S. YANG et al.: CORRECTION OF FLOW CURVES AND CONSTITUTIVE MODELING OF AN AS-CAST N06625 ... 515–524 CORRECTION OF FLOW CURVES AND CONSTITUTIVE MODELING OF AN AS-CAST N06625 NICKEL-BASED ALLOY KOREKCIJA KRIVULJ TE^ENJA IN KONSTITUTIVNO MODELIRANJE LITE NIKLJEVE ZLITINE N06625 Shuo Yang 1 , Yugui Li 1,2* , Yaohui Song 1 , Bin Wang 1 , Yuewu Zheng 1 , Xiaobiao Liu 1 , Yao Li 1 1 School of Materials Science and Engineering, Taiyuan University of Science and Technology, Taiyuan, China 2 School of Mechanical Engineering, Taiyuan University of Science and Technology, Taiyuan, China Prejem rokopisa – received: 2022-05-11; sprejem za objavo – accepted for publication: 2022-08-01 doi:10.17222/mit.2022.496 On a Gleeble-3800 simulator, isothermal hot compression tests of an as-cast N06625 nickel-based alloy were conducted in a wide range of temperatures (950–1200 °C) and strain rates (0.1–50 s –1 ). Considering the influence of friction and adiabatic heat- ing in the test, the flow curves were corrected. A modified Johnson-Cook, strain-compensated Arrhenius-type and modified Hensel-Spittel constitutive models were established based on the corrected flow curves. In a hot compression test, the flow stress is mainly affected by the coupling of the temperature, strain rate and strain, so the consideration of the coupling effect greatly affects the accuracy of the constitutive model. Moreover, different materials have different sensitivities to the three con- ditions. Therefore, the improved model in this paper can improve its universality and accuracy after adding relevant parts to it. The models were compared using the correlation coefficient (R) and average absolute relative error (AARE) to determine their accuracy in predicting the deformation behavior of the above alloy. According to our findings, the strain-compensated Arrhenius-type model provided the greatest forecast accuracy over the whole temperature and strain rate range. Under a certain temperature and strain rate, the modified Johnson-Cook model provided the best prediction accuracy. Keywords: alloy N06625, modification, stress-strain curve, constitutive model Avtorji so izvajali izotermne tla~ne preizkuse na litini Ni vrste N06625 s simulatorjem Gleeble-3800 v temperaturnem obmo~ju med 950 °C in 1200 °C in pri hitrostih deformacije med 0,1 s –1 in 50 s –1 . Ob upo{tevanju vpliva trenja in adiabatnega segrevanja med preizkusi so izvedli korekcije krivulj te~enja. Na osnovi korigiranih krivulj te~enja so dolo~ili modele: model po John- son-Cooku, deformacijsko kompenziran model Arrheniusovega tipa in modificiran Hensel-Spittel konstitutivni model. Med vro~imi tla~nimi preizkusi je bila predvsem napetost te~enja odvisna od kombinacije temperature, hitrosti deformacije in deformacije, kar je mo~no vplivalo na to~nost konstitutivnega modela. Zavedati se moramo, da so razli~ni materiali razli~no ob~utljivi na vsa tri stanja. Zato lahko izbolj{an model, ki je predstavljen, postane univerzalen pristop in tako tudi bolj natan~en model za druge materiale, seveda z dodatkom ustreznih parametrov. Izdelane modele so avtorji medsebojno primerjali z uporabo korelacijskega koeficienta (R) in povpre~ne absolutne relativne napake (AARE; angl.: average absolute relative error). S tem so dolo~ili njihovo natan~nost napovedi z realno deformacijo izbrane zlitine. Na osnovi raziskav ugotavljajo, da deformacijsko kompenziran model Arrheniusovega tipa zagotavlja najbolj{o natan~nost v celotnem temperaturnem obmo~ju in pri vseh izbranih hitrostih deformacije. Pri dolo~eni temperaturi in hitrosti deformacije, izbiri ene same krivulje te~enja, pa John- son-Cookov model zgotavlja najbolj{o natan~nost napovedi. Klju~ne besede: zlitina N06625, modifikacija, krivulja napetost-deformacija, konstitutivni model 1 INTRODUCTION The N06625 nickel-based alloy is a Ni-Cr solid solu- tion strengthened nickel-based alloy with Mo and Nb as the major strengthening elements. Although it is a solid solution strengthened alloy, depending upon the Nb and C percentages, there could be a development of sec- ond-phase particles such as and carbides at particular aging conditions. 1–3 The alloy is extensively used in aero- nautic, aerospace, chemical, nuclear, petrochemical and marine applications due to its good mechanical proper- ties, and resistance to high-temperature corrosion and damage after prolonged exposure to aggressive environ- ments. 3–5 The flow behavior of metals and alloys during ther- mal deformation is a complex and comprehensive inter- action of many factors. During hot deformation, the strain rate, strain and forming temperature affect the microstructure evolution, while the mechanical proper- ties are affected by the microstructure evolution. 6,7 The study of the high-temperature flow behavior of alloys is of great help to designers in formulating a reasonable hot-working process. At present, finite-element simulation software has oc- cupied an irreplaceable position in the actual production process, and the accuracy of simulation results also greatly affects the quality of production. A constitutive model is the core of a finite-element simulation. Choosing the appropriate constitutive model can signifi- cantly improve the prediction accuracy of the simulation. Many scholars have developed various constitutive mod- Materiali in tehnologije / Materials and technology 56 (2022) 5, 515–524 515 UDK 669.15’24’25-196:66.011 ISSN 1580-2949 Original scientific article/Izvirni znanstveni ~lanek MTAEC9, 56(5)515(2022) *Corresponding author's e-mail: liyugui@tyust.edu.cn (Yugui Li) els and their improved models. 6 Among these models, the Johnson-Cook model and the Arrhenius model are widely used for metals and alloys. Badrish 8 carried out a tensile test with Inconel 625 and established the JC-ZA model. The applicability of the model to Inconel 625 was proved by using a variety of statistical parameters. Godasu 9 found that the Arrhenius model can accurately reflect the high-temperature deformation mechanical properties of superalloy 625. Gupta 10 established the Johnson-Cook, a modified Zerilli-Armstrong, a modified Arrhenius-type, and the ANN models of austenitic stain- less steel 316. The correlation coefficient of the ANN model is the highest (0.9930). But due to its difficulty in integrating with the finite-element modeling software and the fact that it is a black-box model, constitutive equations have generally been given a higher preference over ANN. Brown 11 established seven types of constitu- tive models. Taken together, the modified Johnson-Cook, Zener-Hollomon, Hensel-Spittel and modified Hensel- Spittel models provide the best fit with the experimental results. Cai 12 developed a modified Johnson-Cook and strain-compensated Arrhenius-type model of different phase regimes. The results indicated that the accuracy of the modified Johnson-Cook model is higher than that of the strain-compensated Arrhenius-type model at the + phase. Meanwhile, the time required for evaluating the material constants of the modified Johnson-Cook model is much shorter than that of the strain-compensated Arrhenius-type model. Although these studies are avail- able, limited literature shows a comparison of different constitutive models related to the N06625 nickel-base al- loy. The data of a constitutive model are mostly based on isothermal compression tests. In a thermal compression test, there is usually some error in the final stress-strain curve, which is limited by the test conditions. The error mainly comes from two aspects: one is the influence of friction on the flow stress, and the other is the influence of deformation temperature rise on the flow stress. 13 Therefore, many scholars propose to modify the flow stress according to these two factors. The purpose of this study was to develop a constitu- tive model suitable for the high-temperature flow behav- ior of the as-cast N06625 nickel-based alloy. Therefore, isothermal compression tests were carried out in a wide range of temperatures and strain rates. The flow curve was modified to consider the effects of friction and tem- perature rise. Then the coefficients of various constitu- tive models were determined using the modified data. Finally, the correlation coefficient and average absolute relative error were used to examine the validity of the constitutive equations over the entire range of tempera- tures and strain rates, and the prediction capabilities of the three constitutive models were compared. 2 EXPERIMENTAL PART The material used in the test is an N06625 nickel- based alloy ingot. Its chemical composition is shown in Table 1. A hot-compression test sample is prepared by a wire-cut electrical discharge machine 15 mm away from the edge of the ingot to ensure the consistency of the grain of the hot-compression simulation sample. The samples are cylinders with a diameter of 10 mm and a length of 15 mm. Constant-temperature and constant-strain-rate ther- mal-compression simulation tests are carried out on a Gleeble-3800 simulation machine. During the tests, a graphite lubricant is applied on both ends of a sample and tantalum sheets are used to help reduce friction and avoid adhesion between a sample and the equipment. The samples are heated to the preset temperature at a heating rate of 10 °C/s and held at it for 3 min to obtain uniform and equiaxed grains. The samples are then cooled down to (950, 1000, 1050, 1100, 1150 and 1200) °C at a cooling rate of 5 °C/s. The samples are compressed by 60 % at strain rates of (0.1, 1, 10 and 50) s –1 , and cooled rapidly after the deformation. 3 RESULTS AND DISCUSSION 3.1 Correction of the stress-strain curve The flow stress requires friction and temperature cor- rections during deformation. The stress after friction cor- rection is: 14,15 [] f = −− PRh Rh Rh (/ ) exp( / ) / 2 22 21 2 (1) S. YANG et al.: CORRECTION OF FLOW CURVES AND CONSTITUTIVE MODELING OF AN AS-CAST N06625 ... 516 Materiali in tehnologije / Materials and technology 56 (2022) 5, 515–524 Table 1: Chemical composition of the N06625 nickel-based alloy (w/%) Ni Cr Mo Nb Fe Co Si Mn Ti Al C Cu S P 58 20–30 8–10 3.15–4.15 5 1 0.5 0.5 0.4 0.4 0.1 0.07 0.015 0.015 Figure 1: Hot-compression test process = − R H b b 1 1 4 3 2 33 (2) b R R h h =⋅ 4 1 1 1 Δ Δ (3) RRHH 1001 = / (4) Here, f is the friction-corrected flow stress, P is the uncorrected flow stress, R and h are the momentary ra- dius and height of a sample, respectively, μ is the friction coefficient, b is the barrel parameter, R 1 is the average ra- dius of the cylinder after deformation, R is the differ- ence between the maximum and top radius of the cylin- der, H 0 is the initial height of the cylinder, H 1 is the final height, and H 1 is the difference between the initial and final heights of a specimen. The adiabatic-temperature rise during deformation can be calculated using the following Equation (5): 16 ΔT C d = ∫ f 0 (5) S. YANG et al.: CORRECTION OF FLOW CURVES AND CONSTITUTIVE MODELING OF AN AS-CAST N06625 ... Materiali in tehnologije / Materials and technology 56 (2022) 5, 515–524 517 Figure 2: Temperature changes of the samples under different deformation conditions: a) = 0.3–0.9, b) = 0.9 Figure 3: Comparison between the corrected and uncorrected true stress-strain curves: a) 0.1 s –1 ,b)1s –1 ,c)10s –1 ,d)50s –1 where T is the temperature change, is the material density, C p is the heat capacity and is the adiabatic factor. 17 = ≤ 0 1s 1s 1s 0316 .l g < 1s ≥ ⎧ ⎨ ⎪ ⎩ ⎪ (6) The temperature change increases with an increasing strain, as shown in Figure 2a, due to the accumulation of adiabatic-temperature rise and friction. The influence of the heat generated by deformation on the temperature change gradually decreases as deformation temperature increases, as shown in Figure 2b. This is because the lower the deformation temperature, the higher is the rhe- ological stress value. More external energy is required to achieve the same deformation degree, which means that more energy is dissipated in the form of heat, and the temperature-rise phenomenon becomes more obvious. Figure 3 shows the original and corrected stress-strain curves under different deformation condi- tions. The shape of the curve is related to the temperature and strain rate, and the maximum flow stress decreases as the temperature and strain rate increase. At the same time, it should be noted that at lower strain rates and higher temperatures, the difference be- tween the corrected and uncorrected flow stresses is not significant. However, this difference is no longer negligi- ble at higher strain rates and lower temperatures. 3.2 Constitutive model 3.2.1 Modified Johnson-Cook constitutive model Lin 18 proposed a modified Johnson-Cook model con- sidering the interaction of the strain rate and tempera- ture: [] =++ + ⎛ ⎝ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⋅ ⋅ AAAA C m 0111 1l n exp 12 0 + ⎛ ⎝ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ − ⎧ ⎨ ⎪ ⎩ ⎪ ⎫ ⎬ ⎪ ⎭ ⎪ ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ mT T ln () ⎥ (7) where A 0 , A 1 , A 2 , A 3 , m 1 and m 2 are the material con- stants. 950 °C and 1 s –1 are selected as the reference temper- ature and strain rate. Under the reference temperature and strain rate, Equation (7) can be simplified to Equa- tion (8). The values of A 0 , A 1 , A 2 and A 3 can be obtained by fitting and with third-order polynomials. =++ AAAA 0111 (8) At the reference temperature, Equation (7) can be simplified to Equation (9). The average slope obtained S. YANG et al.: CORRECTION OF FLOW CURVES AND CONSTITUTIVE MODELING OF AN AS-CAST N06625 ... 518 Materiali in tehnologije / Materials and technology 56 (2022) 5, 515–524 Figure 4: Comparison of the predicted and tested values of the modified Johnson-Cook constitutive model: a) 0.1 s –1 ,b)1s –1 ,c)10s –1 ,d)50s –1 with the linear fitting of ln# and /(A 0 +A 1 +A 2 2 +A 3 3 )is the value of C. [] =++ + ⎛ ⎝ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ AAAA C 0111 1l n (9) Equation (7) can be converted and natural logarithms taken on both sides. The slopes of ln{ /[(A 0 + A 1 + A 2 2 + A 3 3 )[1+C ln( )]]} and (T–T 0 ) are solved under different conditions, then the slopes and ln( ) are lin- early fitted to obtain the values of m 1 and m 2 . ln () l n AAAA C 0111 1 ++ + ⎛ ⎝ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎧ ⎨ ⎪ ⎪ ⎩ ⎪ ⎪ ⎫ ⎬ ⎪ ⎪ ⎭ ⎪ ⎪ = =+ ⎛ ⎝ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ − mm TT 12 0 ln () (10) So far, the parameters of the modified Johnson-Cook constitutive model have been solved. 3.2.2 Strain-compensated Arrhenius-type constitutive model The Arrhenius constitutive model is shown in Equa- tion (11). 19 So far, the Arrhenius model and improved model have been the most widely used. exp e = − ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ ≤ A Q RT A n 1 2 1 , [] xp( exp sinh( exp , − ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ ≥ Q RT A n for all − ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ ⎧ ⎨ ⎪ ⎪ ⎪ ⎩ ⎪ ⎪ ⎪ Q RT (11) Here, Q is the thermal activation energy and R is the standard molar gas constant. Some scholars use the Zener-Hollomon parameters to further characterize the model: 20 [] ZA e n Q RT = − sinh( ) (12) sinh( ) = ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ Z A n 1 (13) S. YANG et al.: CORRECTION OF FLOW CURVES AND CONSTITUTIVE MODELING OF AN AS-CAST N06625 ... Materiali in tehnologije / Materials and technology 56 (2022) 5, 515–524 519 Figure 5: Comparison of the predicted and tested values of the strain-compensated Arrhenius-type constitutive model: a) 0.1 s –1 ,b )1s –1 , c) 10 s –1 ,d)50s –1 = ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ + ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ + ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎧ ⎨ ⎪ ⎪ ⎩ ⎪ ⎪ ⎫ ⎬ ⎪ ⎪ 1 1 12 1 2 ln Z A Z A nn ⎭ ⎪ ⎪ (14) The following results can be obtained by taking natu- ral logarithms on both sides of Equation (11). ln ln =+− A Q RT 2 (16) [] ln ln ln sinh( =+ − An Q RT (17) Linear fitting is carried out on ln ln and ln , the average slopes are the values of n 1 and , respec- tively. The value of can be obtained from = /n 1 . We can substitute into Equation (17) and then per- form linear fitting on [] ln ln sinh( − . The average slope is n and the average intercept is ln A – Q/RT. [] ln sinh( ln ln =+− n Q nRT A n (18) The value of Q can be obtained from Equation (18). So far, the parameters have been obtained. The flow stress also transforms significantly with the change in the strains; thus, it is necessary to calculate the flow stress in a strain range of 0.05–0.9 at intervals of 0.05. Since the parameters do not correlate, it is necessary to fit their polynomials as a function of strain. Comparing the accu- mulative error values fitted by polynomials of different times of each parameter, it is found that the accumulative error of the polynomial function of 9 times is minor. So far, the parameters of the strain-compensated Arrhenius-type constitutive equation have been solved. 3.2.3 Modified Hensel-Spittle constitutive model The Hensel-Spittel model is a constitutive model with many parameters, proposed by Hensel and Spittel. 21 Spigarelli and El Mehtedi 22 think that this model is not inherently sufficient to describe the change in the strain rate sensitivity at a given temperature and strain. There- fore, some scholars put forward a modified Hensel-Spittel model with the Garofalo equation. 11 sinh( ) ( / =+ Ae e e mT m m m mT m 1234 56 1 (19) exp ,f o r − ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ = Q RT A n 1 1 (20) exp exp( ,f o r − ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ = Q RT A 2 (21) We take the natural logarithms on both sides of Equa- tion (20) and Equation (21). The values of n 1 and are S. YANG et al.: CORRECTION OF FLOW CURVES AND CONSTITUTIVE MODELING OF AN AS-CAST N06625 ... 520 Materiali in tehnologije / Materials and technology 56 (2022) 5, 515–524 Figure 6: Comparison of the predicted and tested values of the modified Hensel-Spittel model: 0.1 s –1 ,b)1s –1 ,c)10s –1 ,d)50s –1 obtained by linearly fitting ln ln and ln ,r e - spectively. The value of can be obtained from = /n 1 . ln ln ln =+− An Q RT 11 (22) ln ln =+− A Q RT 2 (23) The strain and strain rate are assumed to be constant. We take the natural logarithms on both sides of Equation (19) at the same time: [][ ] ln sinh( ) ln( ) =++ + cmm T 115 1 (24) The slope obtained with linear fitting is the value of m 1 + m 5 ln(1 + ). The m 1 + m 5 ln(1 + ) was fitted with y = p 1 + p 2 ln(1 + x) at a constant strain rate. The values of m 1 and m 5 correspond to the values of p 1 and p 2 ,re - spectively. The strain and temperature are assumed to be con- stant. We take the natural logarithms on both sides of Equation (19) at the same time: [] ln sinh( ) ln =+ cm 23 (25) The slope obtained by linearly fitting [] ln sinh( ) ln − is the value of m 3 . The strain rate and temperature are assumed to be constant. We take the natural logarithms on both sides of Equation (19) at the same time: [] ln sinh( ) ln ln( ) =+ ++ cm m/ mT m 32 4 6 1 (26) Equation (26) is fitted with y = c 3 + m 2 lnx + m 4 /x + m 5 T ln(1 + x) to obtain the values of m 2 , m 4 and m 6 . The currently calculated coefficient is substituted into Equa- tion (19) to obtain the value of A under different condi- tions. When drawing the A-strain curve, it is found that the value of A is less affected by the temperature and strain rate and more affected by strain. Therefore, the value of A is transformed into a cubic polynomial about the strain, and the value of A and the strain are fitted with a cubic polynomial. So far, all the parameters of the model are obtained. 3.2.4 Comparisons of constitutive models R and AARE are introduced to evaluate the predic- tion accuracy of the models. R EEPP EE PP ii i N ii i N i N = −− −− = = = ∑ ∑ ∑ () () ()() 1 22 1 1 (27) ARE N EP E ii i i N (%) ( % = − × = ∑ 1 100 1 (28) The average values of R of different models are shown in Table 2. Figure 7 shows the correlation be- tween the test stress and predicted stress of different models. Among the models with strong prediction corre- lation, the data points closer to the regression line are the most. Table 2: Comparison of the constitutive models using R Constitutive model The average value of R MJC 0.9533 SA 0.9771 MHS 0.9334 Although R can represent the linear correlation be- tween the experimental value and predicted value, it is not necessarily true that a higher value of R indicates a better result. Therefore, AARE is introduced to further determine the accuracy of the models. The average values of AARE of different models are shown in Figure 3. Contour maps were drawn using the AARE values of each model (Figure 8). The AARE val- ues of each model at different strain rates and deforma- tion temperatures are shown in Figure 8. The modified Johnson-Cook model’s value of R is 0.9533, and the AARE values range from 2.30 to 25.71. Lin 18 chose a quadratic polynomial in the formula for the modified Johnson-Cook constitutive model for high-strength alloy steel because the curve had the high- est fitting degree with the quadratic polynomial. The flow curve of the N06625 nickel-based alloy is more in line with the trend of the third-order polynomial under reference conditions. As a result, Equation (7) has a cu- bic polynomial. The modified Johnson-Cook constitutive S. YANG et al.: CORRECTION OF FLOW CURVES AND CONSTITUTIVE MODELING OF AN AS-CAST N06625 ... Materiali in tehnologije / Materials and technology 56 (2022) 5, 515–524 521 Figure 7: Comparison of R for: a) modified Johnson-Cook, b) strain-compensated Arrhenius-type, c) modified Hensel-Spittel model model has a good prediction value only at the reference temperature and strain rate, as shown in Figure 4. The value of R of the strain-compensated Arrhenius-type model is 0.9771, and the AARE values range from 2.92 to 34.18. Table 2 shows that at a strain rate of 0.1 s –1 , the value of AARE at 950 °C is the lowest (2.92) and the highest at 1200 °C (34.18). The AARE values range from 4.14 to 19.35 at the other strain rates, and the average value of AARE is greatly reduced. The predicted values of the model under the 0.1 s –1 strain rate deviate from the test values, as shown in Figure 5. The modified Hensel-Spittel model’s value of R is 0.9334, and the AARE values range from 4.54 to 32.28. Taking the peak stress as the dividing line, as shown in Figure 6, the model prediction accuracy of the part after the peak stress of the rheological curve is high, but the prediction result of the part before the peak stress has a large error. It demonstrates that this model does not con- sider the strain adequately. However, the degree of curve position fitting under various temperatures is satisfac- tory. Overall, the strain-compensated Arrhenius-type model accurately predicts the flow behavior of the N06625 nickel-based alloy at high temperatures over a wide temperature and strain-rate range. However, due to a large number of parameters and polynomial equations, this model takes a long time for the calculation. 12 It can be seen from Figure 6 that the AARE increases sharply at 0.1 s –1 and 1200 °C. Before the peak stress, the modi- fied Hensel-Spittel model is unable to accurately predict the part. As a result, this model can be chosen only when the portion of the flow curve after the peak stress is con- sidered. Modeling under fewer deformation conditions is more accurate with the modified Johnson-Cook model. 11 The model’s overall accuracy improves as deformation conditions decrease. A group calculation according to the deformation conditions and setting multiple reference conditions can also reduce the error. In addition, except for ANN, there is almost no model that can accurately simulate the whole flow curve, but the ANN model cannot be combined with the fi- nite-element software at present. This is because the flow stress is affected by the coupling of the temperature, strain rate and strain. Due to the nonlinear relationship between the flow stress and process parameters, it is dif- ficult to establish a desirable constitutive equation, espe- cially for describing the thermal deformation behavior of materials in a wide range of temperatures and strain rates. In recent years, some scholars have used the sub- section method to establish an improved constitutive model. Mei 23 takes the peak stress as the dividing line, from zero to the peak stress as the first stage, fitting the S. YANG et al.: CORRECTION OF FLOW CURVES AND CONSTITUTIVE MODELING OF AN AS-CAST N06625 ... 522 Materiali in tehnologije / Materials and technology 56 (2022) 5, 515–524 Table 3: Comparison of constitutive models using AARE Model Strain rate (s –1 ) Temperature (°C) Average 950 1000 1050 1100 1150 1200 MJC 0.1 11.00 19.38 15.25 17.39 19.59 22.24 13.76 1 2.30 8.45 11.70 12.46 16.19 23.19 10 10.68 7.54 13.56 11.38 13.29 18.66 50 11.09 9.38 5.74 8.87 15.24 25.71 SA 0.1 2.92 9.41 6.91 13.24 21.07 34.18 9.78 1 5.93 4.70 5.02 6.88 9.93 19.35 10 11.16 4.52 5.08 5.37 5.95 12.07 50 14.21 9.19 6.62 4.14 6.92 9.98 MHS 0.1 7.80 9.64 9.18 8.46 9.52 10.02 11.25 1 12.16 7.05 4.54 6.13 7.30 8.86 10 26.46 14.94 6.98 5.17 5.26 6.91 50 32.28 25.37 18.79 8.77 6.56 11.04 Figure 8: Comparison of the AARE for: a) modified Johnson-Cook, b) strain-compensated Arrhenius-type, c) modified Hensel-Spittel model linear equation from the peak stress to the final stress as the second stage, and fitting the nonlinear equation. Liu 24 takes the work-hardening dynamic recovery stage as the first stage and the dynamic-recrystallization stage as the second stage, predicting the flow stress by combining the dynamic-recrystallization critical model and the dynamic model. Although the prediction accuracy of the seg- mented model has been improved, the complexity and time required for the calculation have also increased sig- nificantly. Therefore, it needs to be improved in the fu- ture. 4 CONCLUSIONS A comparative study was carried out on the ability of the modified Johnson-Cook, strain-compensated Arrhenius-type and modified Hensel-Spittel constitutive models to describe the elevated-temperature flow behav- ior of a N06625 nickel-based alloy in a temperature range of 950–1200 °C at strain rates of 0.1–50 s –1 . Based on this study, the following conclusions can be drawn: The true stress-strain curve was drawn based on the data from the thermal-compression simulation test. The original curve was modified to eliminate the influence of friction and adiabatic temperature rise. The errors before and after the flow stress correction cannot be neglected, especially at large strain rates. A modified Johnson-Cook, strain-compensated Arrhenius-type and modified Hensel-Spittel constitutive models were obtained. The strain-compensated Arrhenius-type constitutive model can more accurately predict the high-temperature flow behavior of an entire machining area. The modified Johnson-Cook model has the highest accuracy when only one curve is predicted. The modified Hensel-Spittel model is more suitable for predicting the part after the peak stress of the curve. The value of R of the modified Johnson-Cook model is 0.9533, and the AARE values are between 2.30 and 25.71. The value of R of the strain-compensated Arrhenius-type model is 0.9771, and the AARE values are between 2.92 and 34.18. The value of R of the modi- fied Hensel-Spittel model is 0.9334, and the AARE val- ues are between 4.54 and 32.28. The strain-compensated Arrhenius-type constitutive model requires more calculations and time to obtain re- sults than the other two models. Acknowledgment This project was supported by the Key Core Technol- ogy and Common Technology Research and Develop- ment Project of the Shanxi Province (20201102017), Ex- cellent Innovation Project for Graduate Students from the Shanxi Province (2021Y666), and Key Special Pro- ject of "Science and Technology for Economy 2020" (2020YFF0405969). 5 REFERENCES 1 S. Floreen, G. E. Fruchs, W. J. Yang, The Metallurgy of Alloy 625, Superalloys 718, 625, 706 and Various Derivatives, (1994), 13–37, doi:10.7449/1994/Superalloys_1994_13_37 2 M. Sundararaman, P. Mukhopadhyay, S. 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