The hot deformation behaviors and constitutive modeling of Hastelloy C276

Hastelloy C276 is widely used in the new generation of nuclear power plants, and hot deformation is the optimum way to form the C276 part. In this investigation, the hot deformation and constitutive modeling of Hastelloy C276 alloy are researched, and the processing maps are drawn. The results show that strain rate and hot deformation temperature have remarkable impacts on the deformation behaviors of the Hastelloy C276 alloy. The yield behavior and the flow stress are predicted based on the Arrhenius constitutive equation, and the correlation coefficients are 0.9613 and 0.9837, indicating the high prediction ability of the established constitutive equation. Rising the deformation temperature can decrease the unstable deformation area, and the studied alloy can be deformed at low strain rates. With the increased strain rate, flow localization occurs, which is not suitable for the hot deformation.


Introduction
Due to their excellent durable strength and corrosion resistance at elevated temperatures, nickel-based alloys are often employed in aerospace, nuclear energy, and chemical industries [1][2][3]. Hot deformation is the optimum way to improve the strength of nickel-based superalloys [4][5][6][7][8]. Due to the complex microstructure and mechanical property evolutions, the optimization of the hot processing parameters of nickel-based alloys is the research interesting over the world [9].
As an ultra-high corrosion-resistant superalloy, Hastelloy C276 is an alternative material for the stator and rotor thin-wall shielding sleeves in the new generation of nuclear power plants [10][11][12]. The precipitation kinetics of Hastelloy C276 alloy is complicated in the hot deformation temperature ranges, and the main precipitated phases can affect the tensile, impact-toughness and high-temperature creep properties of the alloy [13]. Therefore, it is important to study the hot-deformation behaviors of the nickel-based Hastelloy C276 alloy.
The hot-deformation process of nickel-based alloys is accompanied by work hardening (WH), dynamic recovery, and recrystallization (DRV/DRX) [12,13]. The hot deformation parameters and initial microstructure will determine the final mechanical properties of nickel-based alloys [2,3,12,14]. Therefore, constitutive models have been established to describe the hot deformation behaviors of nickel-based alloys under various working conditions. The constitutive models of alloys are divided into phenomenological, physical, and artificial neural network models [15,16]. The first kind of model uses macroscopic variables to describe the hotdeformation behaviors of materials, such as high-strength steels [17][18][19], Al alloys [20], superalloys [21][22][23], Mg alloys [24,25], Ti alloys [26,27], and other alloys [28,29]. The second kind of model is established according to the deformation mechanism of the material, and the microscopic aspects, such as dislocation accumulation and grain size evolution, are considered [30][31][32]. The third kind of constitutive model can obtain the material constants using regression analysis and is used by many researchers due to its high prediction accuracy [33][34][35].
The widely used processing map is a method for the optimal selection of thermal deformation parameters [36,37]. Ji et al [38] established a processing map of TA15 titanium alloy. Jiao et al [39] built the processing map Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. of 06Cr19Ni9NbN steel and optimized the compression process parameters. Zhang et al [40] established the strain-compensated model and processing maps to predict the plastic behaviors of Zr-4 alloys. Lypchanskyi et al [41] identified the undesirable and potentially favorable thermal deformation parameters of Nickel-Based Superalloy. Yao et al [42] created the processing maps of the GH909 superalloy. Chen et al [43] constructed a processing map of GH4698 alloy at high-temperature deformation conditions. Jaladurgam and Kanjarla [44] developed a processing map to predict the safe hot workability window of Hastelloy C-276 alloy. Zhang et al [45] reported the microstructural evolution of Ni-Cr-Mo-based C276 superalloy during hot compression, their results revealed that the DRX fraction, high angle grain boundary (HAGB), and Σ3 twin boundary increase with the increased deformation temperature and deformation degree. Zhang et al [46] also studied the hot deformation behavior of C276 superalloy in shifted strain rate compression deformation, and found that the strain rate variation greatly affected the microstructure evolution, leading to different dynamic recrystallization (DRX) tendencies from the compressions with a constant strain rate. Metha et al [47] reported the orientationdependent work hardening behavior of cold-rolled and solution-annealed Hastelloy C-276 alloy. Therefore, the above reports mainly focus on the hot deformation behavior of some typical alloys, and few studies are carried out on Hastelloy C276 alloy. So, the hot deformation of Hastelloy C276 alloy at 900°C-1050°C is investigated and a constitutive model is carried out. The DMM-based processing maps are drawn to find the suitable process parameters, and the established maps are confirmed by microstructure observations.

Material and experiment
In this work, a commercial high-strength superalloy (Hastelloy C-276) was adopted, and the chemical compositions (wt.%) is 0.5Co-15.60Cr-15.86Mo-3.20W-6.10Fe-0.04Si-0.60Mn-0.004C-(Bal.)Ni. The Ø8 mm × 12 mm cylindrical specimens were made from Hastelloy forged bars. The original microstructure of the specimen is shown in figure 1, and a non-heat treatment was performed before the hot deformation. The hot deformation experiments were carried out at four strain rates and four temperatures, and the reduction in height is 60%, i.e., the engineering strain is 60%. The Gleeble-3500 thermo-simulation machine was used to carry out the hot compression tests. During the hot deformation, a 0.1 mm thick tantalum foil was used to minimize friction and sticking. Before hot pressing deformation, the samples were heated up to the given temperature at 10 K per second, and 3 min later, the specimens are deformed at the given deformation temperature. The true stress-strain curves were obtained based on the engineering stress-stain curves obtained by the Gleeble-3500 thermo-simulation machine. The deformed samples were water-cooled after the hot deformation tests to preserve the deformed microstructure. The experimental procedure is shown in figure 2.
The hot compressed deformation samples were cut out by an in-line cutting machine, and the impacts of hot processing on the microstructure were discussed by EBSD experiments. For the EBSD observations, the samples were electrochemically polished in a mixed solution of 10% HClO 4 and 90% C 2 H 5 OH with a voltage of 22 V at the temperature ranges of −30 to 25°C [48]. The EBSD was performed on a JEOL-7001F1 FE-SEM scanning electron microscope equipped with HKL Channel 5 software.  Figure 3 shows the true stress and true strain relationships of Hastelloy C276 alloy in the hot deformation tests. The high-temperature deformation of Hastelloy C276 nickel-based alloy is a complicated evolution between WH and dynamic softening. In the initial stage, the stress rapidly increases to the peak stress. Thanks to the low stacking fault energy of nickel-based alloys, the DRV caused by dislocation climbing, slippage, and annihilation is feeble [49], and the DRV cannot balance the work hardening (WH) effects, where the generation and multiplication of dislocations dominate, leading to a pronounced work hardening phenomenon. Subsequently, DRX occurs when the accumulated dislocation density exceeds the critical value, the flow stress decreases,  dynamic softening dominates with increased deformation, and eventually, the balance between WH, DRV, and DRX is achieved.

Hot deformation characteristics
The processing parameters can affect the flow behaviors of Hastelloy C276 nickel-based alloy during thermal deformation. The stress decreases at a higher temperature, which is attributed to the strong thermal activation of metal atoms at higher deformation temperatures, and the dislocation motion becomes easier [50]. Meanwhile, a higher temperature can promote the motion of grain boundaries, leading to the formation of the DRX grains. On the other hand, the low strain rate can promote grain growth, and the flow stress decreases significantly at a low strain rate. Figure 4 shows the microstructures of the alloy deformed at different temperatures and strain rates. Here, the engineering strain in figure 4 is 0.6, i.e., the true strain is 0.92. Many DRX grains can be found in figure 4, and the DRX degree is high at these deformation conditions. However, the processing parameters have an obvious influence on the microstructures of the studied alloy. The DRX grain sizes are small at low strain rates, figure 4(a), and the DRX grain sizes are smaller at high strain rates, figure 4(b). Due to the high strain rates, there is not enough time for the growth of the DRXed grains. So, the DRX grain sizes at high strain rates are smaller. At the same time, the DRX degree of the alloy deformed at high strain rates is low, as shown in figure 4(b). With the increase of the deformation temperature, the motion of grain boundaries and diffusion of atoms are enhanced, leading to the formation of the DRX grains, figure 4(c). Meanwhile, it can be found from figures 4(b), (c) that the DRXed grain sizes are bigger at high deformation temperatures, and the DRX degree is enlarged too. So, the obvious flow softening occurs at high deformation temperatures and low strain rates, resulting in low flow stress at high deformation temperatures and low strain rates.

Yield behavior and prediction
In figure 3, the stress of the Hastelloy C276 alloy increases to the peaks rapidly at the onset of hot deformation, and no apparent yielding behavior can be found. The yield stress is considered as the stress at 0.2% plastic deformation. Figure 5(a) shows that the yield stress ( y s ) is affected by strain rate and temperature. Increasing  strain rate or decreasing temperature can increase yield stress. According to previous reports [51], the yield stress is modeled as where y s is the yield stress, T is the temperature,  e represents the strain rate, , a b and Q y are the material constants, and R is the gas constant (8.314 J mol −1 K −1 ).
Referencing the calculation method in [52], the material parameters in equation (1) are obtained immediately. Therefore, it can be expressed as:  (2) is suitable for describing the yield stress.
Generally, the hot deformation of Hastelloy C276 alloy can be modeled as [58]: and F ( ) s is a function of stress: Generally, Zener-Hollomon (Z) parameter is used to describe the effects of temperature and strain rate on the hot deformation behaviors of the studied alloy: Consequently, the flow stress is: Using equations (3)-(4), the relationships between s and  e at various temperatures are: where B and C are related to temperature. So, equations (9)-(10) are obtained: Submitting four strain rates and the stresses at a strain of 0.1 into equations (9)-(10), the quantitative relation is shown in figure 6. The values of n andb ¢ are the mean slope values of lines in figure 6, and n and b ¢ are 11.3200 and 0.0440 MPa , 1 respectively. Therefore, α is 0.00388 MPa .  figure 7(a), the n is obtained as 8.4682. The Q (502.9860 kJ mol −1 ) is calculated from the slope of lines in figure 7(b). Based on the interception of lines in figure 7(b), the A is computed, and the value of lnA is 43.0433.
The material values can be calculated at other strains using the same method. The mathematical relationships between the material values and strains are shown in figure 8, and a five-order polynomial is best to describe the mathematical relationships, as shown in equation (13) The good agreements in figure 9 show that the experimental stress is well predicted by the above model. It can be seen that there is little difference between the predicted stress and experimental stress, and the correlation coefficient (R) can be expressed as, where E i and P , i are the experimental and predicted stresses obtained by the model of equation (6), Ē and P are the experimental and predicted mean stresses, respectively.
It can be found from figure 10 that the R is 0.9837, and the predicted stress is well coincided with the experimental stress, indicating the strong predicting ability of the established model. Liu et al [59] predicted the flow stress of Hastelloy C-276 alloy using modified Zerilli-Armstrong, Johnson-Cook, and Arrhenius-type constitutive models, and the Arrhenius-type constitutive model has the highest correlation coefficients (0.984). In their work, the alloy was deformed at 1223-1423 K and 0.01-10 s −1 , which is different from the deformation process described in this work, and similar correlation coefficients are acquired, indicating the high prediction ability of the model established in this work.

Processing maps and microstructural observation
The hot processing map can reveal the microstructure of the material under various deformation conditions and can be used to optimize the thermal processing parameters [58]. During the hot deformation, the energy consumption P of the material includes two parts: the energy G, which is consumed during the deform process of the material, and the energy J, which is consumed to change the microstructure. It is expressed by equation (15): The ratio of G to J is the material strain rate sensitive factor m, which can be expressed as,  figure 11, and the limited deformation conditions and microstructural evolution in the instability deformation regimes can be seen from the maps. Figure 12 shows the processing maps of the Hastelloy C276 alloy at different deformation conditions, the values included as the contours are the energy dissipation efficiency (η), and the unstable areas are marked by gray color. Generally, the mechanisms such as adiabatic shear band, intergranular cracking, dynamic strain aging, kink bands, mechanical twinning, and prior particle boundaries contribute to the flow instability, and the high strain rate and low temperature will lead to the unstable deformation of the studied alloy [60]. The distribution of energy dissipation efficiency (η) and the unstable deformation areas are changed with the strains, as shown in figure 12. When the true stain is 0.3, figure 12(a), there is an unstable deformation region located at low deformation temperature and high strain rate area, and the value of η is less than 0.2. With the increase of true strain, the unstable deformation region is diminished, as shown in figure 12(c), and the studied alloy can be deformed at high strain rates, this phenomenon is related to the microstructure evolution at high strain. Similar results can be found in references [60,61]. With the increase of the strain, the initial grains suffer from large deformation, the dislocation density is increased. At the same time, the large deformation provides a certain time for the nucleation and growth of the DRXed grains, even at a high strain rate, and the DRX degree is enhanced, as shown in figures 3-4. So, the unstable deformation region is diminished with the increased strain.
At higher deformation temperatures, the strong thermal activation of metal atoms is enhanced, the cumulated dislocations are easy to be released, and the DRV/DRX processes are strengthened. So, the homogeneous deformation ability of the alloy is acquired at low and high strain rates. The processing maps in figure 12 also indicate that with the increase of deformation temperature, the stable deformation areas are increased. For example, when the deformation temperature is ranged from 1180-1240 K, the alloy is hard to be deformed at a high strain rate, while when the deformation temperature is increased to the range of 1240-1320 K, the alloy can be deformed at strain rate that no more than 0.05 s −1 (ln 3  e = -). Therefore, the stable deformation areas are increased at high deformation temperatures, and increasing the deformation temperature can decrease the unstable deformation area, as shown in figure 12(c).
The accuracy of the established processing maps can be verified by the IPF maps and the KAM maps, figure 13. The obvious recrystallization phenomena can be found at all deformation conditions, as shown in figures 13(a)-(c). At higher deformation temperatures, the fraction of recrystallization is increased, Figure 11. 3D power dissipation efficiency maps at the true strain of (a) 0.3; (b) 0.6; (c) 0.9.
figures 13(b)-(c). However, with the increase of strain rates, the fraction of recrystallization is low, and the grain sizes are uneven, i.e., the mixed grains. The unstable deformation of the studied alloy is companies with cracks, coarsen grains, mixed grains, and flow localization, which is harmful to the alloy. With the increase of strain rate, flow localization and mixed grains occur, as shown in figure 13(b). Furthermore, the corresponding power dissipation efficiency at the strain rate of 0.1 s −1 is small, figures 11(c) and 12(c), indicating the low recrystallization degree. Figures 13(d)-(f) shows the KAM maps of the alloy deformed at different temperatures and strain rates. The long deformation time at a low strain rate is in favor of the DRV and the DRX process, and the dislocation density is low, figure 13(d). Meanwhile, the high deformation temperature makes for the accelerated softening process, and the dislocation density is low, which is also suitable for hot deformation, figure 13(f). However, for unsuitable deformation conditions, such as high strain rate and low temperature, the multiplicated dislocation is heterogeneous distribution and consumption. The high-density dislocation exists in the material, as marked by the red dots at the top of figure 13(e), which is easy to motivate the flow localization. At the same time, some initial grains are hard to deform, and the dislocation density within these grains is low, which is adverse to the DRX and the DRV process, and the mixed grains occur in figures 13(b) and (e).

Conclusions
In this work, the hot deformation of Hastelloy C276 alloy is studied, and a mathematic model and DMM-based processing maps are drawn to predict the deformation behaviors and find suitable deformation parameters. Some important conclusions are made as follows: (1) The processing parameters have remarkable impacts on the deformation of the studied alloy. The flow stress decreases at a higher temperature, and the flow stress decreases significantly at a lower strain rate.  (3) The hot processing maps of the studied alloy are established and the accuracy of the established processing maps can be verified by the IPF and KAM maps. Increasing the deformation temperature can decrease the unstable deformation area, and the studied alloy can be deformed at low strain rates. With the increase of strain rate, flow localization occurs, which is not suitable for the hot deformation.