UDK 519.61/.64:621.7.016.2 ISSN 1580-2949 Original scientific article/Izvirni znanstveni članek MTAEC9, 49(1)69(2015) MATHEMATICAL MODELLING AND PHYSICAL SIMULATION OF THE HOT PLASTIC DEFORMATION AND RECRYSTALLIZATION OF STEEL WITH MICRO-ADDITIVES MATEMATIČNO MODELIRANJE IN FIZIKALNA SIMULACIJA VROČE PLASTIČNE PREDELAVE IN REKRISTALIZACIJE JEKLA Z MIKRODODATKI Elžbieta Kalinowska-Ozgowicz1, Wojciech Wajda2, Wojciech Ozgowicz1 1Institute of Engineering Materials and Biomaterials, Silesian University of Technology, ul. Konarskiego 18a, 44-100 Gliwice, Poland 2Institute of Metallurgy and Materials Science, Polish Academy of Science, ul. Reymonta 25, Krakow, Poland wojciech.ozgowicz@polsl.pl Prejem rokopisa - received: 2013-10-07; sprejem za objavo - accepted for publication: 2014-02-06 This paper deals with the possibility of optimizing the parameters for a thermomechanical treatment of structural steel with micro-additives, particularly in the processes of the controlled rolling of economical profiles type [240E by means of mathematical modelling of the flow stress (ap) and the physical simulation of this process based on the results of investigations obtained by plastometric hot-torsion tests. In the description of the flow stress the rheological model suggested by C. M. Sellars is applied in the paper, in the form Op = f(£,š,T). Based on this model, the cause of the experimental and theoretical flow stress-strain curves was verified, applying the minimum of the goal function in order to determine most accurately the matching of the analysed curves of the investigated steels. It was found that the applied rheological model achieves a good matching of the experimental and theoretical curves, whereas a physical simulation permits a complementary verification of the optimal parameters of the controlled rolling of the manufactured products. Keywords: flow stress, plastometric torsion test, rheological model, physical simulation, thermomechanical controlled processing (TMCP) Članek obravnava možnost optimizacije parametrov termomehanske obdelave konstrukcijskega jekla z mikrododatki, posebno procese kontroliranega valjanja ekonomskih profilov vrste [240E, z uporabo matematičnega modeliranja napetosti tečenja (Op) in fizikalne simulacije tega procesa na osnovi rezultatov raziskav, dobljenih s preizkusi na vročem torzijskem plastometru. V tem članku je bil za opis napetosti tečenja uporabljen reološki model, ki gaje predlagal C. M. Sellars, v obliki op = f(ß,^,T). Na podlagi tega modela so bile preizkušene eksperimentalne in teoretične krivulje tečenje - raztezek z uporabo minimuma ciljne funkcije, da bi določili najboljše ujemanje analiziranih krivulj preiskovanih jekel. Ugotovljeno je, da uporabljeni reološki model zagotavlja dobro ujemanje eksperimentalnih in teoretičnih krivulj, medtem ko fizikalna simulacija dopolnjuje preverjanje optimalnih parametrov kontroliranega valjanja proizvodov. Ključne besede: napetost tečenja, plastometrični torzijski preizkus, reološki model, fizikalna simulacija, termomehansko kontrolirana predelava (TMCP) tion. One of numerous methods for determining these 1 INTRODUCTION stresses is the hot plastometric torsion test, which also Mathematical modelling is an essential and econo- permits a physical simulation of the sequential rolling deformation. Thus, it becomes possible to determine the mical tool in the application of new techniques for the values of the flow stress in successive roll passes, sepa- rated by interpass times in which thermally activated hot plastic deformation of metals and alloys. It makes it possible not only to reduce the number of technical processes occur, removing the results of the strain experiments, but also to assess the interaction of the hardening4. technological parameters of the investigated processes hardening . determining the usability of the final products. There- The paper presents possibilities for the mathematical fore, it integrates, in most cases, the thermal, mechanical modelling of hot plastic deformation concerning some and microstructural processes. In the future modelling selected products obtained by rolling structural steels will constitute the basis of a system for controlling most with micro-additives in the range of technological technological processes. The constructed models will be parameters simulating such processes in the plastometric applied simultaneously in an off-line simulation and torsion test. Special attention was paid to the verification on-line control of the processes1-3. of the rheological model developed by C. M. Sellars et The determination of the parameters of force in al.5,6 describing the flow stress as a function of the thermomechanically controlled processing (TMCP), deformation and temperature, as well as strain rate and particularly in the controlled rolling of new profiles, the effect of dynamic recrystallization, which dominates requires a knowledge of the values of the flow stress of in the hot deformation for the investigated structural the steel under the specific conditions of the deforma- steels with micro-additives of Nb, V, Ti and N. 2 EXPERIMENTAL PROCEDURE 2.1 Materials o SS(e) a SS(e) - sinh- Z A SS(e) (Ic) The tests were carried out on industrial melts of structural steels with micro-additives, the chemical composition of which is shown in Table 1. Plastometric tests of the steel were performed applying the hot-torsion method. The tested samples had a diameter of 6 mm and a basis length of 10 mm. They were cut out from slabs with dimensions of 200 mm x 220 mm x 5300 mm provided after the hot rolling. By means of continuous torsion, the characteristics of plasticity for the investigated steels were determined in the o - £ system, depending on the temperature of auste-nitization (1150-1200 °C) and deformation (800-1050 °C) as well as on the strain rate within the range 0.15 s-1 to 10 s-1. The testing was conducted on a torsion plasto-meter that was designed at the Institute of Iron Metallurgy in Gliwice, Poland. The conversion of torque-twist to stress-strain was generally based on a Fields and Backofen7 analysis, commonly quoted in8. 2.2 Rheological model In constitutive equations, actually applied for the purpose of modelling the processes of hot plastic working, the effects of dynamic recovery and dynamic recrystalli-zation are not always conveyed explicitly. The equation in which both components are distinctly separated, elaborated at the University of Sheffield by C. M. Sellars et al.5,6, takes the following form: r 0 = 0 0 + (0 SS(e) - 0 ö)1" exp V£ ^^ - R (1) where the respective variables are defined as follows: \0 ■ R =1 (0 SS(e) " 0 SS)11-eXp £ -r— £ C y for £ < £, r £ > £r a - sinh" Z VA0y (1a) (1b) - sinh" ^ Z V. S A, £ r = 031[ q 1 + q 2(0 SS(e))2 ] £ xs " £ C £ C ^ = 1.98 / \Ne Z C 0 „2 V0 SS(e) y :- £ C = C. Z = £ exp Z V0 SS(e) y Qd RT., (1d) (1e) (1f) (1g) (1h) (1i) The variables in Equation (1) are as follows: 0p - flow stress 00 - the maximum stress when the plastic strain £ = 0 0SS(e) - the onset of steady-state conditions in the extrapolated curve 0SS - the onset of steady-state conditions on the experimental flow stress curve £ - plastic deformation £C - strain for the onset of dynamic recrystallization £r - the "transient strain constant" and effectively defines the curvature of the flow-stress curve between 0p and 0SS(e) where the equation saturates £.r - the strain required to reach a fixed amount of softening, measured in terms of A0/A0S. This term effectively defines the rate of softening as a result of the dynamic recrystallization £xs - the strain at the "onset" of steady state when dynamic recrystallization occurs £ - strain rate Z - Zener - Hollomon parameter R - the universal gas constant R. - a term expressing the dynamic recrystallization Qdef - the activation energy for the deformation A, a, n, q, C, N - constants for each characteristic stress 0p. Table 1: Chemical composition of the tested steels Tabela 1: Kemijska sestava uporabljenih jekel Steel Concentration of the element in mass fractions, w/% C Mn Si Cr P S Nb V Ti N Al B1 0.15 1.03 0.25 - 0.018 0.009 0.017 0.05 - 0.0070 0.040 B2 0.16 1.25 0.32 - 0.023 0.019 0.030 0.01 - 0.0060 0.042 S9 0.38 1.52 0.63 0.56 0.027 0.012 0.030 - 0.11 - - S1 0.66 1.02 0.26 0.30 0.029 0.021 - 0.10 - 0.0080 - 1 n 1 0 SS = aSS C Nx 1 The presented model illustrates more distinctly the behaviour of the material in the course of dynamic recrystallization, because it makes it possible to describe the point of deflection (£peak) on the curves o - £ and the procedure of flow stress beginning at the peak strain up to the achievement of the value of the stresses in the steady state (oSS). The coefficients of this constitutive equation are usually determined based on the results of plastometric tests within a wide range of temperature and strain rates. This allows us to model various structural materials over a wide range of conditions for hot plastic deformation. The main difficulty in applying this model lies in the large number of parameters, which have to be unambiguously identified. One of the main factors limiting the application of the mode simulating the processes of plastic working is the difficulty in determining these coefficients, the values of which depend on the kind of applied material. Nevertheless, Equation (1) has been used in this study for the investigated microalloyed steels. For the optimization of the parameters of the applied model of C. M. Sellars5,6, describing stress-strain curves, making use of the goal function 0 in the form: 0 1 - Or (2) where: om - measured stress oc - calculated stress Np - number of measurement points of all the curves in the i-th experiment For the purpose of the minimum of Equation (2), the Nelder and Mead Simplex algorithm was used. The calculations were performed by means of Scilab's calculation packet.9 In results of the optimization for the appropriate parameters of Equation (1) were found. 3 EXPERIMENTAL RESULTS AND DISCUSSION The results of the mathematical modelling of the process of high-temperature plastic deformation for the investigated micro-alloyed steels allowed us to verify analytically the assumed Eqation (1) of the type Op = Op(£, £, T) describing the flow stress (Op) on the experi- mental flow-stress curves, making use of the plasto- metric method for metals and alloys with a low stack- ing-fault energy (SFE), which in the course of hot deformation display the phenomenon of dynamic recry-stallization. Modelling was applied for some selected microalloyed steels of the type HSLA with various contents of the micro-additive Nb (steel B1 and B2), as well as an average-carbon structural steel (0.38 % C) with a binary system of micro-additives of Nb and Ti and a structural rail steel containing about 0.66 % carbon with the micro-additive V (0.10 %). The performed numerical calculations were based on the results of a hot-torsion test, analysing about 80 stress-strain curves recorded with a wide range of external variables, particularly the temperature and the strain rate. A comparison of the experimental flow curves with those in the model concerning the investigated steels is presented in the diagrams in Figures 1 to 4. In order to obtain a universal description of the flow curves, the complete data of measurements Op(£, £, T) concerning the respective kinds of steel were optimized. The influence of the varying distribution of strains on the radius of the twisted sample, as well as the changes in the temperature at the cross-section of a massive sample have, in the process of optimization, not been taken into account. This might perturb the obtained results, particularly the identified parameters of the rheological model, as has been shown in10. The elimination of the effect of the homogeneity of deformation and other errors in the measurements ensures the identification of the parameters of the model, applying the inverse method11,12. The paper10 analyses, however, only the application of this method in the case of various types of hot-compression tests. Future plans will include the application of the inverse method in dealing with the results of hot torsion tests. The numerically determined values of the coefficients assumed in the model, taking into account the Simplex algorithm Figure 1: Comparison of the experimental and theoretical flow curves of microalloyed steel B1, hot deformed in compliance with the torsion method at a strain rate of 8.8 s-1 and at: a) Tdef = 850 °C and b) Tdef = 1000 °C Slika 1: Primerjava eksperimentalne in teoretične krivulje tečenja vroče deformiranega mikrolegiranega jekla B1 v primerjavi z metodo torzije pri hitrosti deformacije 8,8 s-1 pri: a) Tdef = 850 °C in b) Tdef = 1000 °C 2 i =1 Figure 2: Experimental and modelled flow curves of microalloyed steel B2, hot twisted at Tdef = 900 °C and a strain rate of: a) £ = 0.8 s-1 and b) £ = 4.45 s-1 Slika 2: Eksperimentalna in modelirana krivulja tecenja jekla B2, vroče zvijanega pri rdef = 900 °C in hitrosti deformacije: a) £ = 0,8 s-1 in b) £ = 4,45 s-1 concerning the analysed stress-strain curves are shown in Table 2, which also contains the values of the energy of activation of the process of deformation and the final values of the goal function 0, representing the accuracy of the modelling solution. Due to measurement errors and limitations with the model, different values were obtained for the goal function (Equation (2)) concerning the particular types of investigated steels. However, the median square error of the calculation remains within 2-5 % limits. This proves that the assumed model fits well with the experimental results. The flow curves of the investigated steels indicate principally the characteristic procedure of the continuous dynamic recrystallization. Their analysis permits us to state that this process is essentially influenced by the temperatures (Figure 1) and the strain rate (Figure 2), as well as the chemical composition of the steel (Figures 1 to 4). The effect of these factors concerns, first of all, changes in the value of the maximum flow stress and the stresses in the steady state (ass), and also the values of critical deformations (£f), generally assumed to be in compliance with the criterion of the plasticity of steel in the process of hot deformation. It has been generally assumed that the stresses in the flow of the investigated steels increase with the drop in the temperature and the increasing strain rate. It has also been found that the best compatibility in matching the experimental and theoretical curves is achieved in the case of steel S9 (Figure 3), with the Figure 3: Comparison of experimental and modelled flow curves of steel S9 in a hot torsion test: a) Tdef = 900 °C, £ =1 s-1, b) Tdef = 1000 °C, £ =10 s-1 Slika 3: Primerjava eksperimentalne in modelirane krivulje tečenja jekla S9 pri preizkusu vroče torzije: a) Tdef = 900 °C, £ =1 s-1, b) Tdef = 1000 °C, £ = 10 s-1 Figure 4: Experimental and modelled flow curves of steel S1, hot twisted at: a) Tdef = 900 °C, £ =1 s-1, b) Tdef = 1000 °C, £ =10 s-1 Slika 4: Eksperimentalna in modelirana krivulja tečenja jekla S1, vroče zviianega pri: a) Tdef = 900 °C, £ =1 s-1, b) Tdef = 1000 °C, £ =10 s-1 Table 2: Optimal coefficients of the Theological model obtained as a result of Simplex optimization concerning the investigated steels with micro-additives Tabela 2: Optimalni koeficienti reološkega modela, dobljeni kot rezultati optimizacije Simplex preiskovanih jekel z mikrododatki Steel No. Rheology - Oo Activation energy ß/(J/mol) Goal function Coefficients Ao no «o 4.1200 • 1011 0.0525 2.2969 278323.4 0.0489 1.7800 • 1013 0.1223 0.8931 330559.2 0.0399 2.2300 • 108 0.0535 36.5915 297570.7 0.0336 3.7700 • 1010 0.0193 21.2038 296973.7 0.0518 Rheology - strain hardening and dynamic recovery Asse nsse «sse q1 q2 3.19 • 1013 4.2182 0.0028 0.1668 0.0067 • 10-2 6.38 • 1019 4.3216 0.0039 0.0024 •10-2 0.0075 • 10-2 2.81 • 1013 5.3884 0.0049 0.6791 0.0026 • 10-2 1.86 • 1011 2.4098 0.0203 1.2176 0.0001 • 10-2 Rheology - strain hardening and dynamic recrystallization Ass «ss «ss Cc Nc Cx Nx B1 B2 S9 S1 Steel No. B1 B2 S9 S1 Steel No. B1 1.80 • 10 2.4377 0.029 0.000036 0.0244 0.0234 0.279958 B2 1.22 • 1010 2.5560 0.0365 0.0433 0.0079 0.0812 0.1763 S9 5.99 • 108 3.6426 0.0279 0.0944 0.0378 0.6839 0.0957 S1 1.21 • 109 3.9124 0.0302 0.0018 • 10-2 0.0427 1.4927 0.0670 Table 3: Characteristics of the process of rolling the profile [240E Tabela 3: Značilnosti postopka valjanja profila [240E Table of roll passes Profile [240E Slab: 200 mm x 200 mm x 6000 mm; 5o = 382 cm2 Stand Roll Pass No. Working pass No. £s/% £h/% Pass input temp. T/°C 5k/cm2 V1/(m/s) f/s-1 BD 1 4 2.9 2.5 1150-1180 371.0 2.52 1.32 Z1 2 1 17.0 24.6 1140-1170 307.9 4.54 10.30 Z1 3 2 29.4 34.3 - 217.3 5.35 17.56 Z1 4 3 29.3 30.6 - 153.6 6.12 21.86 Z1 5 4 28.3 29.9 — 110.2 6.84 28.36 Z1 6 5 19.8 21.3 1050-1080 88.4 6.89 25.54 Z2 7 6 26.2 27.0 1020-1055 65.2 9.01 47.80 Z2 8 7 25.9 25.9 - 48.3 9.09 54.36 Z2 9 8 19.0 21.5 950-1020 39.1 8.83 52.60 D1 10 9 15.3 12.7 960-1050 33.9 6.97 34.57 D2 11 10 10.0 8.8 920-950 30.5 6.93 29.16 £s, £h indices for the deformation of the rolled channel section [240E 5k - running cross-section of the rolled band vl - linear velocity of the rolling f - strain rate of the rolled band values of goal function ^ = 0.0336, within the entire range of the temperature and deformations. In the case of simulative investigations, some constitutive equations were applied, concerning the modelling of the microstructure13,14, as well as the kinetics of dynamic recrystallization, developed based on data obtained in plastometric tests for a varying size of the primary austenite grain. As in the rheological model, the kinetics of dynamic recrystallization was taken into account, and it was included in the function of the yield stress, expressed by Equation (1). Thus, in the Sellars model the stress depends on the size of the austenite grain. This model predicts a more accurate flow of the material, particularly in the processes of reiterated deformations in the case of hot rolling. The rolling of channel iron [240E of type B2 steel was physically simulated based on the parameters of the rolling mill collected in Table 3. These parameters served as the output data for the physical simulation realized on a torsional plastometer in a test of hot torsion, applying the sequential method at a strain rate amounting to about 4.0 s-1. The analysed flow curves were compared with the o - £ curves that were determined in the course of continuous torsion tests in order to determine the effect of global cycles of deformation on the initiation and progress of the activated thermal processes occurring during the intervals between the roll passes. The results of the kinetic investigations, including an analysis of the share of the decay of strain hardening during the interpass times, are presented in the Figure 5: Deformation schedules for channel section type [240E; rolling simulations of microalloyed steel B2 Slika 5: Zaporedje deformacij v kanalskem delu vrste [240E; simulacija valjanja mikrolegiranega jekla B2 diagrams in Figure 5. The deformations in the respective sequential roll passes were found to be less than the critical values required for the initiation of dynamic recrystallization determined by continuous o - e curves, similar to the case in tests of physical simulations, whereas global deformations in the respective roll passes exceeded the values ecd. Thus we can assume that in the case of the last roll passes on the flushing stand D1 and D2 the strain hardening is accumulated. This character of the deformation changes determines the occurrence of both static and meta-dynamic recrystallization, and consequently a grain refinement of the investigated steels. 4 CONCLUSIONS The mathematical modelling of flow stress in the course of high-temperature deformation allows us to obtain the optimal parameters for the thermomechani-cally controlled processing of the tested structural steels with micro-additives. Sellars's rheological model provides good matching of the experimental and theoretical flow curves in the investigated steels, determined on the basis of hot pla-stometric torsion tests. A correct model describing the behaviour of HSLA steel type B2 during high-temperature plastic deformation process ensured an accurate simulation of the sequential rolling schedule for a selected economical section of the type [240E. A physical simulation of the rolling schedule of a channel section [240E by the means of hot torsion tests ensures the yield stress values op in consecutive roll passes and calculates the force and energy parameters in the controlled rolling process. A kinetic analysis of thermally activated phenomena occurring in the course of high-temperature plastic deformation and interpass times ensures the possibility of shaping the structure of the investigated steels and to determine the mechanical properties of the final products. 5 REFERENCES I P. D. 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