Flow stress modeling, processing maps and microstructure evolution of 05Cr17Ni4Cu4Nb Martensitic stainless steel during hot plastic deformation

The hot deformation behavior of 05Cr17Ni4Cu4Nb stainless steel at the temperatures from 950 °C to 1250 °C and strain rates from 0.01 s−1 to 5 s−1 was investigated on the basis of test data obtained by Gleeble1500D thermo-simulation machine. A two-stage flow stress constitutive model incorporating the effects of the strain rate and temperature on the deformation behavior is proposed. The stress-strain relations of 05Cr17Ni4Cu4Nb steel predicted by the proposed model agree well with the experimental results. Moreover, the hot processing maps are also investigated. Combined with the microstructure evolution analysis, the appropriate hot forming processing parameters of 05Cr17Ni4Cu4Nb steel is proposed in the temperature range of 1000–1100 °C and strain rate range of 0.01–0.02 s−1.


Introduction
With the characteristics of anti-corrosion, resist-to-abrasion and high mechanical strength, 05Cr17Ni4Cu4Nb martensitic stainless steel is widely used in structural components of the equipment in marine construction, chemical industries and power plants with a year-to-year consumption increase [1]. Recently, many studies about 05Cr17Ni4Cu4Nb martensitic stainless steel focus on its heat treatment processes, surface coating methods or the properties during additive manufacturing [2][3][4][5]. Generally, hot forging process is the effective method to form the shape and refine the grains for the parts made of 05Cr17Ni4Cu4Nb steel. To improve the mechanical performances of this material, the designing for the deformation processing parameters of hot forging should be investigated, which requires the further researches on the material flow behaviors, processing maps and the microstructure evolution.
To improve the forming parameters and develop the reliability of hot forging process, finite element(FE) simulation can be effective method to promote the shape-controlling and property-controlling forging technology. To carry out an effective evaluation of the material flow behavior and the stress/strain distribution during hot forging process, it is essential to establish the precise constitutive equation that describes the combined influences of temperature, strain rate, and strain on the flow stress. Generally, two kinds of common methods are adopted in establishing the flow stress constitutive model: phenomenological method and physicalbased method, which have been applied for different types of materials [6][7][8][9][10]. The phenomenological models including Johnson-Cook (JC) model [11], Arrhenius equation [12], etc are mainly based on the mathematical equations in order to fit or regress the experimental data, therefore, most of the parameters in phenomenological models lack physical meanings. During the hot deformation process, the internal microstructure of metals generally evolves extensively and significantly influences the constitutive behavior of materials [13]. Investigations of the hot deformation behaviors for different martensitic steels(30Cr15Mo1N [14], AISI403/ 403Nb [15], high nitrogen martensitic steel [16], etc) have been carried out carefully, and it was found that the dynamic recrystallization(DRX) occur during the hot deformation conditions. However, the constitutive Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. models in the literatures were established on the basis of Arrhenius equation, which were not quite suitable to give a reasonable description on the mechanism of DRX. To solve this problem, the physical-based models are established on the framework of deformation mechanism, which take into account the evolution of dislocation density and recrystallization kinetics [17][18][19]. Generally, the true stress-true strain relations of steels in hot deformation exhibit the characteristics of working hardening, recovery and DRX obviously. Therefore, the physical-based two-stage constitutive model was proposed to describe the hardening, softening and steady behavior in different stages of the true stress-true strain curves. The researches in literatures exihibit the good applicability and higher accuracy of the two-stage physical-based constitutive model for various kinds of alloys, including the Ni-based superalloys [20,21], steels [22][23][24][25], etc In addition, the hot flow stress behavior of martensitic steels(AISI 410 [26], 2Cr11Mo1VNbN [27], FB2 rotor steel [28], etc) were described by the two-stage physical-based model in the literatures, and the characteristics of the flow stress were captured. Although the physical-based model has been used in several kinds of alloys, researches are still lack in analying the hot deformation behavior of 05Cr17Ni4Cu4Nb martensitic stainless steel. In the presented work, the two-stage physical-based model is adopted to establish the stress-strain relations of this steel, considering the experimental flow stress curves of this steel exhibit the typical DRX feature. The DRX fraction is implied in the presented twostage model, which is related to the peak strain e p and critical strain e c and determined from the data of saturated stress s sat and steady state stress s ss obtained from the experimental flow stress curves. Considering the variables in the equations represent corresponding physical meanings, the presented two-stage model gives a more reasonable interpretation to the constitutive properties of 05Cr17Ni4Cu4Nb steel including the dependences of the deformation temperature, strain rate and strain history.
The optimization of the plastic deformation conditions of steels is gaining increasing attention in the research and development of hot forming technologies [29]. Based on the dynamic materials model(DMM), the processing map technology has been established as an effective approach to investigate the deformation  mechanisms of materials and find the optimal processing parameters. It has been applied in various kinds of materials including steels [30,31], alloys of aluminum, magnesium, etc Xu et al [32] determined that the optimum condition for 25Cr3Mo3NiNb steel was at 1077-1177°C and 0.001-0.03 s −1 . Cai et al [33] suggested 1120°C-1160°C and 0.03-0.1 s −1 as the optimum process parameters by analyzing the processing maps. As one kind of the most widely used materials, stainless steels are studied by many researchers, and the hot deformation mechanism and optimum processing parameters are revealed according to the processing maps [34][35][36][37][38]. Ren et al [39] researched the hot deformation characteristics of AISI420 stainless steel, and specified that the hot deformation should be carried out at the temperature conditions of 1007-1087°C and the strain  rate of 0.01-0.05 s −1 when the strain is no less than 0.5. During the hot forming process, deformation defects caused by the instable plastic flow of the materials can be reduced and avoided on the basis of the investigation of the processing maps.
In this paper, the effects of thermo-mechanical parameters on the hot deformation behavior of 05Cr17Ni4Cu4Nb martensitic stainless steel are investigated. Based on the experimental data, a two-stage flow stress constitutive equations have been established. By using the processing map approach associated with the microstructure investigation, the appropriate processing conditions for this material are determined.

Experiment material and methods
The uniaxial isothermal compression tests of 05Cr17Ni4Cu4Nb steel were performed by using the Gleeble1500D thermo-mechanical simulator. j8 mm × 12 mm cylindrical specimens for the tests were manufactured. The experiments procedures are illustrated in figure 1(a). The compression tests were carried out under various temperatures(950°C, 1050°C, 1150°C, 1250°C) and strain rates(0.01 s −1 , 0.1 s −1 , 1 s −1 , 5 s −1 ) conditions. During the hot deformation test, the slices of graphite were used between the specimen and anvil to reduce the friction. The deformed specimens were immediately quenched in water after the hot compressive deformation with a true stain of 0.8. Then, the specimens were split and the axial sections were polished and etched in boiled aqueous solution of sulphuric acid with potassium permanganate to capture the micrographs (sulphuric acid 10% by volume 100 g+potassium permanganate 1 g). The optical and SEM observations are carried out. SEM study was carried out on the Shimadzu SS550-SEM scanning electron microscope with the operation voltage 15 kV. Figure 1

Flow features and hot deformation mechanism
In the compressive tests, the friction and heat of deformation may influence the flow stress data. According the measurements of different specimens, the values of the barreling coefficient = B hR h R M 2 0 0 2 reported by Roebuck [40] are calculated. h 0 and h are the heights of the specimen before and after deformation, respectively. R 0 and R M are the initial radius before deformation and the maximum radius after deformation of the specimen. In this study, the values of B for different specimens are no more than 1.1, thus, the experimental flow stress curves are valid and the effect of friction on the flow stress is ignored. Besides, the heat deformation effect was removed by using the interpolation method according to the fitted relationship between s and T 1 . / Figure 2 shows the true stress-true strain curves of 05Cr17Ni4Cu4Nb steel at the temperature range of 950°C−1250°C and the strain rate range of 0.01 s −1 -5s −1 . The flow stress curves exhibit the combined effects of the strain, strain rate and temperature. In the initial stage of deformation, the flow stress rises rapidly with the increasing of strain due to the work hardening(WH) caused by dislocation generation and multiplication [41,42]. With the increasing of strain, the effect of WH is offset partially by dynamic softening, including dynamic recovery(DRV) and dynamic recrystallization(DRX). After reaching the peak value, the flow stress gradually decreases to a relatively steady state due to the equilibrium achieved between WH and dynamic softening. Figure

Establishment of Constitutive relations
According to presented researches, the evolvement of dislocation density r can be represented as [23][24][25]28,43]: where k 1 and k 2 are the coefficients related to working hardening and dynamic recovery. When the strain e is zero, r equals to the initial dislocation density r . 0 According to equation (1), the dislocation density r can be expressed as follows: In the above description, the framework of the flow stress model for 05Cr17Ni4Cu4Nb steel has been established. Generally, the relations among strain rate e  (s −1 ), temperature T (K), activation energy Q(J/mol) and Zener-Holloman parameter can be described in different ways as: MPa −1 . According to equation (7), the values of n and A ln is associated with the mean slope and intercept of e In against as ln sinh ( ( )) in figure 4(c). Thus, the mean values of n and A are determined as 5.251 and 7.21×10 14 s −1 . Then, the partial differentiation of equation (12) is: where Q is related to the slope of the plot for as ln sinh ( ( )) against T 1 ( figure 4(d)). Based on the calculated data, Q is determined as 426822.1 J/mol. Generally, the activation energy of hot deformation is considered to be influenced by various factors such as element composition, initial microstructure, experimental conditions, etc It is found that the higher deformation activation energy often appears in the hot deformation of alloys with higher yield strength [49]. In literatures, the values of Q are determined as 376.5 kJ/mol [50] and 892.35 kJ/mol for 34CrNiMo6 steel and 3Cr20Ni10W2 steel [49], respectively. With the rising of the alloying elements fraction, it indicates that the deformation activation energy tends to enlarge obviously. The alloying elements enhance the strength of steel by increasing the obstruction of the dislocation moment and grain boundary migration, and thus lead to the increment of the hot deformation activation energy.
In hot plastic deformation, the working hardening rate is defined as: figure 5 shows the q s curves under different deformation conditions. When q = 0, s are corresponding to s p and s , ss and the value of strain which relates to s p in the flow stress curves is the peak strain e . p According to the test data, the relationship between e ln p and Z ln is illustrated in figure 6(a), and the mathematical model of e p can be determined as:  Thus, s c can also be determined. Then, according to the experimental experiment flow stress data, e , c which is corresponding to s , c can be obtained. In figure 6(b), the relationship between e c and Z is established approximately as: The initial stress s 0 can be determined directly from the experimental data, and the relationship between s 0 and Z is shown in figure 6(e). The following expression can be specified:  According to the above analysis, the parameters in the proposed two-stage constitutive equation of 05Cr17Ni4Cu4Nb steel have been determined on the basis of the hot compression test data. According to this model, the flow stress at high temperatures can be calculated and predicted. Figure 8 shows the comparison between the predicted results of this model and the experimental data of hot compression tests. The predicted flow stress values are in good accordance with the experimental data. To further verify the applicability of the proposed constitutive model, the correlation coefficient R and the average absolute relative error AARE are adopted, which are expressed as equations (23) and ( where E i and P i are the experimental data and the predicted values, respectively. N is the amount of data used in the investigation. According to equations (23) and (24), the values of R and AARE for the presented constitutive equation are 0.994 and 5.29%, respectively, which indicate that the proposed model can give a high prediction accuracy for the flow stress of this steel. The steels generally obey the Von-Mises yield criterion and can be treated as the isotropic materials during the plastic deformation at high temperature, therefore, the proposed two-stage constitutive model is available to apply in the numerical simulations of different kinds of hot forging processes and provide credible predictions of the flow behavior of 05Cr17Ni4Cu4Nb steel.

Processing maps in hot deformation
The processing map is a widely used approach to optimize the processing parameters during hot forging. The processing map method considers the workpiece as a power dissipater during the plastic deformation. The total dissipated power P absorbed by the workpiece can be divided into two parts, the content G for temperature rise and the co-content J for microstructure dissipation. The total dissipated power can be described as: The ideal power dissipation J max occurs when m equals 1.0 in equation (26). The parameter h, which represents the power dissipation efficiency, is calculated by: The instability parameter x can be expressed by using the following relation [28]: The negative value of instability parameter x indicates the occurrence of unstable plastic deformation. In this paper, h and x are calculated and demonstrated as the processing maps for 05Cr17Ni4Cu4Nb steel in hot forming process, as shown in figure 9. In figures 9(a)-(d), the contour line represents the value of h and the shaded region represents the instability conditions in which x < 0. The three-dimensional isosurfaces of processing maps are demonstrated in figures 9(e) and (f).
According to the contour lines in figures 9(a)-(d), the distributions of h are changed with the increasing of strain. When strain reaches 0.4, the peak value of h occurs at two regions. One is at (1050°C, 0.01 s −1 ), while the other is at (1200°C, 5 s −1 ). The high value of h indicates that more energy has been dissipated for microstructure evolvement (such as DRX) and induces relatively fine grains. In the meanwhile, considering the occurrence possibility of grain coarsening at high temperature and large resistance to plastic flow at large strain rate, the appropriate parameters of hot forming can be selected from the region near the first peak value condition (1050°C, 0.01 s −1 ) of the h when the strain is more than 0.4.
According to shade region shown in figures 9(a)-(d), it indicates that the change of strain also influences the distribution of instability parameter x. With the increasing of strain, the region of x < 0, in which the instable flow may occur, firstly shrinks obviously and then fluctuates slightly. The instability area is relatively small at the strain of 0.4( figure 9(b)). When e > 0.4, the area of instability region is demonstrated at the temperature 950-1060°C and the strain rate 0.22-5 s −1 .

Microstructure evolution in hot deformation
Microstructures of 05Cr17Ni4Cu4Nb specimens compressed to the true strain 0.8 were observed by optical microscope and SEM. The evolution of microstructures under different deformations is aggregated in figure 10. The microstructure is significantly influence by deformation temperature and strain rate, and the grain morphology shows the typical DRX occurs. Figure 11(a) shows the grain diameters of the steel after the deformation at different conditions, which demonstrates that the average grain size enlarges with the rising of T and the decreasing of e.  Figure 11(b), (c) and (d) demonstrated the microstructures of 05Cr17Ni4Cu4Nb steel under the deformation conditions of (950°C, 5 s −1 ), (1050°C, 5 s −1 ), and (1050°C, 1 s −1 ). It shows that intergranular cracks and voids appear and may lead to the instable deformation, which verify the rationality of the predicted instability region in figure 9(d). Figure 11(e) demonstrated the microstructure of this steel under