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Integrated model for thermo-mechanical controlled process in rod (or bar) rolling
Journal of Materials Processing Technology 125±126 (2002) 678±688
Integrated model for thermo-mechanical controlled process in rod (or bar) rolling Y. Leea,*, S. Choia, P.D. Hodgsonb a
Plate, Rod and Welding Group, POSCO Technical Research Laboratories, Pohang 790-785, South Korea School of Engineering and Technology, Deakin University, Pigdons Road, Geelong, Vic. 3217, Australia
b
Received 12 December 2001; received in revised form 8 February 2002; accepted 8 February 2002
Abstract This study presents an integrated model for computing the thermo-mechanical parameters (cross-sectional shape of workpiece, the pass-bypass strain and strain rate and the temperature variation during rolling and cooling between inter-stands) and metallurgical parameters (recrystallisation behaviour and austenite grain sizeÐAGS), to assess the potential for developing ``Thermo-Mechanical Controlled Process'' technology in rod (or bar) rolling, which has been a well-known technical terminology in strip (or plate) rolling since 1970s. The advantage of this model is that metallurgical and mechanical parameters are obtained simultaneously in a short computation time compared with other models. The model has been applied to a rod mill to predict the exit cross-sectional shape, area and AGS per pass by incorporating the equations for AGS evolution being used in strip rolling. At the ®nishing train of rod mills, the strain rates reach as high as 1000±3000 s 1 and the inter-pass times are around 10±60 ms. The results show that the proposed model is an ef®cient tool for evaluating the effects of process-related parameters on product quality and dimensional tolerance of the products in rod (or bar) rolling. The results of the simulation demonstrated that the equation for AGS evolution being used in strip rolling might have limitations when applied directly to rod rolling at a high strain rate. # 2002 Elsevier Science B.V. All rights reserved. Keywords: Rod (or bar) rolling; Thermo-mechanical controlled process; Integrated model; Numerical simulation; Austenite grain size (AGS); High strain rate
1. Introduction Signi®cant advances have been made in predicting the thermo-mechanical parameters (cross-sectional shape, strain, strain rate and temperature) and the metallurgical parameters, during hot rolling of material to control the mechanical properties of the products as well as the dimensional tolerances. Most of previous works [1±3], however, have concentrated on the strip (or plate) rolling process because the thermo-mechanical parameters can be determined using the assumption of plane strain deformation. Little research has been undertaken for rod (or bar) rolling since the workpiece between the grooved rolls is neither in the condition of plane strain nor plane stress. There is a strong demand to develop a mathematical model that is reliable, accurate and non-iterative to obtain the thermo-mechanical parameters associated with rod rolling *
Corresponding author. Tel.: 850-54-220-6058; fax: 850-54-220-6911. E-mail address: [email protected] (Y. Lee).
process. The reason is that once these parameters are obtained, then the recrystallisation behaviour and austenite grain size (AGS) evolution model developed for strip rolling can be applied directly to rod rolling. There have been several attempts [4±6] to do this. Maccagno et al. [4], in a study of the AGS evolution associated with rod rolling sequence, calculated the strains by multiplying the area strains by a constant factor. Strains were obtained by taking the natural logarithm of the ratio of the fractional reductions in cross-sectional area through the rolling stands. The strain at a pass was then assumed to be 1.7 times the area strain for the roughing stands, and 2.5 times the area strain for the ®nishing stands. Kemp [5] proposed that the strain per pass should be factors of 1.5± 2 times the area strains in the roughing stands and factors of 2±3 times the area strains in all subsequent stands. In the work of Maccagno et al. and Kemp, however, a mathematical rationale for the use of the constant factors was missing. Lehnert and Cuong [6] suggested a model that calculates the strain, strain rate in rod rolling and AGS, based on the assumption of plane strain deformation. The three-dimensional
0924-0136/02/$ ± see front matter # 2002 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 4 - 0 1 3 6 ( 0 2 ) 0 0 3 4 7 - 3
Y. Lee et al. / Journal of Materials Processing Technology 125±126 (2002) 678±688
deformation zone was subdivided into longitudinal strips of equal width to roll gap direction and each strip analysed separately. No experimental veri®cation, however, was provided to support the assumption. This approach might be applicable once the exit cross-sectional shape at a pass can be correctly predicted. However, a method to obtain the exit cross-sectional shape was not provided. Recently, Lee [7] has proposed an approximate model that calculates strain at a given pass in a round±oval (or oval± round) pass rolling sequence. The proposed model is based on the elementary theory of plasticity and the equivalent rectangle approximation method. Lee et al. [8] also performed FE simulation to show the validity of the approximate model. They compared the strains calculated from the approximate model with those from FE model. The strain in the FE simulation was de®ned as the averaged effective plastic strain at the exit of the roll gap (at the pass). Lee et al. [9] also veri®ed the approximate model experimentally by conducting the tensile test of the rolled specimen and compared the predicted AGS for a four-pass bar rolling sequence with measured AGS from hot torsion test. These approaches [7±9] based on the ``average'' concept have some limitations since it does not show the ``distribution'' over the cross-section, such as the strain, strain rate and AGS. Karhausen et al. [10], Kuziak et al. [11] and Yanagimoto et al. [12] have presented three-dimensional FE analysis (FEA) for the microstructure evolution in hot bar rolling, coupled with the experiment-based AGS prediction model. FEA is very effective for computing the distribution of effective plastic strain, strain rate and temperature across the section as well as predicting the surface pro®le of deformed workpiece and AGS. However, this approach requires at least half an hour to run FE code for a single pass since three-dimensional analysis is required. Thus, considering computational time for the bar (or rod) mills that consists of a large number of passes (stands) and the complicated mechanical/thermal boundary conditions (the friction condition at the roll/material interface and heat transfer coef®cients dependent on the temperature and contact pressure), the ``average'' model still remains as a useful tool to calculate the thermo-mechanical parameters and consequently AGS per pass. This study presents a systematic procedure for computing the thermo-mechanical (cross-sectional shape of workpiece, strain, strain rate and temperature variation during rolling and cooling between inter-stands) and metallurgical parameters (recrystallisation behaviour and AGS) for a rod mill where the strain rates can be as high as 3000 s 1 (in the ®nishing stands) and the associated deformation times are of the order of 1 ms, and the inter-pass times are around 15± 60 ms. At the ®nishing train of the rod mill, there is signi®cant heat generation due to the high strain rate plastic deformation of the workpiece. The prediction of this relies on an accurate representation of the constitutive behaviour under the imposed deformation conditions. In the present
679
study, it was assumed Shida's constitutive equation [13] could be used to calculate the plastic deformation of materials at high strain rates. The validity of this assumption has been examined using split Hopkinson-Pressure Bar (SHPB) test technique [14], which measures the stress±strain relation of material at high strain rate (1000±4000 s 1). The high temperatures at SHPB test were obtained by enclosing the specimen in a clam-shell radiant-furnace [15]. There has been some debate regarding the AGS evolution during rod rolling at very high strain rates. It is unlikely that this will be the same as in plate and strip rolling. This paper presents an attempt to predict the AGS under high strain rate deformation in the ®nishing train and the effect of watercooling in the cooling zone and equalisation zone between stands (passes). A one-dimensional heat transfer approach together with an equivalent circular approximation method of the workpiece at each pass was used to calculate the temperature evolution of the material during rolling and cooling. 2. Modelling of thermo-mechanical parameters The following ®ve sub-sections give the procedure to calculate the thermo-mechanical parameters (strain and strain rate per pass and temperature evolution of material during rolling). 2.1. Prediction of cross-sectional shape To determine the pass-by-pass strain and strain rate, it is necessary to ®rst predict the stress-free surface pro®le of material that does not contact the roll directly at the exit of roll gap. Recently, Lee et al. [16] developed an analytical model, which is robust and non-iterative in computation, to predict the surface pro®le of the exit cross-section of a workpiece for the oval±round (or round±oval) pass sequence most widely used pass sequence in present rod (or bar) mills. The advantage of this model is that only the geometric consideration is required in the formulation. Hence, it greatly simpli®es the problem of obtaining the ®nal rolled shape. 2.2. Determination of equivalent rectangular cross-section Since the cross-sectional shape of the deformed workpiece at a pass is not rectilinear, one encounters dif®culties in formulating the equation for the calculation of the strain within a frame of Cartesian co-ordinates. To overcome this problem, the equivalent rectangle approximation method that transforms the non-rectangular cross-section into a rectilinear one was employed. There are many ways to achieve this objective while the net cross-sectional area is maintained. Maintaining the cross-sectional area is presumed to be a necessary, but not a suf®cient requirement for equivalence. There are generally three methods to establish equivalent rectangular
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Fig. 1. (a) Schematic description of round±oval pass rolling (front view). (b) Application of equivalent rectangle approximation method to round±oval pass rolling. G represents the roll gap and Reff the effective roll radius. H i and H p indicate incoming and outgoing equivalent cross-sectional height. Incoming and outgoing equivalent cross-sectional widths are represented by W i and W p .
sections; namely, method of maximum height, method of maximum width and the method of width±height ratio. The effective (or mean) roll radius necessary in the calculation of the rolling speed at a pass has been calculated from the method of maximum width [17,18]. Therefore, in this study, the cross-sections are approximated into rectangles using the method of maximum width. Fig. 1 shows an example of the method of maximum width applied to the round±oval pass rolling, in which an incoming stock is of a round shape and stock with an oval shape is produced. Sections A, B and C, respectively, correspond to the positions where a section of the undeformed workpiece is about to be rolled, being deformed and that of the deformed workpiece leaving the roll groove. The equivalent rectangle approximation described in Fig. 1 can also be applied to another type of passes such as oval±round, box±oval pass and diamond±square pass, etc. 2.3. Strain at a pass (pass-by-pass strain) The strain de®ned as the maximum average effective plastic strain at a given pass can be then calculated from the rectilinear shape transformed. The calculation, however, should include the non-linear change of draught, spread and elongation of the material being deformed. The assumption introduced to overcome this problem is the hypothesis of parallelepiped deformation [17]. According to this hypothesis, a cube of material subjected to a load will change to a rectangular prism and its angle and sides will remain orthogonal with this prior to deformation. It should be noted that the calculation is focused on bulk deformation of the workpiece. Derivation procedure for the
equation that computes the strain at a pass is described in Refs. [7±9]. 2.4. Strain rate at a pass (pass-by-pass strain rate) As in drawing and forging processes, the strain rate in bar (or rod) rolling changes at various stages of deformation. The strain rate is a maximum at the entrance to the roll (or in its vicinity) and decreases along the roll bite, ®nally becoming zero at the outlet. For this reason, it is necessary to introduce an ``average'' strain rate for a given pass. The ``average'' strain rate can be de®ned as the strain over a time interval which can be calculated from: ep e_ p ; tp
(1)
where tp represents the time interval taken for a section A to pass through to a section C (Fig. 1). Hereafter the ``average'' strain rate is referred to as the strain rate. The time interval can be expressed as tp
60L s; 2pNReff
(2)
where Reff, Lp and N are the effective roll radius, the effective projected contact length of the grooved roll and workpiece, and roll rpm at a given pass, respectively. The effective roll radius at a given pass is calculated by the method of maximum width [17,18]. The effective projected contact length is expressed as s Hp G L Rmax (3) H i H p : 2
Y. Lee et al. / Journal of Materials Processing Technology 125±126 (2002) 678±688
2.5. Temperature evolution during rod (or bar) rolling The temperature of the workpiece during rolling depends on various factors such as rolling speed, initial temperature of the billet, plastic deformation of the workpiece, the crosssectional shape of the workpiece at each pass, cooling condition in the individual passes and distribution of cooling and equalisation zone between stands (passes). To take care of the combined actions of these parameters, the model for the temperature evolution of the workpiece during rod (or bar) rolling has been formulated based on the following assumptions: (i) Uniform initial temperature of the billet (or workpiece). (ii) No longitudinal temperature gradient (i.e., infinitely long rod). (iii) Uniform heat generation across the cross-section of workpiece due to plastic deformation in the roll gap. (iv) Circular cross-sectional shape at each pass. Assumptions (i) and (ii) are typical for this type of problem and assumption (iii) is introduced because the strain was de®ned as maximum average effective (equivalent) plastic strain at a given pass and the strain rate as the strain divided by the time required for the workpiece to undergo this strain in the roll gap. Finally, the background for assumption (iv) is explained as follows. The workpiece deformed at each pass is not of a circular cross-sectional shape (even at round pass). In this case, the heat transfer model that should be solved is a two-dimensional problem with a curved geometric boundary condition. At present, there is no systematic procedure for solving such problems using the ®nite difference method. One way to overcome this dif®culty is to employ the equivalent circle approximation method that transforms a non-circular crosssectional shape of workpiece into a circular one while the net cross-sectional area is maintained. The problem can be then reduced to an axi-symmetric heat transfer problem. Under these assumptions, temperature variation within the rod is governed by the following one-dimensional axi-symmetric heat transfer equation: @ @T @T k ; (4) q_ rCp @r @r @t where t is the time that takes for an volume element to travel the distance in the rolling direction and r the radius of the rod. k, r and Cp are thermal conductivity, density and speci®c heat of the material, respectively. Note that q_ is the volumetric rate of heat generation within the rod due to the plastic deformation of the workpiece in the roll gap and can be expressed as: q_ w
se_ ; rbCp
(5)
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where w and b (4.185 kJ/kcal) are the faction of plastic deformation work converted into heat generation and the mechanical equivalent of heat, respectively. s is the ¯ow stress of the workpiece. Typically w 0:9 is used [19]. It was assumed the heat generated due to friction at the interface of roll and workpiece was compensated by heat loss due to the roll coolant applied on the workpiece as well as the work rolls. (In rod rolling process, the coolant on work roll is not wiped out but drops on the workpiece during rolling.) Solution of Eq. (4) is not possible analytically because of the variation of k, Cp and q_ with temperature. Hence, an implicit ®nite difference technique was applied to Eq. (4). The solution method of Eq. (4) using the ®nite difference method with boundary conditions is well described in [20,21]. The values of k and Cp have been taken to be a function of the temperature of the material and were acquired from the literature [22,23]. The density of material was assumed to be constant over the temperature range of interest. To predict the temperature variation of workpiece under the action of water-cooling in the cooling zone and equalisation zone between stands (passes), we followed Morales et al.'s approach [24,25]. They developed a mathematical model to predict the cooling behaviour of steel rods under the action of water-cooling in rod mill. They surveyed the various water-cooling (spray, jet, mist) models and examined the in¯uence of operation parameters such as rod size, rolling speed, rod temperature and water ¯ow-rate on the temperature distribution within the rod before it enters into the transformation conveyer. 3. Constitutive equation at high temperature and strain rate It is well known that signi®cant heat generation occurs due to the high strain rate plastic deformation of the workpiece in the ®nishing train of rod mill. Thus, the constitutive equation of material plays a crucial role in computing the temperature evolution of workpiece during rod rolling. In the present study, it was assumed that Shida's constitutive equation [13] could predict the ¯ow stress of materials subjected to high strain rates. Shida's constitutive equation gives the ¯ow stress of carbon steels as a function of the strain, the strain rate, temperature and the carbon content. It is applicable in the range of carbon content: 0.07±1.2%, temperature: 700±1200 8C and strain: up to 0.7, but might not be applicable when the range of strain rate goes beyond 100 s 1. It takes account of the ¯ow stress behaviour of the steels in austenite, ferritic and in the two-phase regions. This is a strength of Shida's equation in comparison with the other constitutive equations [26]. Shida's main dif®culty in developing the constitutive equation in high strain rate lay in the lack of experimental facilities to provide data for the plastic deformation of materials under very high strain rates.
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The validity of this model has been examined using the SHPB test [14], which measures the stress±strain relation of the material at very high strain rate (1000±4000 s 1). The high temperatures at SHPB test in this study were obtained by enclosing the specimen in a clam-shell radiant-furnace. This test was conducted at Department of Mechanical Design and Production Engineering of Yonsei University in Korea. Fig. 2 shows the measured and predicted stress± strain curves of a low carbon steel (0.1%C) at different temperatures and similar strain rates. In all cases, the measured ¯ow stress is higher than predicted one, although the general shapes of the predicted and measured curves are similar. The difference between the predicted and measured stresses is approximately 20%. For the current work, it was assumed that the Shida's equation could be applied with a 20% increase. However, it is also clear that much more work is required in this area.
4. Application The recrystallisation model and AGS evolution model generally used in hot strip rolling together with the thermomechanical parameters (strain, strain rate at a pass and temperature of workpiece during rolling) have then been applied to POSCO no. 3 rod mill to simulate the variation of AGS of workpiece during rolling. The equations used in this study are listed in Appendix A. The average temperature at each pass was taken as the representative temperature of the workpiece. In the simulation, the thermo-mechanical model calculates the thermo-mechanical parameters of the workpiece during rolling once the information regarding the rolling schedule (roll groove geometry and inter-pass time), initial temperature and initial size of the workpiece are given. This creates an output ®le of the thermo-mechanical parameters
Fig. 2. Flow stress±strain curves of low carbon steels (0.1%C) at different temperatures.
Y. Lee et al. / Journal of Materials Processing Technology 125±126 (2002) 678±688
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Fig. 3. Schematic diagram of the integrated thermo-mechanical processing model for rod (or bar) rolling.
for the whole process. The metallurgical model then picks up this ®le and computes recrystallisation behaviour and AGS evolution for the whole process. This procedure is described in Fig. 3. 5. Results and discussion 5.1. Exit cross-sectional shape and temperature history of workpiece during rolling Fig. 4 illustrates the predicted surface pro®les of exit cross-section and those measured for the seven stands
(passes) in the roughing train (nos. 1±7) of POSCO no. 3 rod mill. The predicted exit cross-sectional shape slightly overestimates that measured at pass no. 1. At pass no. 2, the measured surface pro®le is in good agreement with the predicted one. At the round pass, i.e., pass nos. 3, 5 and 7, skewed round pro®les are observed (Fig. 4(c), (e) and (g)). This means that the workpiece was not rotated exactly 908. It was twisted more than 908 or less than 908. This kind of problem occurs easily in horizontal±horizontal type rolling sequence. Meanwhile, at the oval pass rolling, a relatively stable rolling is observed, as shown in Fig. 4(b), (d) and (f). Overall, the predicted surface pro®les of exit cross-section are in agreement with those measured.
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Fig. 4. Predicted and measured surface profiles and areas of the exit cross-section for the roughing train in POSCO no. 3 rod mill.
Y. Lee et al. / Journal of Materials Processing Technology 125±126 (2002) 678±688
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5.2. Evolution of AGS
Fig. 5. Time±temperature history of workpiece during rolling in POSCO no. 3 rod mill: (a) roughing train; (b) intermediate finishing train (#14±19) and finishing train (#20±29).
Fig. 5 shows the predicted temperature history and measured ones for two types of carbon steels (0.67 and 0.80%C). Due to lack of number of pyrometers installed in the rod mill, comparison was only carried out at limited locations. Fig. 5(a) shows that at pass no. 2 (designated as #2), the measured surface temperatures are lower than those predicted. Meanwhile, opposite results were observed at after #11. Fig. 5(b) demonstrates the heat generation attributable to the high strain rate plastic deformation in the intermediate ®nishing train (#14±19) and ®nishing train (#20±29). Measured temperatures at before #20 and after #29 are higher than the predicted ones. However, the general level of agreement is encouraging the given dif®culty.
For rod mills, the strain rates in the ®nishing stands can be as high as 1000±3000 s 1. The associated deformation times are of the order of 1 ms and the inter-pass times are around 10±60 ms. This has raised questions regarding the AGS evolution that takes place during rolling associated with high strain rates. It is unlikely these will be the same as in plate and strip rolling. This paper presents a simulation to study whether we can apply directly the equation for AGS evolution generally used in strip rolling to rod rolling. Table 1 shows AGS per pass for the entire process when the ®nish rolling speed is 100 m/s for the production of 5.5 mm diameter rod. The static recrystallisation occurred in four of the last 10 stands at the ®nishing train. It seems that material experienced work hardening at the four stands (passes). It can be deduced that the critical strain triggering meta-dynamics recrystallisation (MDRX) was larger than the calculated strain at six stands of the last 10 stands at the ®nishing train. This result is quite unusual because other work [4] has suggested that the higher strain rate in the ®nishing train of rod (or bar) mills favour meta-dynamic recrystallisation. In passes 21 and 29, there was a large change in the fraction of statically recrystallised material. A small amount of the RXN fraction is attributable to very short inter-pass time and small strain at passes 21 and 29, compared with others. It should be noted that the fraction of static RXN depends strongly on incubation time, i.e., interpass time and strain at a pass. The smaller applied strain the larger t50. The larger t50 compared with the inter-pass time leads small amount of fraction of static RXN. (See equations in Appendix A.) Table 2 shows AGS evolution when the water-cooling during rolling is in operation. When the material is processed with water-cooling, the AGS becomes smaller, as expected. However, in this case, it is observed that static recrystallisation take places at nine stands of the ®nishing train. When the model predicts static recrystallisation, there are some inter-pass periods where low fractions of recrystallisation occur. For meta-dynamic recrystallisation, there is always complete recrystallisation in the following inter-pass period. At this stage, it is unclear whether it is reasonable to expect partial recrystallisation at some passes in the ®nishing train of rod (or bar) rolling process. Hence, further work is required to develop appropriate models for these types of deformation conditions. It is likely to apply to both the static and meta-dynamic recrystallisation reactions. The actual grain sizes are also an area for further study, which should include both laboratory and direct measurement of the AGS of the material during rolling. It is clear that the complete integrated model developed here provides an important framework for the further development of thermomechanically controlled process technology in rod (or bar) rolling and as a tool to direct future fundamental research in this ®eld.
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Table 1 Thermo-mechanical and metallurgical parameters during rod rollinga Std. no.
Strain
Strain rate (s 1)
Temperature (8C)
RXN fraction (%)
Mode
IPT (s)
AGS (mm)
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29
0.347 0.435 0.461 0.491 0.320 0.488 0.354 0.513 0.398 0.514 0.437 0.454 0.436 0.503 0.471 0.480 0.462 0.439 0.461 0.577 0.416 0.440 0.434 0.435 0.477 0.453 0.467 0.348 0.383
0.36 0.63 0.78 1.37 1.26 2.59 2.53 5.01 5.00 9.34 10.85 17.48 19.12 38.47 41.00 73.58 78.80 118.69 129.12 317.44 387.60 475.46 503.72 795.32 1061.86 1684.12 1868.80 2392.93 3094.85
1049.9 1014.8 986.0 957.2 962.6 939.4 937.3 934.7 896.6 931.4 900.8 921.1 898.7 893.4 898.6 913.2 924.7 934.9 947.1 944.4 953.0 963.5 971.8 986.8 999.2 1012.5 1025.5 1035.6 1049.0
1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 0.985 1.000 0.992 0.985 0.990 1.000 0.248 1.000 1.000 1.000 1.000 1.000 1.000 0.943 0.500
MDRX MDRX MDRX MDRX MDRX MDRX MDRX MDRX MDRX MDRX MDRX MDRX MDRX MDRX SRX SRX SRX SRX SRX SRX SRX MDRX MDRX MDRX MDRX MDRX MDRX SRX SRX
5.800 27.220 33.237 15.923 11.578 39.634 6.676 5.472 4.041 3.294 2.473 6.024 1.551 8.781 0.591 0.676 0.521 0.426 0.343 3.002 0.059 0.049 0.039 0.032 0.025 0.030 0.016 0.013 0.011
62.11 47.65 40.37 31.04 31.65 29.90 24.52 22.48 17.87 20.24 16.05 20.65 15.28 18.94 17.82 20.99 20.93 21.35 22.02 31.41 21.65 8.19 8.32 8.12 8.06 7.75 7.99 19.75 16.72
a
IPT: inter-pass time; finish rolling speed: 100 m/s; size: 5.5 mm diameter.
Table 2 Thermo-mechanical and metallurgical parameters during rod rolling when water-cooling between stands is ona Std. no.
Strain
Strain rate (s 1)
Temperature (8C)
RXN fraction (%)
Mode
IPT (s)
AGS (mm)
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
0.347 0.435 0.461 0.491 0.320 0.488 0.354 0.513 0.398 0.514 0.437 0.454 0.436 0.503 0.471 0.480 0.462 0.439 0.461 0.577 0.416 0.440 0.434 0.435 0.477
0.36 0.63 0.78 1.37 1.26 2.59 2.53 5.01 5.00 9.34 10.85 17.48 19.12 38.47 41.00 73.58 78.80 118.69 129.12 317.44 387.60 475.46 503.72 795.32 1061.86
1049.9 1014.8 986.0 957.2 962.6 939.4 937.3 934.7 896.6 931.4 900.8 921.1 898.7 800.9 877.9 845.6 825.7 838.8 848.9 796.5 804.8 815.6 825.0 842.5 857.9
1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 0.971 0.930 0.942 0.985 0.939 1.000 0.165 0.470 1.000 0.814 0.792
MDRX MDRX MDRX MDRX MDRX MDRX MDRX MDRX MDRX MDRX MDRX MDRX MDRX SRX SRX SRX SRX SRX SRX SRX SRX SRX MDRX SRX SRX
5.800 27.220 33.237 15.923 11.578 39.634 6.676 5.472 4.041 3.294 2.473 6.024 1.551 8.781 0.591 0.676 0.521 0.426 0.343 3.002 0.059 0.049 0.039 0.032 0.025
62.11 47.65 40.37 31.04 31.65 29.90 24.52 22.48 17.87 20.24 16.05 20.65 15.28 16.70 14.91 10.55 8.41 10.22 9.49 14.05 10.90 5.61 3.48 5.20 5.66
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687
Table 2 (Continued ) Std. no.
Strain
Strain rate (s 1)
26 27 28 29
0.453 0.467 0.348 0.383
1684.12 1868.80 2392.93 3094.85
a
Temperature (8C) 875.4 894.2 908.6 927.1
RXN fraction (%)
Mode
IPT (s)
0.801 0.792 0.560 0.862
SRX SRX SRX SRX
0.030 0.016 0.013 0.011
AGS (mm) 6.29 6.94 6.49 8.65
Water-cooling tubes are placed after Std. no. 13 and after Std. no. 19. IPT: inter-pass time; finish rolling speed: 100 m/s; size: 5.5 mm diameter.
strip rolling, and a proper constitutive equation describing the deformation behaviour of the material at high strain rate (up to 3000 s 1) and high temperature is available. However, the results of the simulation demonstrate that the equations for AGS evolution, based on those used in strip rolling, may have limitations when applied directly to rod rolling where material experiences very high strain rates in the ®nishing train.
6. Conclusions An integrated model has been developed which computes the thermo-mechanical (cross-sectional shape of workpiece, the pass-by-pass strain and strain rate and the temperature variation during rolling and cooling between inter-stands) and metallurgical parameters (recrystallisation behaviour and AGS) during high speed rod rolling. This might have the potential to be used for developing ``Thermo-Mechanical Controlled Process'' technology in rod (or bar) rolling. This study demonstrates that, on an industrial scale, a quantitative description of the thermo-mechanical and metallurgical parameters of the material during each stage of rod (or bar) manufacturing process is possible if the proposed model can be mutually integrated with the recrystallisation model and AGS evolution model being used in
Acknowledgements One of authors (Y. Lee) would like to acknowledge K.J. Park and Professor O.K. Min (Yonsei University, Korea) who performed the split Hopkins-Pressure Bar test at high temperature.
Appendix A Equations and constants used in numerical simulation [4]. Model
Equation
Parameters
Critical strain
ec 5:6 10 4 do0:3 Z 0:17 n t x 1 exp 0:693 t0:5 QRXN p q t0:5 to e do exp RT Qdef Z e_ exp RT " 1:5 # t x 1 exp 0:693 t0:5 Qmd t0:5 kmd Z nmd exp RT
do 250 mm
Strain recrystallisation
Zener±Hollomon parameter Meta-dynamic recrystallisation
Recrystallised grain size Static Meta-dynamic Grain growth
dSRX Ae
a b do
dMDRX kZ
exp
45000 8:31T
n1 to 2:3 10 15 , p 2:5, q 1, QRXN 230 kJ/mol Qdef 300 kJ/mol, R 8:31
kmd 1:1, nmd 0:8, Qmd 230 kJ/mol
A 343, a 0:5, b 0:4
0:23
k 2:6 104
Static (if tip > 1 s)
m dm dSRX k tip
Meta-dynamic (if tip > 1 s)
m dm dMDRX k tip
400000 8:31T 400000 2:65t0:5 exp 8:31T
4:32t0:5 exp
m 7, k 1:5 1027 m 7, k 8:2 1025
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Y. Lee et al. / Journal of Materials Processing Technology 125±126 (2002) 678±688
Appendix A. (Continued ) Model
Equation
Static (if tip < 1 s)
m d m dSRX k tip
Meta-dynamic (if tip < 1 s)
m d m dMDRX k tip
Partial RXN
113000 8:31T 113000 2:65t0:5 exp 8:31T
4:32t0:5 exp
Parameters m 2, k 4:0 107 m 2, k 1:2 107
4=3
doi1 Xi dRXi 1 Xi 2 doi , eaccuml ei1 1 Xi ei i1
References [1] C.M. Sellars, Modeling microstructural development during hot rolling, Mater. Sci. Technol. 6 (1990) 1072±1081. [2] T. Sakai, J.J. Jonas, Dynamic recrystallization: mechanical and microstructural conditions, Acta Metall. 32 (1984) 189±209. [3] P.D. Hodgson, R.K. Gibbs, A mathematical model to predict the mechanical properties of hot rolled C±Mn and micro alloyed steels, ISIJ Int. 32 (1992) 1329±1338. [4] T.M. Maccagno, J.J. Jonas, P.D. Hodgson, Spreadsheet modeling of grain size evolution during rod rolling, ISIJ Int. 36 (1996) 720± 728. [5] I.P. Kemp, Model of deformation and heat transfer in hot rolling of bar and sections, J. Iron Mak. Steel Mak. 17 (1990) 139±143. [6] W. Lehnert, N.D. Cuong, Integrated model for calculating microstructural and forming parameters of steel during rolling in continuous mills, ISIJ Int. 32 (1995) 1100±1108. [7] Y. Lee, Calculating model of mean strain in rod rolling process, ISIJ Int. 39 (1999) 961±963. [8] Y. Lee, H.J. Kim, S.M. Hwang, Analytic model for the prediction of mean effective strain in rod rolling process, J. Mater. Process. Technol. 114 (2001) 81±93. [9] Y. Lee, S. Choi, P.D. Hodgson, Analytical model of pass-by-pass strain in rod (or bar) rolling and its applications to prediction of austenite grain size, Mater. Sci. Eng. A 336 (2002) 177±189. [10] K. Karhausen, R. Kopp, M.M. De Souza, Numerical simulation method for designing thermomechanical treatments, illustrated by bar rolling, Scan. J. Metall. 20 (1991) 351±363. [11] R. Kuziak, M. Glowacki, M. Pietrzyk, Modelling of plastic flow, heat transfer and microstructural evolution during rolling of eutectoid steel rods, J. Mater. Process. Technol. 60 (1996) 589± 596. [12] J. Yanagimoto, T. Ito, J. Liu, FE-based analysis for the microstructure evolution in hot bar rolling, ISIJ Int. 40 (2000) 65±70.
[13] S. Shida, Empirical formula of flow stress of carbon steelsÐ resistance to deformation of carbon steels at elevated temperature, 2nd Report, J. JSTP 10 (1969) 610±617 (in Japanese). [14] J.D. Campbell, Dynamic Plasticity of Metals, Springer, Udine, 1970. [15] W.S. Lee, G.W. Yeh, The plastic deformation behavior of AISI4340 alloy steel subjected to high temperature and strain rate loading conditions, J. Mater. Process. Technol. 71 (1997) 224±234. [16] Y. Lee, S. Choi, Y.H. Kim, Mathematical model and experimental validation of surface profile of a workpiece in round±oval±round pass sequence, J. Mater. Process. Technol. 81 (2000) 87±96. [17] Z. Wusatowski, Fundamentals of Rolling, Pergamon Press, London, 1969, pp. 107±109. [18] Y. Lee, An analytical study of mean roll radius in rod rolling, ISIJ Int. 41 (2001) 1416±1418. [19] G.I. Taylor, H. Quinney, The latent energy remaining in a metal after cold working, Proc. R. Soc. London A 143 (1934) 307±326. [20] P. Braun-Angott, H. Roholff, A method for the calculation of temperature during hot rolling of bar and rod, Steel Res. 64 (1993) 350±355. [21] B. Carnahan, H.A. Luther, W.O. Wilkes, Applied Numerical Methods, Wiley, New York, 1979, pp. 466±471. [22] W.C. Chen, I.V. Samarasekera, E.B. Hawbolt, Fundamental phenomena governing heat transfer during rolling, Metall. Trans. A 24 (1993) 1307±1320. [23] The British Iron and Steel Research Association, Physical Constants of Some Commercial Steels at Elevated Temperatures, Butterworths, Guildford, Surrey, UK, 1953, pp. 3±14. [24] R.D. Morales, M. Toledo, A.G. Lopez, A.C. Tapia, C. Petersen, Mathematical model for on-line heat treatment of steel bars, Steel Res. 62 (1991) 433±440. [25] R.D. Morales, A.G. Lopez, I.M. Olivares, Heat transfer analysis during water spray cooling of steel rods, ISIJ Int. 30 (1990) 48±59. [26] J.G. Lenard, M. Pietrzyk, L. Cser, Mathematical and Physical Simulation of the Properties of Hot Rolled Products, Elsevier, Netherlands, 1999, pp. 73±75.
Integrated model for thermo-mechanical controlled process in rod (or bar) rolling Y. Leea,*, S. Choia, P.D. Hodgsonb a
Plate, Rod and Welding Group, POSCO Technical Research Laboratories, Pohang 790-785, South Korea School of Engineering and Technology, Deakin University, Pigdons Road, Geelong, Vic. 3217, Australia
b
Received 12 December 2001; received in revised form 8 February 2002; accepted 8 February 2002
Abstract This study presents an integrated model for computing the thermo-mechanical parameters (cross-sectional shape of workpiece, the pass-bypass strain and strain rate and the temperature variation during rolling and cooling between inter-stands) and metallurgical parameters (recrystallisation behaviour and austenite grain sizeÐAGS), to assess the potential for developing ``Thermo-Mechanical Controlled Process'' technology in rod (or bar) rolling, which has been a well-known technical terminology in strip (or plate) rolling since 1970s. The advantage of this model is that metallurgical and mechanical parameters are obtained simultaneously in a short computation time compared with other models. The model has been applied to a rod mill to predict the exit cross-sectional shape, area and AGS per pass by incorporating the equations for AGS evolution being used in strip rolling. At the ®nishing train of rod mills, the strain rates reach as high as 1000±3000 s 1 and the inter-pass times are around 10±60 ms. The results show that the proposed model is an ef®cient tool for evaluating the effects of process-related parameters on product quality and dimensional tolerance of the products in rod (or bar) rolling. The results of the simulation demonstrated that the equation for AGS evolution being used in strip rolling might have limitations when applied directly to rod rolling at a high strain rate. # 2002 Elsevier Science B.V. All rights reserved. Keywords: Rod (or bar) rolling; Thermo-mechanical controlled process; Integrated model; Numerical simulation; Austenite grain size (AGS); High strain rate
1. Introduction Signi®cant advances have been made in predicting the thermo-mechanical parameters (cross-sectional shape, strain, strain rate and temperature) and the metallurgical parameters, during hot rolling of material to control the mechanical properties of the products as well as the dimensional tolerances. Most of previous works [1±3], however, have concentrated on the strip (or plate) rolling process because the thermo-mechanical parameters can be determined using the assumption of plane strain deformation. Little research has been undertaken for rod (or bar) rolling since the workpiece between the grooved rolls is neither in the condition of plane strain nor plane stress. There is a strong demand to develop a mathematical model that is reliable, accurate and non-iterative to obtain the thermo-mechanical parameters associated with rod rolling *
Corresponding author. Tel.: 850-54-220-6058; fax: 850-54-220-6911. E-mail address: [email protected] (Y. Lee).
process. The reason is that once these parameters are obtained, then the recrystallisation behaviour and austenite grain size (AGS) evolution model developed for strip rolling can be applied directly to rod rolling. There have been several attempts [4±6] to do this. Maccagno et al. [4], in a study of the AGS evolution associated with rod rolling sequence, calculated the strains by multiplying the area strains by a constant factor. Strains were obtained by taking the natural logarithm of the ratio of the fractional reductions in cross-sectional area through the rolling stands. The strain at a pass was then assumed to be 1.7 times the area strain for the roughing stands, and 2.5 times the area strain for the ®nishing stands. Kemp [5] proposed that the strain per pass should be factors of 1.5± 2 times the area strains in the roughing stands and factors of 2±3 times the area strains in all subsequent stands. In the work of Maccagno et al. and Kemp, however, a mathematical rationale for the use of the constant factors was missing. Lehnert and Cuong [6] suggested a model that calculates the strain, strain rate in rod rolling and AGS, based on the assumption of plane strain deformation. The three-dimensional
0924-0136/02/$ ± see front matter # 2002 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 4 - 0 1 3 6 ( 0 2 ) 0 0 3 4 7 - 3
Y. Lee et al. / Journal of Materials Processing Technology 125±126 (2002) 678±688
deformation zone was subdivided into longitudinal strips of equal width to roll gap direction and each strip analysed separately. No experimental veri®cation, however, was provided to support the assumption. This approach might be applicable once the exit cross-sectional shape at a pass can be correctly predicted. However, a method to obtain the exit cross-sectional shape was not provided. Recently, Lee [7] has proposed an approximate model that calculates strain at a given pass in a round±oval (or oval± round) pass rolling sequence. The proposed model is based on the elementary theory of plasticity and the equivalent rectangle approximation method. Lee et al. [8] also performed FE simulation to show the validity of the approximate model. They compared the strains calculated from the approximate model with those from FE model. The strain in the FE simulation was de®ned as the averaged effective plastic strain at the exit of the roll gap (at the pass). Lee et al. [9] also veri®ed the approximate model experimentally by conducting the tensile test of the rolled specimen and compared the predicted AGS for a four-pass bar rolling sequence with measured AGS from hot torsion test. These approaches [7±9] based on the ``average'' concept have some limitations since it does not show the ``distribution'' over the cross-section, such as the strain, strain rate and AGS. Karhausen et al. [10], Kuziak et al. [11] and Yanagimoto et al. [12] have presented three-dimensional FE analysis (FEA) for the microstructure evolution in hot bar rolling, coupled with the experiment-based AGS prediction model. FEA is very effective for computing the distribution of effective plastic strain, strain rate and temperature across the section as well as predicting the surface pro®le of deformed workpiece and AGS. However, this approach requires at least half an hour to run FE code for a single pass since three-dimensional analysis is required. Thus, considering computational time for the bar (or rod) mills that consists of a large number of passes (stands) and the complicated mechanical/thermal boundary conditions (the friction condition at the roll/material interface and heat transfer coef®cients dependent on the temperature and contact pressure), the ``average'' model still remains as a useful tool to calculate the thermo-mechanical parameters and consequently AGS per pass. This study presents a systematic procedure for computing the thermo-mechanical (cross-sectional shape of workpiece, strain, strain rate and temperature variation during rolling and cooling between inter-stands) and metallurgical parameters (recrystallisation behaviour and AGS) for a rod mill where the strain rates can be as high as 3000 s 1 (in the ®nishing stands) and the associated deformation times are of the order of 1 ms, and the inter-pass times are around 15± 60 ms. At the ®nishing train of the rod mill, there is signi®cant heat generation due to the high strain rate plastic deformation of the workpiece. The prediction of this relies on an accurate representation of the constitutive behaviour under the imposed deformation conditions. In the present
679
study, it was assumed Shida's constitutive equation [13] could be used to calculate the plastic deformation of materials at high strain rates. The validity of this assumption has been examined using split Hopkinson-Pressure Bar (SHPB) test technique [14], which measures the stress±strain relation of material at high strain rate (1000±4000 s 1). The high temperatures at SHPB test were obtained by enclosing the specimen in a clam-shell radiant-furnace [15]. There has been some debate regarding the AGS evolution during rod rolling at very high strain rates. It is unlikely that this will be the same as in plate and strip rolling. This paper presents an attempt to predict the AGS under high strain rate deformation in the ®nishing train and the effect of watercooling in the cooling zone and equalisation zone between stands (passes). A one-dimensional heat transfer approach together with an equivalent circular approximation method of the workpiece at each pass was used to calculate the temperature evolution of the material during rolling and cooling. 2. Modelling of thermo-mechanical parameters The following ®ve sub-sections give the procedure to calculate the thermo-mechanical parameters (strain and strain rate per pass and temperature evolution of material during rolling). 2.1. Prediction of cross-sectional shape To determine the pass-by-pass strain and strain rate, it is necessary to ®rst predict the stress-free surface pro®le of material that does not contact the roll directly at the exit of roll gap. Recently, Lee et al. [16] developed an analytical model, which is robust and non-iterative in computation, to predict the surface pro®le of the exit cross-section of a workpiece for the oval±round (or round±oval) pass sequence most widely used pass sequence in present rod (or bar) mills. The advantage of this model is that only the geometric consideration is required in the formulation. Hence, it greatly simpli®es the problem of obtaining the ®nal rolled shape. 2.2. Determination of equivalent rectangular cross-section Since the cross-sectional shape of the deformed workpiece at a pass is not rectilinear, one encounters dif®culties in formulating the equation for the calculation of the strain within a frame of Cartesian co-ordinates. To overcome this problem, the equivalent rectangle approximation method that transforms the non-rectangular cross-section into a rectilinear one was employed. There are many ways to achieve this objective while the net cross-sectional area is maintained. Maintaining the cross-sectional area is presumed to be a necessary, but not a suf®cient requirement for equivalence. There are generally three methods to establish equivalent rectangular
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Fig. 1. (a) Schematic description of round±oval pass rolling (front view). (b) Application of equivalent rectangle approximation method to round±oval pass rolling. G represents the roll gap and Reff the effective roll radius. H i and H p indicate incoming and outgoing equivalent cross-sectional height. Incoming and outgoing equivalent cross-sectional widths are represented by W i and W p .
sections; namely, method of maximum height, method of maximum width and the method of width±height ratio. The effective (or mean) roll radius necessary in the calculation of the rolling speed at a pass has been calculated from the method of maximum width [17,18]. Therefore, in this study, the cross-sections are approximated into rectangles using the method of maximum width. Fig. 1 shows an example of the method of maximum width applied to the round±oval pass rolling, in which an incoming stock is of a round shape and stock with an oval shape is produced. Sections A, B and C, respectively, correspond to the positions where a section of the undeformed workpiece is about to be rolled, being deformed and that of the deformed workpiece leaving the roll groove. The equivalent rectangle approximation described in Fig. 1 can also be applied to another type of passes such as oval±round, box±oval pass and diamond±square pass, etc. 2.3. Strain at a pass (pass-by-pass strain) The strain de®ned as the maximum average effective plastic strain at a given pass can be then calculated from the rectilinear shape transformed. The calculation, however, should include the non-linear change of draught, spread and elongation of the material being deformed. The assumption introduced to overcome this problem is the hypothesis of parallelepiped deformation [17]. According to this hypothesis, a cube of material subjected to a load will change to a rectangular prism and its angle and sides will remain orthogonal with this prior to deformation. It should be noted that the calculation is focused on bulk deformation of the workpiece. Derivation procedure for the
equation that computes the strain at a pass is described in Refs. [7±9]. 2.4. Strain rate at a pass (pass-by-pass strain rate) As in drawing and forging processes, the strain rate in bar (or rod) rolling changes at various stages of deformation. The strain rate is a maximum at the entrance to the roll (or in its vicinity) and decreases along the roll bite, ®nally becoming zero at the outlet. For this reason, it is necessary to introduce an ``average'' strain rate for a given pass. The ``average'' strain rate can be de®ned as the strain over a time interval which can be calculated from: ep e_ p ; tp
(1)
where tp represents the time interval taken for a section A to pass through to a section C (Fig. 1). Hereafter the ``average'' strain rate is referred to as the strain rate. The time interval can be expressed as tp
60L s; 2pNReff
(2)
where Reff, Lp and N are the effective roll radius, the effective projected contact length of the grooved roll and workpiece, and roll rpm at a given pass, respectively. The effective roll radius at a given pass is calculated by the method of maximum width [17,18]. The effective projected contact length is expressed as s Hp G L Rmax (3) H i H p : 2
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2.5. Temperature evolution during rod (or bar) rolling The temperature of the workpiece during rolling depends on various factors such as rolling speed, initial temperature of the billet, plastic deformation of the workpiece, the crosssectional shape of the workpiece at each pass, cooling condition in the individual passes and distribution of cooling and equalisation zone between stands (passes). To take care of the combined actions of these parameters, the model for the temperature evolution of the workpiece during rod (or bar) rolling has been formulated based on the following assumptions: (i) Uniform initial temperature of the billet (or workpiece). (ii) No longitudinal temperature gradient (i.e., infinitely long rod). (iii) Uniform heat generation across the cross-section of workpiece due to plastic deformation in the roll gap. (iv) Circular cross-sectional shape at each pass. Assumptions (i) and (ii) are typical for this type of problem and assumption (iii) is introduced because the strain was de®ned as maximum average effective (equivalent) plastic strain at a given pass and the strain rate as the strain divided by the time required for the workpiece to undergo this strain in the roll gap. Finally, the background for assumption (iv) is explained as follows. The workpiece deformed at each pass is not of a circular cross-sectional shape (even at round pass). In this case, the heat transfer model that should be solved is a two-dimensional problem with a curved geometric boundary condition. At present, there is no systematic procedure for solving such problems using the ®nite difference method. One way to overcome this dif®culty is to employ the equivalent circle approximation method that transforms a non-circular crosssectional shape of workpiece into a circular one while the net cross-sectional area is maintained. The problem can be then reduced to an axi-symmetric heat transfer problem. Under these assumptions, temperature variation within the rod is governed by the following one-dimensional axi-symmetric heat transfer equation: @ @T @T k ; (4) q_ rCp @r @r @t where t is the time that takes for an volume element to travel the distance in the rolling direction and r the radius of the rod. k, r and Cp are thermal conductivity, density and speci®c heat of the material, respectively. Note that q_ is the volumetric rate of heat generation within the rod due to the plastic deformation of the workpiece in the roll gap and can be expressed as: q_ w
se_ ; rbCp
(5)
681
where w and b (4.185 kJ/kcal) are the faction of plastic deformation work converted into heat generation and the mechanical equivalent of heat, respectively. s is the ¯ow stress of the workpiece. Typically w 0:9 is used [19]. It was assumed the heat generated due to friction at the interface of roll and workpiece was compensated by heat loss due to the roll coolant applied on the workpiece as well as the work rolls. (In rod rolling process, the coolant on work roll is not wiped out but drops on the workpiece during rolling.) Solution of Eq. (4) is not possible analytically because of the variation of k, Cp and q_ with temperature. Hence, an implicit ®nite difference technique was applied to Eq. (4). The solution method of Eq. (4) using the ®nite difference method with boundary conditions is well described in [20,21]. The values of k and Cp have been taken to be a function of the temperature of the material and were acquired from the literature [22,23]. The density of material was assumed to be constant over the temperature range of interest. To predict the temperature variation of workpiece under the action of water-cooling in the cooling zone and equalisation zone between stands (passes), we followed Morales et al.'s approach [24,25]. They developed a mathematical model to predict the cooling behaviour of steel rods under the action of water-cooling in rod mill. They surveyed the various water-cooling (spray, jet, mist) models and examined the in¯uence of operation parameters such as rod size, rolling speed, rod temperature and water ¯ow-rate on the temperature distribution within the rod before it enters into the transformation conveyer. 3. Constitutive equation at high temperature and strain rate It is well known that signi®cant heat generation occurs due to the high strain rate plastic deformation of the workpiece in the ®nishing train of rod mill. Thus, the constitutive equation of material plays a crucial role in computing the temperature evolution of workpiece during rod rolling. In the present study, it was assumed that Shida's constitutive equation [13] could predict the ¯ow stress of materials subjected to high strain rates. Shida's constitutive equation gives the ¯ow stress of carbon steels as a function of the strain, the strain rate, temperature and the carbon content. It is applicable in the range of carbon content: 0.07±1.2%, temperature: 700±1200 8C and strain: up to 0.7, but might not be applicable when the range of strain rate goes beyond 100 s 1. It takes account of the ¯ow stress behaviour of the steels in austenite, ferritic and in the two-phase regions. This is a strength of Shida's equation in comparison with the other constitutive equations [26]. Shida's main dif®culty in developing the constitutive equation in high strain rate lay in the lack of experimental facilities to provide data for the plastic deformation of materials under very high strain rates.
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The validity of this model has been examined using the SHPB test [14], which measures the stress±strain relation of the material at very high strain rate (1000±4000 s 1). The high temperatures at SHPB test in this study were obtained by enclosing the specimen in a clam-shell radiant-furnace. This test was conducted at Department of Mechanical Design and Production Engineering of Yonsei University in Korea. Fig. 2 shows the measured and predicted stress± strain curves of a low carbon steel (0.1%C) at different temperatures and similar strain rates. In all cases, the measured ¯ow stress is higher than predicted one, although the general shapes of the predicted and measured curves are similar. The difference between the predicted and measured stresses is approximately 20%. For the current work, it was assumed that the Shida's equation could be applied with a 20% increase. However, it is also clear that much more work is required in this area.
4. Application The recrystallisation model and AGS evolution model generally used in hot strip rolling together with the thermomechanical parameters (strain, strain rate at a pass and temperature of workpiece during rolling) have then been applied to POSCO no. 3 rod mill to simulate the variation of AGS of workpiece during rolling. The equations used in this study are listed in Appendix A. The average temperature at each pass was taken as the representative temperature of the workpiece. In the simulation, the thermo-mechanical model calculates the thermo-mechanical parameters of the workpiece during rolling once the information regarding the rolling schedule (roll groove geometry and inter-pass time), initial temperature and initial size of the workpiece are given. This creates an output ®le of the thermo-mechanical parameters
Fig. 2. Flow stress±strain curves of low carbon steels (0.1%C) at different temperatures.
Y. Lee et al. / Journal of Materials Processing Technology 125±126 (2002) 678±688
683
Fig. 3. Schematic diagram of the integrated thermo-mechanical processing model for rod (or bar) rolling.
for the whole process. The metallurgical model then picks up this ®le and computes recrystallisation behaviour and AGS evolution for the whole process. This procedure is described in Fig. 3. 5. Results and discussion 5.1. Exit cross-sectional shape and temperature history of workpiece during rolling Fig. 4 illustrates the predicted surface pro®les of exit cross-section and those measured for the seven stands
(passes) in the roughing train (nos. 1±7) of POSCO no. 3 rod mill. The predicted exit cross-sectional shape slightly overestimates that measured at pass no. 1. At pass no. 2, the measured surface pro®le is in good agreement with the predicted one. At the round pass, i.e., pass nos. 3, 5 and 7, skewed round pro®les are observed (Fig. 4(c), (e) and (g)). This means that the workpiece was not rotated exactly 908. It was twisted more than 908 or less than 908. This kind of problem occurs easily in horizontal±horizontal type rolling sequence. Meanwhile, at the oval pass rolling, a relatively stable rolling is observed, as shown in Fig. 4(b), (d) and (f). Overall, the predicted surface pro®les of exit cross-section are in agreement with those measured.
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Fig. 4. Predicted and measured surface profiles and areas of the exit cross-section for the roughing train in POSCO no. 3 rod mill.
Y. Lee et al. / Journal of Materials Processing Technology 125±126 (2002) 678±688
685
5.2. Evolution of AGS
Fig. 5. Time±temperature history of workpiece during rolling in POSCO no. 3 rod mill: (a) roughing train; (b) intermediate finishing train (#14±19) and finishing train (#20±29).
Fig. 5 shows the predicted temperature history and measured ones for two types of carbon steels (0.67 and 0.80%C). Due to lack of number of pyrometers installed in the rod mill, comparison was only carried out at limited locations. Fig. 5(a) shows that at pass no. 2 (designated as #2), the measured surface temperatures are lower than those predicted. Meanwhile, opposite results were observed at after #11. Fig. 5(b) demonstrates the heat generation attributable to the high strain rate plastic deformation in the intermediate ®nishing train (#14±19) and ®nishing train (#20±29). Measured temperatures at before #20 and after #29 are higher than the predicted ones. However, the general level of agreement is encouraging the given dif®culty.
For rod mills, the strain rates in the ®nishing stands can be as high as 1000±3000 s 1. The associated deformation times are of the order of 1 ms and the inter-pass times are around 10±60 ms. This has raised questions regarding the AGS evolution that takes place during rolling associated with high strain rates. It is unlikely these will be the same as in plate and strip rolling. This paper presents a simulation to study whether we can apply directly the equation for AGS evolution generally used in strip rolling to rod rolling. Table 1 shows AGS per pass for the entire process when the ®nish rolling speed is 100 m/s for the production of 5.5 mm diameter rod. The static recrystallisation occurred in four of the last 10 stands at the ®nishing train. It seems that material experienced work hardening at the four stands (passes). It can be deduced that the critical strain triggering meta-dynamics recrystallisation (MDRX) was larger than the calculated strain at six stands of the last 10 stands at the ®nishing train. This result is quite unusual because other work [4] has suggested that the higher strain rate in the ®nishing train of rod (or bar) mills favour meta-dynamic recrystallisation. In passes 21 and 29, there was a large change in the fraction of statically recrystallised material. A small amount of the RXN fraction is attributable to very short inter-pass time and small strain at passes 21 and 29, compared with others. It should be noted that the fraction of static RXN depends strongly on incubation time, i.e., interpass time and strain at a pass. The smaller applied strain the larger t50. The larger t50 compared with the inter-pass time leads small amount of fraction of static RXN. (See equations in Appendix A.) Table 2 shows AGS evolution when the water-cooling during rolling is in operation. When the material is processed with water-cooling, the AGS becomes smaller, as expected. However, in this case, it is observed that static recrystallisation take places at nine stands of the ®nishing train. When the model predicts static recrystallisation, there are some inter-pass periods where low fractions of recrystallisation occur. For meta-dynamic recrystallisation, there is always complete recrystallisation in the following inter-pass period. At this stage, it is unclear whether it is reasonable to expect partial recrystallisation at some passes in the ®nishing train of rod (or bar) rolling process. Hence, further work is required to develop appropriate models for these types of deformation conditions. It is likely to apply to both the static and meta-dynamic recrystallisation reactions. The actual grain sizes are also an area for further study, which should include both laboratory and direct measurement of the AGS of the material during rolling. It is clear that the complete integrated model developed here provides an important framework for the further development of thermomechanically controlled process technology in rod (or bar) rolling and as a tool to direct future fundamental research in this ®eld.
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Table 1 Thermo-mechanical and metallurgical parameters during rod rollinga Std. no.
Strain
Strain rate (s 1)
Temperature (8C)
RXN fraction (%)
Mode
IPT (s)
AGS (mm)
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29
0.347 0.435 0.461 0.491 0.320 0.488 0.354 0.513 0.398 0.514 0.437 0.454 0.436 0.503 0.471 0.480 0.462 0.439 0.461 0.577 0.416 0.440 0.434 0.435 0.477 0.453 0.467 0.348 0.383
0.36 0.63 0.78 1.37 1.26 2.59 2.53 5.01 5.00 9.34 10.85 17.48 19.12 38.47 41.00 73.58 78.80 118.69 129.12 317.44 387.60 475.46 503.72 795.32 1061.86 1684.12 1868.80 2392.93 3094.85
1049.9 1014.8 986.0 957.2 962.6 939.4 937.3 934.7 896.6 931.4 900.8 921.1 898.7 893.4 898.6 913.2 924.7 934.9 947.1 944.4 953.0 963.5 971.8 986.8 999.2 1012.5 1025.5 1035.6 1049.0
1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 0.985 1.000 0.992 0.985 0.990 1.000 0.248 1.000 1.000 1.000 1.000 1.000 1.000 0.943 0.500
MDRX MDRX MDRX MDRX MDRX MDRX MDRX MDRX MDRX MDRX MDRX MDRX MDRX MDRX SRX SRX SRX SRX SRX SRX SRX MDRX MDRX MDRX MDRX MDRX MDRX SRX SRX
5.800 27.220 33.237 15.923 11.578 39.634 6.676 5.472 4.041 3.294 2.473 6.024 1.551 8.781 0.591 0.676 0.521 0.426 0.343 3.002 0.059 0.049 0.039 0.032 0.025 0.030 0.016 0.013 0.011
62.11 47.65 40.37 31.04 31.65 29.90 24.52 22.48 17.87 20.24 16.05 20.65 15.28 18.94 17.82 20.99 20.93 21.35 22.02 31.41 21.65 8.19 8.32 8.12 8.06 7.75 7.99 19.75 16.72
a
IPT: inter-pass time; finish rolling speed: 100 m/s; size: 5.5 mm diameter.
Table 2 Thermo-mechanical and metallurgical parameters during rod rolling when water-cooling between stands is ona Std. no.
Strain
Strain rate (s 1)
Temperature (8C)
RXN fraction (%)
Mode
IPT (s)
AGS (mm)
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
0.347 0.435 0.461 0.491 0.320 0.488 0.354 0.513 0.398 0.514 0.437 0.454 0.436 0.503 0.471 0.480 0.462 0.439 0.461 0.577 0.416 0.440 0.434 0.435 0.477
0.36 0.63 0.78 1.37 1.26 2.59 2.53 5.01 5.00 9.34 10.85 17.48 19.12 38.47 41.00 73.58 78.80 118.69 129.12 317.44 387.60 475.46 503.72 795.32 1061.86
1049.9 1014.8 986.0 957.2 962.6 939.4 937.3 934.7 896.6 931.4 900.8 921.1 898.7 800.9 877.9 845.6 825.7 838.8 848.9 796.5 804.8 815.6 825.0 842.5 857.9
1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 0.971 0.930 0.942 0.985 0.939 1.000 0.165 0.470 1.000 0.814 0.792
MDRX MDRX MDRX MDRX MDRX MDRX MDRX MDRX MDRX MDRX MDRX MDRX MDRX SRX SRX SRX SRX SRX SRX SRX SRX SRX MDRX SRX SRX
5.800 27.220 33.237 15.923 11.578 39.634 6.676 5.472 4.041 3.294 2.473 6.024 1.551 8.781 0.591 0.676 0.521 0.426 0.343 3.002 0.059 0.049 0.039 0.032 0.025
62.11 47.65 40.37 31.04 31.65 29.90 24.52 22.48 17.87 20.24 16.05 20.65 15.28 16.70 14.91 10.55 8.41 10.22 9.49 14.05 10.90 5.61 3.48 5.20 5.66
Y. Lee et al. / Journal of Materials Processing Technology 125±126 (2002) 678±688
687
Table 2 (Continued ) Std. no.
Strain
Strain rate (s 1)
26 27 28 29
0.453 0.467 0.348 0.383
1684.12 1868.80 2392.93 3094.85
a
Temperature (8C) 875.4 894.2 908.6 927.1
RXN fraction (%)
Mode
IPT (s)
0.801 0.792 0.560 0.862
SRX SRX SRX SRX
0.030 0.016 0.013 0.011
AGS (mm) 6.29 6.94 6.49 8.65
Water-cooling tubes are placed after Std. no. 13 and after Std. no. 19. IPT: inter-pass time; finish rolling speed: 100 m/s; size: 5.5 mm diameter.
strip rolling, and a proper constitutive equation describing the deformation behaviour of the material at high strain rate (up to 3000 s 1) and high temperature is available. However, the results of the simulation demonstrate that the equations for AGS evolution, based on those used in strip rolling, may have limitations when applied directly to rod rolling where material experiences very high strain rates in the ®nishing train.
6. Conclusions An integrated model has been developed which computes the thermo-mechanical (cross-sectional shape of workpiece, the pass-by-pass strain and strain rate and the temperature variation during rolling and cooling between inter-stands) and metallurgical parameters (recrystallisation behaviour and AGS) during high speed rod rolling. This might have the potential to be used for developing ``Thermo-Mechanical Controlled Process'' technology in rod (or bar) rolling. This study demonstrates that, on an industrial scale, a quantitative description of the thermo-mechanical and metallurgical parameters of the material during each stage of rod (or bar) manufacturing process is possible if the proposed model can be mutually integrated with the recrystallisation model and AGS evolution model being used in
Acknowledgements One of authors (Y. Lee) would like to acknowledge K.J. Park and Professor O.K. Min (Yonsei University, Korea) who performed the split Hopkins-Pressure Bar test at high temperature.
Appendix A Equations and constants used in numerical simulation [4]. Model
Equation
Parameters
Critical strain
ec 5:6 10 4 do0:3 Z 0:17 n t x 1 exp 0:693 t0:5 QRXN p q t0:5 to e do exp RT Qdef Z e_ exp RT " 1:5 # t x 1 exp 0:693 t0:5 Qmd t0:5 kmd Z nmd exp RT
do 250 mm
Strain recrystallisation
Zener±Hollomon parameter Meta-dynamic recrystallisation
Recrystallised grain size Static Meta-dynamic Grain growth
dSRX Ae
a b do
dMDRX kZ
exp
45000 8:31T
n1 to 2:3 10 15 , p 2:5, q 1, QRXN 230 kJ/mol Qdef 300 kJ/mol, R 8:31
kmd 1:1, nmd 0:8, Qmd 230 kJ/mol
A 343, a 0:5, b 0:4
0:23
k 2:6 104
Static (if tip > 1 s)
m dm dSRX k tip
Meta-dynamic (if tip > 1 s)
m dm dMDRX k tip
400000 8:31T 400000 2:65t0:5 exp 8:31T
4:32t0:5 exp
m 7, k 1:5 1027 m 7, k 8:2 1025
688
Y. Lee et al. / Journal of Materials Processing Technology 125±126 (2002) 678±688
Appendix A. (Continued ) Model
Equation
Static (if tip < 1 s)
m d m dSRX k tip
Meta-dynamic (if tip < 1 s)
m d m dMDRX k tip
Partial RXN
113000 8:31T 113000 2:65t0:5 exp 8:31T
4:32t0:5 exp
Parameters m 2, k 4:0 107 m 2, k 1:2 107
4=3
doi1 Xi dRXi 1 Xi 2 doi , eaccuml ei1 1 Xi ei i1
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