow stresses acquired from tests performed at temperatures of T1 and T2, respectively; T1 is the reference temperature (taken as 20°C), and T2 (T2 > T1) is the experimental temperature.For U71Mn rail steel, Figure12 shows the temperature sensitivity na as a function of true strain at 5000 s−1 under varying temperature ranges. It shows that the na increases as temperature rises. A significant observation is that the na 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.Figure 12
Temperature sensitivity at 5000 s−1 under various temperature ranges.


## 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 Figure5, it displays that the strain hardening rate is evident under quasistatic compressive condition, which can be explicated by the fact that the deformation-generated 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](4)ΔT=ζρsCv∫0εσdε,where ρs is the density of the studied material (adopted as 7.8 g/cm3), Cv 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 stress-strain curve.Table3 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.Table 3
Variation of adiabatic temperature increment under various compressive conditions.
Temperature (°C)
Strain rate (s−1)
Temperature increment ΔT (°C)
ε = 0.05
ε = 0.1
ε = 0.15
ε = 0.2
ε = 0.25
20
1500
14.02
32.74
51.66
—
—
5000
15.59
35.68
55.76
—
—
10000
16.47
37.86
59.22
80.02
100.36
200
1500
12.11
29.18
46.46
—
—
5000
12.60
30.28
47.94
—
—
10000
13.31
31.63
50.20
68.33
86.12
400
1500
10.15
24.23
—
—
—
5000
10.63
25.33
39.96
—
—
10000
11.58
26.81
41.99
56.89
71.60
600
1500
9.44
22.47
—
—
—
5000
10.30
24.12
39.09
—
—
10000
9.89
23.72
37.94
52.14
66.33
800
1500
2.66
6.74
—
—
—
5000
3.03
7.48
12.85
—
—
10000
3.12
8.15
14.14
20.92
28.22


## 3.4. Establishment of the Johnson–Cook Model

As an empirically-based 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 as(5)σ=A+Bεn1+Clnε˙∗1−T∗m,where ε represents the plastic strain, σ denotes the Von Mises flow stress, ε˙∗=ε˙/ε˙0 represents the dimensionless strain rate (ε˙ means the true strain rate, while ε˙0 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 T∗=T−Tr/Tm−Tr represents the homologous temperature where Tm denotes the melting point (taken as 1469°C for the studied steel) and Tr denotes the reference temperature (taken as 20°C, T ≥ Tr).The adiabatic temperature riseΔT 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 T∗ in equation (5) is transformed to(6)T∗=T+ζ/ρsCv∫0εσdε−TrTm−Tr.On the basis of the experimentally obtained stress-strain curves given in Figures5 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(7)σ=A+Bεn.Taking the logarithmic transformation on the both sides of equation (7), then equation (7) will transform to(8)lnσ−A=lnB+nlnε.Based on the results of quasistatic experiment from Figure5, the curve of lnσ−A vs. lnε can be plotted by the substitution of A into equation (8) and by utilizing the values of the stress-strain curve under varying strains from the hardening part of the stress-strain 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(9)σ=A+Bεn1+Clnε˙∗.Employing the results of stress-strain 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 σ/A+Bεn−1 vs. lnε˙∗ plot.At 0.001 s−1, equation (5) can be transformed as(10)σA+Bεn1+Clnε˙∗=1−T∗m.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 ln1−σ/A+Bεn1+Clnε˙∗ vs. lnT∗ 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. Table4 lists the original JC model parameters. Therefore, the expression of the JC constitutive equation of U71Mn rail steel is given as follows:(11)σ=1231.54+2501.96ε1.5571+0.01284lnε˙0.0011−T−2014490.85624.Table 4
Material constants in the original JC model for U71Mn.
Parameter
A (MPa)
B (MPa)
n
C
m
Values
1231.54
2501.96
1.557
0.01284
0.85624Figure13 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:(12)σ=A+Bεn1+Clnε˙∗1−m0T∗maT+b,where a, b, and m0 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:(13)σ=1231.54+2501.96ε1.5571+0.01284lnε˙0.0011−1.27T−2014490.85624−0.48T+361.8.Figure 13
Comparison between the experimental and predicted stress using the original JC model at 200°C and 10000 s−1.The comparison results between the measured and predicted flow stress employing the modified JC model are presented in Figure14. 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 as(14)R=∑i=1i=Nσexpi−σ¯expσpi−σ¯p∑i=1i=Nσexpi−σ¯exp2∑i=1i=Nσpi−σ¯p2,AARE%=1N∑i=1i=Nσexpi−σpiσexpi×100,where σexp is the experimentally-obtained flow stress, σp represents the predicted flow stress, σ¯exp denotes the mean values of σexp, and σ¯p means the mean values of σp. 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.Figure 14
Comparison between the experimental and predicted stress using the modified JC model predictions: (a) 1500 s−1; (b) 5000 s−1; (c) 10000 s−1.
(a)
(b)
(c)Figure 15
Correlation between the measured and predicted flow stresses under varying deformation conditions. (a) Original JC model; (b) modified JC model.
(a)
(b)


## 4. Conclusions

In this paper, the true stress-strain 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.



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*Source: 6862391-2019-03-06.xml*