Constitutive modeling for elevated temperature flow behavior of a novel Cr-Si alloyed hot stamping steel

The flow behavior of a novel Cr-Si alloyed hot stamping steel (Cr-Si steel) at elevated temperature was investigated via isothermal compression tests on a Gleeble-3500 thermomechanical simulator with a temperature range of 600 ∼ 900 °C and a strain rate range of 0.1 ∼ 10 s−1. Subsequently, the Arrhenius-type constitutive model, comprising strain compensation, was established in accordance with the friction and adiabatic heating corrected stress-strain curves. Furthermore, the predictability and prediction accuracy of the constitutive model were verified. The results reveal that at a constant strain rate, the flow stress of the Cr-Si steel initially increases as the strain increases before tending to stabilize, owning to the combined effects of work hardening and dynamic recovery. The peak flow stresses decrease as the temperatures increase and the strain rates decrease. The constitutive model can accurately predict the elevated temperature constitutive relationship of the Cr-Si steel during the hot stamping process.


Introduction
Hot stamping steels, as one of the advanced high-strength steels (AHSSs), have been widely used in automotive body structural applications, due to their ultra-high strength levels, high dimension accuracy and low springback [1][2][3]. Generally, the hot stamping steel sheets should be heated in a furnace above the austenitization temperature, then transferred to a stamping die and quenched to achieve excellent mechanical properties [1]. However, oxidation and decarburization occur on the surface of the hot stamping steel sheets during the hot stamping process [4,5].
In order to improve the high-temperature oxidation resistance of the hot stamping steel, surface coating technologies are adopted for steel surfaces, including Al-Si coating and Zn coating [5,6]. Nevertheless, the issues associated with the Al-Si coating are low bendability, high costs and international patent restrictions [5,7]. While the Zn coating leads to liquid metal embrittlement (LME) cracking during the resistance spot welding (RSW) process [8,9]. In recent years, except for the coating technologies, the high-temperature oxidation resistance of the hot stamping steel has been investigated and several methods, such as integrating the hot stamping process with the medium-Mn steel [10] or adding Cr and Si elements [11][12][13], has been proposed. Although such high content of Mn element (above 6.0%) in steel significantly improves the mechanical properties after the hot stamping process, and interacts with Cr element to decrease the hot stamping temperature to 750 ∼ 810°C, but that is detrimental to the welding performance [10]. The existence of Cr and Si improves the oxidation resistance of steels by impeding the diffusion of iron cations [14,15]. Whereas, when the Si content is above 1.2%, the net-like Fe 2 SiO 4 forms in the innermost layer of the oxide scales during the hot rolling process and is difficult to remove [16]. Hence, in our studies, a novel uncoated Cr-Si steel with low Si Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. content is developed. The oxide scales exhibit a thickness of less than 5 μm, which is only 1/10 that of traditional 22MnB5 steel.
Besides high-temperature oxidation, the variation of temperature field, stress field and microstructure transformation are complicated during the hot stamping process. Therefore, a constitutive modeling containing process parameters such as flow stress, strain rate, and temperature must be established to investigate the flow behavior of hot stamping steel at elevated temperatures. For traditional 22MnB5 steel, the constitutive modeling of the flow behavior at elevated temperature has been widely studied, including phenomenological constitutive models (Johnson-Cook, Arrhenius-type) and physical-based constitutive models (Zerilli-Armstrong, Norton-Hoff) [17][18][19][20]. The chemical compositions of the Cr-Si steel have modified, therefore, the deformation behaviors at elevated temperatures are different from the traditional 22MnB5 steel. However, there is no report about the flow behavior and constitutive model for Cr-Si steel at elevated temperatures.
In this study, a constitutive model of the novel Cr-Si steel comprising strain compensation was established using the friction and adiabatic heating corrected stress-strain curves. Furthermore, the predictability and prediction accuracy of the constitutive model were examined on the basis of the correlation coefficient (R) and the average absolute relative error (AARE).

Experiments
The chemical compositions of the 22MnB5 and the Cr-Si steels are shown in table 1. The A c1 and A c3 temperatures of the Cr-Si steel are 680°C and 872°C, respectively, that are slightly higher than those of the 22MnB5 steel (A c1 : 658°C, A c3 : 857°C). The steels were melted into ingot in a vacuum induction furnace and forged into slabs. Then the slabs were processed by the hot rolling, cold rolling, and annealing to obtain the cold rolled annealed sheets. Finally, the sheets were heated to 900°C for 3 min and cooled to room temperature at 30°C s −1 for hot stamping. After hot stamping process, the Cr-Si steel reaches a level of 1.5 GPa in tensile strength and exhibits a higher total elongation (9.16%) than the 22MnB5 steel (6.11%), as shown in figure 1. Nevertheless, the elevated temperature flow behaviors of the steel after adding Cr and Si elements are still not clear and worth investigating.
Cylindrical specimens of Φ8 mm × 12 mm in size were selected from the slab of the Cr-Si steel for isothermal compression tests. Which were conducted on the Gleeble-3500 thermomechanical simulator. Temperature controlling curves during the isothermal compression tests are shown in figure 2. The specimens were heated to 1200°C with a heating rate of 5°C s −1 and held for 3 min to homogenize the microstructure. Then cooled to the deformation temperature (600 ∼ 900°C) with a cooling rate of 10°C s −1 and held for 30 s. The tested strain rates were 0.1 s −1 , 1 s −1 and 10s −1 , and the total deformation amount was 80%. In order to control the temperature  accurately, the thermocouple wires need to be welded before the tests. Meanwhile, both ends of the specimens should be coated with graphite lubricant, and stick tantalum metal sheet to reduce friction between the mold and the specimens.

Results and discussion
3.1. Ture stress-strain curves at elevated temperature Figure 3 shows the true stress-strain curves of the Cr-Si steel under different temperatures and strain rates. Initially, the flow stress of the steel increases rapidly with the increase of strains, due to the dominant work hardening. With the strain increases, the dislocation density, energy storage, and driving force of recovery are increase. The stress increases slowly because the dynamic recovery effect exceeds the work hardening effect [21]. Finally, the flow stresses reach a steady state as a result of the dynamic equilibrium between dynamic recovery and work hardening. Under constant strain rate, the flow stress initially increases as the strain increases before tending to stabilize. The peak flow stress decreases as the temperature increases and the strain rate decreases [22,23].

Corrections of friction and adiabatic heating on flow stress curves
During the isothermal compression process, the presence of friction and adiabatic heating would considerably increase the flow stress, despite the use of the graphite lubricant [24]. Therefore, it is necessary to correct the friction and adiabatic heating temperature of the experimental data. The corrected flow stress (s f ) considering the friction effect can be expressed as [25,26]: Where σ is the experimental flow stress (MPa), ε is the experimental strain, r 0 is the initial radius of specimens (mm), h 0 is the height of specimens (mm). The friction factor ( f ) is related to the deformation temperature and strain rate, and can be expressed as follows [25]: Where r and h are the average radius and height of cylinder after deformation, respectively. The barrel parameter (b) can be expressed as: Where Δh and Δr are the reduction of height and difference between the maximum and top radius of cylinder after deformation, these parameters can be obtained through measurement.
In addition, the flow stress (s ) considering the temperature rise caused by adiabatic heating can be corrected according to the equation (4) [24]: Where r is the density of the specimen (g·cm −3 ), C p is the specific heat of the specimen (J·g −1 ·K −1 ), T is the absolute temperature (K) during hot stamping process. The adiabatic correction factor (B) can be calculated using the equation (5): Where x w is the half height of the specimen, K w is the thermal conductivity of the specimen, HTC is the interface heat-transfer coefficient, x D is the distance from the die surface to the die interior where the temperature is constant, K D is the thermal conductivity of die,  e is the strain rate (s −1 ). The parameters used in equations (4) and (5) are calculated using JMatPro software, as shown in table 2. The corrected frictionless and non-adiabatic heating true stress-strain curves derive from equation (1)-(5) are shown in figure 4. It can be seen that the corrected flow stresses are lower than the experimental flow stresses. When compared with the uncorrected flow stress, the work hardening phenomena is eliminated after the correction of the friction and the adiabatic heating effects.   (4) and (5).

Establishment of constitutive model for the Cr-Si steel
Arrhenius-type equation, particularly at elevated temperatures, is the most comprehensive equation describing the relationships between flow stress, strain rate and temperature [19]. Previous studies showed that the Zener-Hollomon parameter (Z) in an exponential equation can be used to express the effect of strain rate and deformation temperature [27]: Where Q is the deformation activation energy (kJ·mol −1 ), R is the universal gas constant (8.3145 J·mol −1 ·K −1 ). The flow stress s (MPa) during isothermal compression mainly depends on the temperature T (K) and the strain rate  e (s −1 ). Therefore, The Arrhenius-type equation can be used to express the relationships between them [28,29]: Taking the natural logarithm of both sides of equation (10) to equations (13) and (14):    is 405.3760 kJ·mol −1 , which is higher than that of traditional 22MnB5 steel [17,20]. Therefore, the addition of Cr and Si elements are clearly related to the deformation activation energy. The Cr and Si elements can influence the diffusion of other elements as well as the glide or climb of dislocations, thus altering the activation energy of hot deformation [30]. Hence, the constitutive equation can be acquired as equation (15): The Arrhenius-type equation only considers the peak stress corresponding to the strain. In order to improve the prediction accuracy, it is critical to take the effect of other strains on the material parameters into account [31]. As a result, for strains ranging from 0.1 to 0.75, the solutions of α, n, Q, and lnA are determined. The relationships between the solutions and the true strains are shown in figure 7. It is indicated that the fifth-order polynomial fit curve is consistent with the effect of strain on the constants.
The final multinomial relationships between α, n, Q, lnA and strains are shown in equations (16) - (19). Table 4 displays the coefficients of the polynomial functions for α, n, Q, lnA that were fitted to the equations.   Figure 8 illustrates the comparison between the experimental values and the predicted results under different strain rates and temperatures. The comparison results indicate that the constitutive model can accurately predict the constitutive relationship of the Cr-Si steel at elevated temperatures over the whole strain range.
In order to further describe the accuracy of the constitutive model, the predictability and prediction accuracy of the constitutive model were examined on the basis of the correlation coefficient (R) and the average absolute relative error (AARE). These indices can be expressed as follows [17][18][19]: where E i is the experimental values, P i is the predicted results,¯Ē P and are the average value of E i and P , i respectively, N is the amount of the experimental data. Figure 9 depicts the R and AARE calculation results based on the experimental and predicted values. The values of the R and AARE are 0.9729 and 6.0045%, respectively. The results reveal that the constitutive model can accurately predict the constitutive relationship of the Cr-Si steel during the hot stamping process.

Conclusion
In this study, the flow behaviors of the novel Cr-Si steel with excellent oxidation resistance at elevated temperatures are investigated by conducting the isothermal compression tests with a temperature range of 600 ∼ 900°C and a strain rate range of 0.1 ∼ 10 s −1 . The following conclusions can be made:  (2) The peak flow stresses decrease as the temperatures increase and the strain rates decrease. The deformation activation energy Q for the Cr-Si steel is 400.6908 kJ·mol −1 , which is higher than that of the 22MnB5 steel.
(3) The constitutive model can accurately predict the constitutive relationship of the Cr-Si steel during the hot stamping process. The values of the R and AARE are 0.9729 and 6.0045%, respectively.