A Characterization for the Deformation Behavior of As-Cast P 91 Alloy Steel and Utilization in Hot Extrusion Process

The constitutive models of as-cast P91 alloy steel were investigated by unidirectional thermal compression tests, which were conducted over a wide range of temperature and strain rate on Gleeble-3500 thermal simulation machine. The constitutive models were coupled with finite element method to simulate the hot extrusion process. Furthermore, the industrial hot extrusion experiment on actual pipe was conducted on 500MN vertical extrusion machine and the load-displacement data during the extrusion were recorded. Comparisons between the numerical simulation and industrial experimental results were carried out and confirmed that the developed constitutive models can be used in the simulation and industrial research of compact hot extrusion process.


Introduction
Heavy caliber thick-wall pipes are widely used in the fields such as large supercritical thermal power, nuclear power, and petrochemical engineering.The current production process of heavy caliber thick-wall pipe is the punching and hot extrusion (PHE) process shown in Figure 1.This PHE process has several defects including long production cycle, serious waste in material, and investments of large scale equipment.In order to solve these disadvantages, a new compact hot extrusion (CHE) process shown in Figure 2 was proposed by Li et al. [1].During the CHE process, the as-cast billet is demolding under hot extrusion temperature and directly extruded on the extrusion machine.The short forming process realizes the integration on shape forming and performance improving.Thus, the hot deformation behaviors of as-cast billet during CHE process are required to be further investigated.
Investigations and great progress on the behaviors of asforged P91 alloy steel have been done based on the experimental research and simulation results [7][8][9][10][11].However, the reports of hot deformation performance on as-cast P91 alloy steel have not been seen yet.
The object of this study is to establish an innovative constitutive model which can accurately be applied to the simulation and experimental research on the CHE process.The hot deformation behaviors of as-cast P91 alloy steel were investigated by hot compression tests.In addition, the constitutive models of as-cast P91 alloy steel were coupled with the finite element analysis of hot extrusion on heavy caliber thick-wall pipe.The comparisons between the simulation and the industrial hot extrusion experiment were then discussed with the load-displacement data.on a Gleeble-3500 thermomechanical simulator.Cylindrical specimens were machined with 8 mm in diameter and 12 mm in height.

Experimental
The specimens were deformed at the reduction of 60% in height.Different deformation temperatures (from 1173 to 1473 K) and different strain rates (from 0.01 to 5 s −1 ) were used in the hot compression tests.The specimens were firstly preheated to 1473 K at a rate of 5 K/s and held for 5 min at isothermal conditions for fully austenitizing.Then the specimens were cooling to the deformation temperature at the rate of 5 K/s and held for 30 s for the homogenization of temperature and compressed at the selected conditions.Finally, the compressed specimens were rapidly quenched in water.

Experimental Results and Analysis.
The true strain-stress curves of as-cast P91 alloy steel under different deformation conditions are depicted in Figure 3.It can be obtained that the flow stress curves are sensitively dependent on the strain rate and the temperature.It increases with the increase of strain rate and decreases with the increase of temperature.The dynamic recrystallization (DRX) occurs under higher temperature and lower strain rate.
During the hot deformation process, the flow stress is sensitive to the work hardening and the dynamic softening [10].At the beginning of the deformation, the flow stress increases with the increasing strain.The material is under an unstable status due to the increasing dislocation density.The stored energy turns into a driving force for the dislocation migration.When the strain reaches to the critical strain, the dynamic recovery occurs and the flow stress increases owing to the confliction between softening and hardening.Moreover, with the strain continuing to increase, the work hardening and softening reach to a certain balance and the flow stress reaches to a plateau as well.Thus, the critical strain and work hardening are the significant parameters for the flow stress and should be considered in the constitutive models.

Constitutive Model for As-Cast P91 Alloy Steel
The experimental true stress-strain curves were fitted with the logarithmic from Hollomon equation ( =  ⋅   ), where  is the flow stress,  is the strength coefficient,  is the plastic strain, and  is the work hardening exponent.The work hardening rate  (/) in this investigation can be expressed from the experimental researches as [12,13] where  is a parameter which effected by the deformation temperature and strain rate and  is a constant.The relationships between work hardening rate and flow stress can be shown in Figure 4 (the deformation temperature was 1473 K),   is the critical stress,   is the peak stress, and   is the saturation stress.
It can be obtained in Figure 4, by extending the imaginary line with -axis, and the intersection (where  = 0,  =   ) was gotten.From (1), constant  can be found as  = −  .In Figure 4, the flow stress has a linear relationship between work hardening rate and flow stress before critical stress, and  is the slope of the line, so the point of critical strain and stress is the inflection point of the curves.From (1), the flow stress can be expressed as With the condition of  =   and  =   , the constant  can be described as Substituting ( 3) into (2) the constitutive model of flow stress can be obtained as The parameters in this model will be established and determined in the following parts.The values of   ,   , and   under different deformation conditions can be obtained and illustrated in Table 1.

Calculations of the Activation Energy and 𝑍 Parameter.
Arrhenius equations were represented [14] which are used in describing the relationship by hyperbolic sine, and it can be expressed with the Zener-Hollomom () parameter as follows:  where  is the activation energy (kJ/mol),  is the constant (8.314J⋅mol −1 ⋅K −1 ),  is deformation temperature (K), and  and  are material constants.According to (5), the strain rate can be expressed as where () is the function of the stress which can be described in different conditions [15]; the functional relations between flow stress and strain rate can be described as follows: where  and  1 are the materials dependent constants.Substituting the value of   into (5)∼( 7) and taking natural logarithm of (7), the relationships between ln ε and ln , ln ε and  are illustrated in Figures 5 and 6, respectively. 1 ( 1 = 8.78215) and  ( = 0.06965) are determined from the slopes of straight line in Figures 5 and 6, where , , and  1 are related by  = / 1 , and  can be obtained as 0.00793.
Taking natural logarithm of ( 6), activation energy  can be expressed as Similarly, taking natural logarithm on both sides of ( 5), the relationship between ln sinh() and ln ε can be obtained.Using the linear plot of ln sinh() versus ln ε (Figure 7), the value of  can be calculated ( = 6.2154).The value of  can be obtained by using the same method (in Figure 8, the linear fit between ln sinh() and 1000/); the average value of  is 328.1 kJ/mol.The determination of parameter  can be obtained from (9).The linear plot of ln[( −   )/(  −   )] versus ( −   ) can be seen in Figure 9, and  is determined as the slopes of the lines.As mentioned above, l is sensitive to strain rate and deformation temperature which is similar with parameter .So establish a relationship between  and  as

Calculation of
where , , and  are the material parameters.The values of  can be obtained in (6), and plotting  and  in Figure 10.

Calculations of 𝜀
and   .The critical stress   and critical strain   cannot be directly obtained from the experiment data.Kim et al. [16][17][18] established the relationships between   ,   , and  under hot deformation, and it can be expressed as The values of  1 ,  1 ,  2 , and  2 can be obtained by using linear regression as shown in Figure 11.The values of  1 ,  1 ,  2 , and  2 are 0.035, −1.03, 10.55, and −312.61,respectively.Hence, the critical strain and stress can be represented as   = 0.035 ln  − 1.03   = 10.55 ln  − 312.61. (13) 3.4.Calculation of   .Saturated stress   cannot be directly obtained from the strain-stress curves as critical stress.
It can be obtained according to Figure 4. From ( 5), the mathematical model of saturated stress is shown in (14): Taking the nature logarithm on both sides of (14) gives The relationship of   and  can be expressed as The value of constants  2 and  2 can be obtained using linear regression as presented in Figure 12.In summary, the model for   can be represented as follows   = 126.103⋅ sinh −1 (5.63 × 10 −16 ) 6.25 .( Figure 13 shows the comparisons of the experimental and predicted flow stress by the proposed constitutive model.It is undeniable that there have deviations between the experimental data and the calculations of the value fitting.So taking the correlation coefficient  to study the accuracy of the calculated data,  can be obtained by the following equation: where   exp is the experimental data,    is the predicted data,  exp is the experimental average data, and   is the average predicted data.The correlation coefficient  under different deformation conditions is illustrated in Table 2.The smaller the distinctness between  and 1, the lesser the error of the constitutive model.From Table 2, the result indicated that the proposed model can give an accurate and precise estimate of flow stress for as-cast P91 alloy steel.

Utilization in the Research on the Hot Extrusion of CHE Process
The hot extrusion is a process with the characteristics of high strain rate and large deformation degree, and the deformation conditions are more complicated than the hot compression tests.It is vital to study the flow stress during the hot extrusion process.Thus, the application of the as-cast P91 alloy steel constitutive model on the CHE hot extrusion process should be further studied and explored.The finite element (FE) method is the efficient and economical alternative method for studying the flow stress behavior during hot deformation [19][20][21].Hence, the constitutive model of the as-cast p91 alloy steel was encoded into the DEFORM platform by the FOR-TRAN language.The hot extrusion numerical simulation results were compared with the industrial experimental research.
4.1.The Thermal-Mechanical FE Model.According to the geometric symmetry of the billet and extrusion pipe, the 3D axisymmetric simplified FE models were established in Figure 14.The thermal-mechanical FE model combined with the constitutive model of as-cast P91 alloy steel was encoded into DEFORM platform.During the simulation, the ascast billet moves with the movement of terrace die while the die and needle keep static.The material of the billet was as-cast P91 steel, and the die was H-13 die steel.The required simulation conditions for the reliability evaluation of the thermal-mechanical coupled FE model are shown in Table 3.Heat convection coefficient between billet and extrusion tolls (W⋅m −2 ⋅K −1 ) 1100 Heat convection coefficient between billet and air (W⋅m −2 ⋅K −1 ) 20

Industrial Hot Extrusion Experiment on Heavy Caliber
Thick-Wall Pipe.The hot extrusion experiment was carried out by the 500 MN vertical hot extrusion equipment (Figure 15) in the factory.The hot extrusion process was investigated by using the same extrusion parameters (Table 3) as FE analysis.The billet was presprayed with glass lubricant.The 80 mm in height glass lubrication gasket was placed between the billet and the terrace die.During the extrusion process, the load-displacement data of the terrace die were measured.

The Comparisons between FE Analysis and Hot
Extrusion Experiment. Figure 16(a) shows the experimental billet; Figure 16(b) is the simulated initial temperature distribution of the billet.It can be seen that the billet temperature decreases due to the heat convection and heat emissivity.The temperature of the surfaces is lower than the inner.Figure 17 shows the extrusion production pipe of this experiment, and the dimensional accuracy of the pipe meets the requirements of the industrial.The distributions of temperature field, effective strain, and stress during the hot extrusion process are illustrated in Figure 18.It can be found that the heat is generated during the extrusion process, and the temperature increases in the large deformation area (the conical surface and the die orifice).The temperature increases due to the transfer of the plastic deformation heat and the dynamic friction heat.The effective strain and stress field during the hot extrusion process indicated that the extrusion force increased during the process.When the material flows out of the die orifice, the effective strain remains stable while the effective stress reaches to the peak value.Furthermore, it also can be seen in Figure 18(c) that the values of the effective stress are larger than zero; this illustrates that the stress state during the hot extrusion is the compressive stresses rather than the tensile stresses.It shows that the hot cracking will not appear on the extruded pipe which is verified by the industrial experiment shown in Figure 17.
The comparisons of the load-displacement curves between the measured and simulated results are illustrated in Figure 19.The results showed that the maximum load value of the experiment is 430 MN, while the simulation value is 410 MN.The difference was due to the simplification of the FE analysis; the actual industrial extrusion conditions are more complicated.Thus the error is in a reasonable deviation range.The computed results present a good estimate of the relationship between load and displacement during the hot extrusion.
The simulation and experimental results indicated that the proposed constitutive model of as-cast P91 alloy steel can give an accurate and precise estimate of the flow stress and it can be used to the analyze of the CHE process.In other words, the constitutive models established from the compressive tests can be applied to the industrial research and simulation.

Conclusions
The deformation characteristics of as-cast P91 alloy steel were investigated by means of the unidirectional hot compression experiments over a practical range of temperatures and strain rates.According to the results, the flow stress is sensitive to the work hardening and the critical strain.A parameter  was proposed to express the constitutive models.The thermal-mechanical finite element model was further developed by encoding the constitutive models into DEFORM platform.According to the actual thick-wall pipe, the hot extrusion process was simulated and experimented, respectively.Comparisons between the simulated and experimental results were carried out and confirmed that the developed constitutive models can be used in the industrial and simulation research of compact hot extrusion process.

Figure 3 :
Figure 3: The true stress-strain curves of different deformation temperatures at strain rate of 0.1 s −1 (a); different strain rates at deformation temperature of 1473 K (b).

Figure 4 :
Figure 4: The relationships between work hardening rate and flow stress.

Figure 17 :
Figure 17: The extruded pipe under the extrusion process (a); after the extrusion (b).

Table 1 :
The values of   ,   ,   , and   under different deformation conditions.

Table 2 :
The correlation coefficients under different deformation conditions.

Table 3 :
Hot extrusion parameters setting for the thermalmechanical coupled FE model.