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Investigation of interfacial behaviors between the strip and roll in hot strip rolling by finite element method
Tribology International 38 (2005) 413–422 www.elsevier.com/locate/triboint
Investigation of interfacial behaviors between the strip and roll in hot strip rolling by finite element method Cheng Gang Sun* POSCO Technical Research Laboratories, Stainless Steel Research Group, Pohang 790-785, South Korea Received 25 November 2003; received in revised form 30 July 2004; accepted 14 September 2004 Available online 23 November 2004
Abstract A finite element (FE)-based approach is presented for the precision analysis of the interfacial thermo-mechanical behavior of the roll and strip in the entire tandem mill during hot strip rolling. The validity of the proposed model is examined through the comparison with measurements. Then, the effect of various process parameters on the detailed aspects of interfacial thermo-mechanical behavior of the roll and strip is investigated via a series of process simulations. q 2004 Elsevier Ltd. All rights reserved. Keywords: Finite element method; Roll; Strip; Hot strip rolling; Interfacial thermo-mechanical behavior
1. Introduction In hot strip rolling, a hot slab is passed through the roll gap several times, with its thickness being progressively reduced to achieve the final strip dimensions. In order for precision control of the product quality as well as to enhance the production economy in most modern rolling process, the interfacial thermo-mechanical behavior between the strip and roll, which may be strong dependent on frictional behavior and heat transfer at the interface, must be well understood. Incorrect information related to these two phenomena may lead to the problems in the prediction of roll force, strip shape and flatness, and strip temperatures in the hot rolling process. Precise modeling such as interfacial behavior, however, is a difficult task. In the light of heat transfer at contact surface, it is not only dependent on the process parameters in terms of temperature, reduction ratio and speed, but also on the surface geometry and properties such as hardness and roughness of the roll and strip. As a result, a simplified expression was generally assumed for the interface heat transfer coefficient (hlub). Through the past decades, there had been a lot of investigations to * Tel.: C82 54 220 6293; fax: C82 54 220 6915. E-mail address: [email protected] 0301-679X/$ - see front matter q 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.triboint.2004.09.002
determine the hlub, however, a noticeable variation was observed due to the widely varied experiment conditions, for example, 2–20, 10–50, 23–81 and 200 kW/m2K in Refs. [1–4], respectively. Such evaluations restricted to a limited laboratory condition may inevitably lead to the lower accuracy in prediction of hlub in the industry rolling process. Recently, based on the published hlub between two nominally flat, but microscopically rough surface and its relationship to the surface roughness, fractional contact area of two surfaces and the interfacial pressure, Hlady et al. [5] developed a model that described hlub as a function of the rolling pressure, surface hardness, as well as of the harmonic conductivity and surface roughness parameters of the roll and strip. A suitable application of such model for the estimation of hlub in the industry hot rolling process was recently made by Sun et al. [6] As far as the coefficient of friction was concerned, it was generally considered to be strongly dependent on process and material parameters according to the adhesion hypothesis [7] in hot rolling process. In the sense of process parameters, the relative velocity between the roll and strip, the surface temperature at the entry and the reduction ratio are primly important; on the other hand, the material parameters related to the contact surfaces may include the deformation resistance of
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Nomenclature f h hlub hw , he k n p T Te Tw 3ij 3 0 ij 3 3_ 3_ij 3_ij0 u un, u t
body force vector traction vector interface heat transfer coefficient convective heat transfer coefficient thermal conductivity, bulk modulus outward unit normal vector hydrostatic pressure temperature environment temperature temperature of the flowing medium strain tensor deviatoric strain tensor effective strain effective strain rate strain rate tensor deviatoric strain rate tensor velocity vector, displacement vector normal and tangential components of u
the strip, the roughness of the strip and roll and the thickness of the scale, etc. When the lubricants are used, the viscosity and its dependence on the temperature and pressure should also to be taken into account. As a result, though remarkable investigations had been made to explore the individual effect of the reduction [8], rolling speed [9], temperature [10], as well as material’s resistance and thickness of scale layer [11] on coefficient of friction in the past, the functional dependence of the process and material parameters had not yet been presented until the recent work by Lenard and Leon [12], who developed an empirical model accounting for the aforementioned dependence based on the experiment, in which the growth of the scale and thickness of the layer of scale were properly controlled. Although the friction and heat transfer at the interface are disparate phenomena, there may exist a definite relationship between them, which can be summarized as follows: coefficient of friction is influenced by the temperature distributions of strip at the interface, since the deformation resistance is strongly dependent on temperatures. On the other hand, temperature distributions in the strip are affected by heat generation due to interfacial friction, resulting in strong correlations between the mechanical and the thermal behaviors of the strip and roll at the contact surface. However, very few studies have been undertaken on the investigation of the interdependence between the friction and heat transfer at the contact surfaces. In this paper, the individual interfacial models were well incorporated into finite element models to constitute an integrated process model for demonstrating the interfacial thermo-mechanical behavior of strip and roll. The validity of the proposed
sij s sn st s 0 ij dij m x U G rc s3 q ur H1, H2 R
stress tensor flow stress normal stress tangential stress deviatoric stress tensor kronecker delta coefficient of Coulomb friction penalty constant analysis domain boundary of the analysis domain heat capacity per unit volume Stefan–Bolzmann constant multiplied by emissivity heat flux per unit area roll speed entry and exit thickness of the strip roll radius
model was examined through comparison with measurements. Then, a series of process simulation were conducted to investigate the effect of the process parameters on the interfacial behavior.
2. Finite element (FE) process models 2.1. FE model for analysis of rigid-viscoplastic deformation A steady-state, rigid-viscoplastic deformation behavior of the strip was adopted in the present investigation. Equation of motion (neglecting all the acceleration terms): sij;jCfiZ0
(1)
Constitutive relationship: sij Z Kpdij C sij0
(2)
2s 3_ 33_ ij
(3)
sij0 Z
where the flow stress s of the strip deforming at elevated temperatures may be represented by s Z f ð3; 3_; TÞ
(4)
Incompressibility condition: ui;i Z 0
(5)
Traction boundary conditions: sij nj Z hi ;
on Ghi
(6)
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415
Velocity prescribed boundary conditions: ui Z u i ;
on Gui
(7)
Boundary conditions at the roll–strip interface— I. Normal stress. A penalty algorithm was applied for the determination of the normal stress distributions at the roll–strip interface. For this purpose, the velocity prescribed boundary condition un Z u D n;
on Gc
(8)
was replaced by sn Z Kxðun K u D n Þ;
on Gc
(9)
where sn denotes the normal stress, un and uD n (Z0) denote the normal component of the velocity vector of the strip and that of the roll, respectively. Considering that the magnitude of sn should be finite in order to maintain stress equilibrium, it is evident that the contact condition represented by Eq. (9) is strictly enforced as x becomes very large. Boundary conditions at the roll–strip interface— II. Tangential stress. The tangential stress distributions at the roll–strip interface due to friction may be modeled by the Coulomb friction model. It should be noted that the sticking zone is not known a priori before solving the boundary value problem. An approach that may be conceived is to develop and apply a special trial and error procedure to locate the sticking zone. Another possible approach, which was adopted in this investigation, is to replace the Coulomb friction model by a hypothetical model. The model closely resembles the Coulomb friction model but bears no difference between the sticking zone and the sliding zone as far as the mathematical expression of the frictional stress at each zone is concerned, so that the need for locating the sticking zone may be removed. A step-like function which varies continuously with the magnitude of Du, the velocity vector of the strip relative to that of the work roll, was used for this purpose, as follows st Z Kmsn gðjDujÞ
(10)
where st denotes the tangential stress acting along the direction of Du, and 2 jDuj gðjDujÞ Z tanK1 (11) p a a, a very small constant. As shown in Fig. 1, the inverse tangent function can closely approximate the Coulomb friction at the sliding zone (which is, g(jDuj)Z1 when jDujO0), as well as the Coulomb friction at the sticking zone (which is 0%g(jDuj)%1 when jDujZ0), provided the constant a is sufficiently small. The detailed finite element formulation may be seen in Refs. [6,13].
Fig. 1. Modified Coulomb friction model, stZKmsng(jDuj).
2.2. FE model for analysis of heat transfer in the strip and in the roll The governing equation for steady-state heat transfer in the strip as well as that in the roll is given by rcui T;i Z ðkT; Þ;i C Q
(12)
where r is the density; ui represent the components of the velocity vector and Q represents the heat dissipated by plastic deformation. Note that QZ s 3_ in the strip and QZ0 in the roll, when the roll deformation is purely elastic. During rolling, both the roll and the strip may be subject to various thermal boundary conditions, as follows T Z T;
on GT
kT;i ni Z Khw ðT K Tw Þ;
(13) on Gh
kT;i ni Z Ks3ðT 4 K Te4 Þ K he ðT K Te Þ; kT;i ni Z q;
on Gq
(14) on Ge
(15) (16)
where GT, Gh, Ge, Gq represent the temperature prescribed surface, the surface where convection heat transfer occurs, the surface exposed to the environment, the heat flux prescribed surface, respectively. Detailed formulation may be found in Refs. [6,13]. In the strip and also in the roll, a large amount of heat is transported to the downstream by convection due to high processing speed. It is well known that when the convection term governs heat transfer more significantly than the diffusion term, solutions based on the standard Galerkin formulation are often corrupted by spurious node to node oscillations. To remove such a numerical instability, proper modification of the standard Galerkin formulation is necessary. The detailed description may be also seen in Refs. [6,13].
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3. Integrated FE model for the analysis of the roll–strip system An integrated FE model for the analysis of the thermo-mechanical behavior of the roll–strip system at a single mill stand consisted of three basic FE models—a model for the analysis of thermo-viscoplastic deformation of the strip (Model A), a model for the analysis of steady-state thermal behavior of the strip (Model B), and a model for the analysis of steady-state thermal behavior of the work roll (Model C). Interaction among the thermal and mechanical behaviors of the strip and the thermal behavior of the work roll may be summarized as follows: the heat transfer and plastic deformation occurring in the strip affect each other, since the flow stress is temperature-dependent while plastic deformation generates heat; and the thermal behavior of the strip is coupled to that of the work roll due to roll–strip contact. In the present investigation, the interaction was taken into account by adopting an iterative solution scheme, as shown in Fig. 2. The heat transfer coefficient at the roll–strip interface used in conjunction with the present FE model was an empirical equation derived by Hlady et al [5] hlub Z
kZ
k C
Pr s S
kr kS kr C kS
1:7 (17)
(18)
Fig. 2. An integrated FE process model for the prediction of interfacial thermo-mechanical behavior of roll–strip system occurring at a mill stand.
where CZ35!10K3 mm, kr and kS are the thermal conductivity of the roll and that of the strip, respectively, Pr is the mean roll pressure, and s S is the mean flow stress in the skin of the strip at the bite region. A simpler empirical formula for the coefficient of friction at contact surfaces between roll and strip was developed by Lenard and Leon [12] m Z 0:363
Pr s m
K 0:360
(19)
where s m is the mean flow stress at the bite region. Due to the effect of the interface heat transfer coefficient and friction coefficient on the thermo-mechanical behavior, an additional iteration loop was required, as shown in Fig. 2.
4. Simulation strategy and computational conditions Investigated was the thermo-mechanical behavior of the strip occurring in the finishing mill of POSCO No. 3 hot strip mill, Kwangyang works. The finishing mill consisted of seven mill stands (from F1 to F7), with each inter-stand being 5.8 m long. With respect that the tandem mill has an extremely large line length compared to the strip thickness, finite element simulation considering the entire finishing mill as a single analysis domain is impractical in the light of the computational efficiency. An alternative choice would be to divide the finish mill into several sub zones. As shown in Fig. 3a, each zone may be classified into one of the following four types: the first zone, which represents a region located in front of the first mill stand, the last zone, which represents a region located between the last mill stand and somewhere in front of the run-out-table for water cooling, an inter-stand zone, and a mill stand zone occupied by the roll–strip system. As shown in Fig. 3b, simulation may be performed for each zone in sequence, starting from the first zone until the simulation for the last zone. Note that the temperatures predicted at the current zone were employed as the inlet boundary conditions for the next thermal simulation. The thermal and mechanical properties of the roll and strip and a variety of process variables selected for the present investigation were summarized in Tables 1 and 2, respectively. The thermal and mechanical boundary conditions employed in simulation for each zone are shown in Fig. 4. Fig. 5 shows the finite element meshes representing the work roll and the strip in a mill stand zone, and a finite element mesh representing the strip in other zones. Note that the mesh density distributions in the work roll were designed so as to take into account the occurrence of the large temperature gradients at the bite region.
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Fig. 3. (a) Division of a finishing mill into several sub zones, (b) computational procedure.
Table 1 Thermal and mechanical properties of the roll and strip Strip material
JS-SPHC
Chemical composition (%)
Flow stress
CZ0.068 MnZ0.021 SiZ0.001 PZ0.015 SZ0.007
From Shida [22]
Roll material
rc
k Temperature (8C)
(W/mm 8C)
Temperature (8C)
(J/mm3 8C)
0.0 200.0 400.0 600.0 800.0 1000.0
0.0595 0.0532 0.0491 0.0368 0.0285 0.0276
100.0 200.0 400.0 600.0 750.0 900.0
0.003781 0.004079 0.004677 0.005926 0.008685 0.006650
rc
k
HSS
Shell
Ni-Grain
Core Shell Core
Temperature (8C)
(W/mm 8C)
110.0 213.0 398.0 502.0 530.0
0.0224 0.0246 0.0262 0.0272 0.0273 0.027 0.0235 0.027
Temperature (8C)
(J/mm3 8C) 0.004572
0.004248 0.00405 0.004248
k, thermal conductivity; rc, specific heat capacity. When T!700.0 8C, TZ700.0 8C was assumed for the evaluation of the flow stress. The thickness of shell is 50 mm both for HSS and Ni-Grain.
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Table 2 Process conditions Case No.
Stand No.
D (mm)
H0 (mm)
r (%)
VR (m/ min)
RDT (8C)
W (mm)
1
F1 F2 F3 F4 F5 F6 F7 F1 F2 F3 F4 F5 F6 F7
835.0 820.0 781.0 681.0 635.0 640.0 660.0 835.0 796.0 782.0 681.0 626.0 652.0 664.0
46.31 21.68 11.51 6.88 4.31 3.17 2.61 33.98 16.50 8.56 5.17 3.30 2.32 1.84
53.2 46.9 40.2 37.4 26.5 17.7 11.1 51.4 48.1 39.6 36.2 29.7 20.7 13.6
67.0 126.0 212.0 331.0 462.0 580.0 665.0 62.0 120.0 200.0 310.0 441.0 571.0 672.0
1020
1219
1034
1000
2
D, diameter of work roll; H0, strip entry thickness; r, reduction ratio. VR, rolling speed; RDT, roughing mill delivery temperature; W, width. F1–F3 stands, HSS roll; other stands, Ni-Grain roll. Carbon content for case 2 is 0.059%.
5. Results and discussion 5.1. Temperatures and rolling forces through the entire mill stand Illustrated in Fig. 6 were the distributions of strip temperatures at surface and center along the entire finishing mill. The surface temperatures of the strip were dramatically decreased as the strip passed through a mill stand zone, due to contact with the rather chilled work roll. However, the surface temperatures were quickly restored, as the strip passed through an inter-stand zone, due to the profuse heat supply from the relatively hot mass in the interior of the strip. On the contrary, as may be seen from Fig. 6, the temperatures on the center of the strip were increased as the strip passed through a mill stand, due to heat generation by plastic deformation. However, the center temperatures were decreased as the strip passed through an inter-stand zone, due to the heat loss to the skin of the strip. Rolling forces for two different rolling conditions through the entire mill were examined and shown in Fig. 7. A good agreement between the predictions and measurements was found except at the last two stands associated with the lower predictions. This may be attributed to the retained effective strain arising from the incomplete recrystallization happened at the inter-stands before entering into these two stands according to Refs. [14,15]. The coefficients of friction and interface heat transfer coefficients determined for the entire mill stand are depicted in Fig. 8, and the similar distributions were found for each case, which were lower at both front and rear stands but higher at media stands, consistent with the observation in Ref. [12]. This can be addressed from the reality that the interface heat transfer coefficient and coefficient of friction are strong function of the ratio
Fig. 4. Thermal boundary conditions for (a) the work roll in a mill stand zone, for (b) the strip in a mill stand zone, and for (c) the strip in other zones, and mechanical boundary conditions for (d) the strip in a mill stand zone.
between mean pressure and mean surface effective stress ðPr =s S Þ, as well as that between mean pressure and mean effective stress ðPr =s m Þ in the bite region as described in Eqs. (17) and (19), as illustrated in Fig. 9. Also to be noted was that the maximum points both for m and hlub were located at stand 4 for case 1 as well as stand 5 for case 2, which may be dependent on the process conditions. 5.2. The effect of the reduction As illustrated in Fig. 10a, when the reduction ratio is increased, the separating force can be inevitably increased, which may be contributed by the rise of the contact area and the pass strain, and thus, the roll pressures, increased. Hence the real area of contact may approach the apparent area, indicating that more asperities can be flattened at a rate that
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Fig. 7. Comparison between the predicted and measured rolling forces through the entire mill.
Fig. 5. Finite element meshes for (a) the work roll, for (b) the strip in a mill stand zone, for (c) the strip in other zones. The thickness of the strip was exaggerated.
depends on the deformation resistance of the strip. On the other hand, the rise of the reduction ratio will greatly affect the flow stress of strip not only due to the increment of effective strain but also because of the interaction between the temperature increase due to the heat generation by the plastic deformation and the temperature decrease due to the heat loss by the longer contact time with work roll. As a result, since the dramatic increase of the roll pressure due to the rise of the reduction ratio is much higher comparing to the variance of the flow stress affected by the temperatures and pass strain, both the ratio of Pr to s m and that of Pr to s S
Fig. 6. Strip temperatures at the surface and center through the entire mill. Note that the measurements were made only at the surface of strip.
were noticeably increased, as shown in Fig. 10b. Consequently both m and hlub were remarkably increased, as displayed in Fig. 10c, which can also be interpreted according to the adhesion theory of friction. The more asperities, the more of adhesive bonds formed between the asperities of the roll and rolled strip will be present, indicating the increasing resistance to relative motion as more of them would have to be broken, which definitely reflected the experiment observations published in Refs. [8,12,16]. 5.3. The effect of the rolling speed Increasing the rolling speed may certainly increase the strain rate of the rolled material, resulting in the rise of the flow stress of strip; on the contrary, the increase of speed may improve the strip temperatures, resulting in the decrease of the flow stress. As illustrated in Ref. [6], despite such negative effect, the flow strength of the strip is generally increased as the increase of the speed for low carbon steel, which led to the rise of the rolling force,
Fig. 8. m and hlub through the entire mill.
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Fig. 9. Pr =s s and Pr =s m through the entire mill.
Fig. 11. Effect of rolling speed on (a) rolling force, (b) Pr =s S and Pr =s m , and (c) m and hlub.
Fig. 10. Effect of reduction ratio on (a) rolling force, (b) Pr =s S and Pr =s m , and (c) m and hlub.
as shown in Fig. 11a. However, it can be seen in Fig. 11b that a decreasing effect in terms of the ratio of Pr to s m as raising the rolling speed was observed, indicating the same negative effect of speed on coefficient of friction as depicted in Fig. 11c, which agreed with the experimental observation as described in Refs. [9,17]. In addition, it can be demonstrated from the adhesion theory of friction that the formation of adhesive bonds may take time. Therefore, as the speed is increased, the time available for such formation is less, indicating the general decrease of the coefficient of friction with increasing the rolling speed. At the contact surfaces between the roll and strip, the surface temperatures were increased not only due to the less heat loss to the chilled work roll because of the less contact time but also due to the more heat generation by the frictional force because of the higher relative velocity
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have been ideally made at the location with several depths under the surface, then the surface temperatures were predicted by the inverse analysis. As a consequence, a sound evaluation of the effect of the temperatures on m and hlub by the experiment became almost impossible. Therefore, some contradict results regarding to the effect of temperatures on the friction coefficient may be found in the publications, for example, an invariable effect [18], a decreased effect [19], and an increased effect [20] during hot ring compression when the temperatures were increased. It should be noted that such discrepancy may be resulted from the diverse conditions including the status of scale formation and lubrication coherent to the experiments. In the sense of the thermal and mechanical dependence, raising the strip temperatures may greatly decrease the flow stress of strip, leading to the decrease of rolling force, as shown in Fig. 12a. However, the decrease of the roll pressure may reduce the real area of the contact at the interface, indicating the decrease of the frictional force and heat transfer. Consequently, the negative effect of the temperatures on m and hlub as the rise of strip temperatures, as shown in Fig. 12c, can be well expressed in terms of the ratio of Pr to s m and Pr to s S , respectively, as illustrated in Fig. 12b. It should be noted that such negative effect coincided with the evaluation through the hot rolling experiment [10] and industry analysis in Refs. [12,21] for low carbon and ferritic stainless steel.
6. Concluding remarks
Fig. 12. Effect of temperature on (a) rolling force, (b) Pr =s S and Pr =s m , and (c) m and hlub.
between the roll and strip, resulting in the increase of harmonic conductivities between the roll and strip. As a result, though the ratio of Pr to s S was decreased with respect to the increase of the speed, as shown in Fig. 11b, the hlub was raised as the improvement of the speed, as illustrated in Fig. 11c.
A finite element-based approach was presented for the precise prediction of the interfacial thermo-mechanical behavior of the roll and strip through the entire tandem mill in the hot strip rolling. It was demonstrated through the present investigation that the model was effective for the sound prediction of the interfacial interdependence between the thermal and mechanical behaviors of roll–strip system and, consequently for the successful exploration of the detailed aspect of the effect of the process parameters. However, the model addresses only a part of interfacerelated problems, which including the mechanism of scale formation, the effect of scale and lubricant, the effect of temperature, viscosity and pressure on lubrication, etc. They should constitute a part of the future works to be done to achieve the sound product quality.
5.4. The effect of the temperature As aforementioned that the friction and heat transfer at the interface between the roll and strip are strongly interdependent, however, unfortunately, measuring the surface temperatures is still unfeasible both in the laboratory and industry process. Generally, measurements
Acknowledgements The authors are grateful to POSCO for the financial support, with which the present investigation was possible and for permission to publish the results.
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References [1] Malinowski Z, Lenard JG, Davies ME. A study of the heat-transfer coefficient as a function of temperature and pressure. J Mater Proc Technol 1994;41(2):125–42. [2] Chen BK, Thompson PF, Choi SK. Temperature distribution in the rollgap during hot flat rolling. J Mater Proc Technol 1992;30(1):115–30. [3] Murata K, Morise H, Mitsutssuka M, Haito H, Kumatsu T, Shida S. Heat transfer between metals in contact and its application to protection of rolls. Trans ISIJ 1984;24(9):B309. [4] Sellars CM. Computer modeling of hot-working processes. Mater Sci Technol 1985;1(4):325–32. [5] Hlady CO, Brimacombe IJ, Samarasekera IV H, Hawbolt EB. Heat transfer in the hot rolling of metals. Metall Mater Trans B 1995; 26B(10):1019–28. [6] Sun CG, Yun CS, Chung JS, Hwang SM. Investigation of thermomechanical behavior of a work roll and of roll life in hot strip rolling. Metall Mater Trans A 1998;29A(9):2407–24. [7] Bowden FP, Tabor D. The friction and lubrication of solids. Oxford: Clarendon Press; 1950. [8] Karagiozis AN, Lenard JG. The effect of material properties on the coefficient of friction in cold rolling. In: Proceedings of Eurotrib’95 1985. p. 1–7. [9] Zhang S, Lenard JG. Reduction of the roll force during lubricated cold rolling of aluminium strips. J Synth Lubrication 1996;12:303–21. [10] Munther P, Lenard JG. Tribology during hot, flat rolling of steels. Ann CIRP 1995;44(1):213–6. [11] Li YH, Sellars CM. Modeling surface temperatures during hot rolling of steel. In: Proceedings of modeling of metal rolling processes, vol. 3 1997. p. 178–86.
[12] Lenard JG, Leon BN. The coefficient of friction during hot rolling of low carbon steel strips. ASME J Tribol 2002;124(10):840–5. [13] Hwang SM, Sun CG, Ryoo SR, Kwak WJ. An integrated FE process model for precision analysis of thermo-mechanical behaviors of rolls and strip in hot strip rolling. Comput Methods Appl Mech Eng 2002; 191:4015–33. [14] Hodgson PD, Gibbs RK. A mathematical model to predict the mechanical properties hot rolled C–Mn and microalloyed steels. ISIJ Int 1992;32(12):1329–38. [15] Jonas JJ. The hot strip mill as an experimental tool. ISIJ Int 2000; 40(8):731–8. [16] Hum B, Colquhoun HW, Lenard JG. Measurements of friction during hot rolling of aluminum strips. J Mater Proc Technol 1996;60: 331–8. [17] Yuen WYD, Popelianski Y, Prouten M. Variations of friction in the roll bite and their effects on cold strip rolling. Iron Steelmaker 1996; 22(2):33–9. [18] Wang F, Lenard JG. An experimental study of interfacial friction-hot ring compression. J Eng Mater Technol 1992;114(1):13–18. [19] Venugopal S, Srinivasan G, Venkadesan S, Seetharaman V. A note on the determination of the friction factor by means of the reductioncapacity test. J Mech Work Technol 1989;19:261–6. [20] Pawelski O, Rasp W, Hoerster C. The ring compression test as simulation test for the investigation of friction in hot metal forming. Steel Res 1989;60:395–402. [21] Jin W, Piereder D, Lenard JG. A study of the coefficient of friction during hot rolling of a ferritic stainless steel. J Soc Tribol Lubrication Eng 2002;58(11):29–37. [22] Shida S. Empirical formula of flow-stress of carbon steels. J Jpn Soc Technol Plast 1969;10(8):610–7.
Investigation of interfacial behaviors between the strip and roll in hot strip rolling by finite element method Cheng Gang Sun* POSCO Technical Research Laboratories, Stainless Steel Research Group, Pohang 790-785, South Korea Received 25 November 2003; received in revised form 30 July 2004; accepted 14 September 2004 Available online 23 November 2004
Abstract A finite element (FE)-based approach is presented for the precision analysis of the interfacial thermo-mechanical behavior of the roll and strip in the entire tandem mill during hot strip rolling. The validity of the proposed model is examined through the comparison with measurements. Then, the effect of various process parameters on the detailed aspects of interfacial thermo-mechanical behavior of the roll and strip is investigated via a series of process simulations. q 2004 Elsevier Ltd. All rights reserved. Keywords: Finite element method; Roll; Strip; Hot strip rolling; Interfacial thermo-mechanical behavior
1. Introduction In hot strip rolling, a hot slab is passed through the roll gap several times, with its thickness being progressively reduced to achieve the final strip dimensions. In order for precision control of the product quality as well as to enhance the production economy in most modern rolling process, the interfacial thermo-mechanical behavior between the strip and roll, which may be strong dependent on frictional behavior and heat transfer at the interface, must be well understood. Incorrect information related to these two phenomena may lead to the problems in the prediction of roll force, strip shape and flatness, and strip temperatures in the hot rolling process. Precise modeling such as interfacial behavior, however, is a difficult task. In the light of heat transfer at contact surface, it is not only dependent on the process parameters in terms of temperature, reduction ratio and speed, but also on the surface geometry and properties such as hardness and roughness of the roll and strip. As a result, a simplified expression was generally assumed for the interface heat transfer coefficient (hlub). Through the past decades, there had been a lot of investigations to * Tel.: C82 54 220 6293; fax: C82 54 220 6915. E-mail address: [email protected] 0301-679X/$ - see front matter q 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.triboint.2004.09.002
determine the hlub, however, a noticeable variation was observed due to the widely varied experiment conditions, for example, 2–20, 10–50, 23–81 and 200 kW/m2K in Refs. [1–4], respectively. Such evaluations restricted to a limited laboratory condition may inevitably lead to the lower accuracy in prediction of hlub in the industry rolling process. Recently, based on the published hlub between two nominally flat, but microscopically rough surface and its relationship to the surface roughness, fractional contact area of two surfaces and the interfacial pressure, Hlady et al. [5] developed a model that described hlub as a function of the rolling pressure, surface hardness, as well as of the harmonic conductivity and surface roughness parameters of the roll and strip. A suitable application of such model for the estimation of hlub in the industry hot rolling process was recently made by Sun et al. [6] As far as the coefficient of friction was concerned, it was generally considered to be strongly dependent on process and material parameters according to the adhesion hypothesis [7] in hot rolling process. In the sense of process parameters, the relative velocity between the roll and strip, the surface temperature at the entry and the reduction ratio are primly important; on the other hand, the material parameters related to the contact surfaces may include the deformation resistance of
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Nomenclature f h hlub hw , he k n p T Te Tw 3ij 3 0 ij 3 3_ 3_ij 3_ij0 u un, u t
body force vector traction vector interface heat transfer coefficient convective heat transfer coefficient thermal conductivity, bulk modulus outward unit normal vector hydrostatic pressure temperature environment temperature temperature of the flowing medium strain tensor deviatoric strain tensor effective strain effective strain rate strain rate tensor deviatoric strain rate tensor velocity vector, displacement vector normal and tangential components of u
the strip, the roughness of the strip and roll and the thickness of the scale, etc. When the lubricants are used, the viscosity and its dependence on the temperature and pressure should also to be taken into account. As a result, though remarkable investigations had been made to explore the individual effect of the reduction [8], rolling speed [9], temperature [10], as well as material’s resistance and thickness of scale layer [11] on coefficient of friction in the past, the functional dependence of the process and material parameters had not yet been presented until the recent work by Lenard and Leon [12], who developed an empirical model accounting for the aforementioned dependence based on the experiment, in which the growth of the scale and thickness of the layer of scale were properly controlled. Although the friction and heat transfer at the interface are disparate phenomena, there may exist a definite relationship between them, which can be summarized as follows: coefficient of friction is influenced by the temperature distributions of strip at the interface, since the deformation resistance is strongly dependent on temperatures. On the other hand, temperature distributions in the strip are affected by heat generation due to interfacial friction, resulting in strong correlations between the mechanical and the thermal behaviors of the strip and roll at the contact surface. However, very few studies have been undertaken on the investigation of the interdependence between the friction and heat transfer at the contact surfaces. In this paper, the individual interfacial models were well incorporated into finite element models to constitute an integrated process model for demonstrating the interfacial thermo-mechanical behavior of strip and roll. The validity of the proposed
sij s sn st s 0 ij dij m x U G rc s3 q ur H1, H2 R
stress tensor flow stress normal stress tangential stress deviatoric stress tensor kronecker delta coefficient of Coulomb friction penalty constant analysis domain boundary of the analysis domain heat capacity per unit volume Stefan–Bolzmann constant multiplied by emissivity heat flux per unit area roll speed entry and exit thickness of the strip roll radius
model was examined through comparison with measurements. Then, a series of process simulation were conducted to investigate the effect of the process parameters on the interfacial behavior.
2. Finite element (FE) process models 2.1. FE model for analysis of rigid-viscoplastic deformation A steady-state, rigid-viscoplastic deformation behavior of the strip was adopted in the present investigation. Equation of motion (neglecting all the acceleration terms): sij;jCfiZ0
(1)
Constitutive relationship: sij Z Kpdij C sij0
(2)
2s 3_ 33_ ij
(3)
sij0 Z
where the flow stress s of the strip deforming at elevated temperatures may be represented by s Z f ð3; 3_; TÞ
(4)
Incompressibility condition: ui;i Z 0
(5)
Traction boundary conditions: sij nj Z hi ;
on Ghi
(6)
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Velocity prescribed boundary conditions: ui Z u i ;
on Gui
(7)
Boundary conditions at the roll–strip interface— I. Normal stress. A penalty algorithm was applied for the determination of the normal stress distributions at the roll–strip interface. For this purpose, the velocity prescribed boundary condition un Z u D n;
on Gc
(8)
was replaced by sn Z Kxðun K u D n Þ;
on Gc
(9)
where sn denotes the normal stress, un and uD n (Z0) denote the normal component of the velocity vector of the strip and that of the roll, respectively. Considering that the magnitude of sn should be finite in order to maintain stress equilibrium, it is evident that the contact condition represented by Eq. (9) is strictly enforced as x becomes very large. Boundary conditions at the roll–strip interface— II. Tangential stress. The tangential stress distributions at the roll–strip interface due to friction may be modeled by the Coulomb friction model. It should be noted that the sticking zone is not known a priori before solving the boundary value problem. An approach that may be conceived is to develop and apply a special trial and error procedure to locate the sticking zone. Another possible approach, which was adopted in this investigation, is to replace the Coulomb friction model by a hypothetical model. The model closely resembles the Coulomb friction model but bears no difference between the sticking zone and the sliding zone as far as the mathematical expression of the frictional stress at each zone is concerned, so that the need for locating the sticking zone may be removed. A step-like function which varies continuously with the magnitude of Du, the velocity vector of the strip relative to that of the work roll, was used for this purpose, as follows st Z Kmsn gðjDujÞ
(10)
where st denotes the tangential stress acting along the direction of Du, and 2 jDuj gðjDujÞ Z tanK1 (11) p a a, a very small constant. As shown in Fig. 1, the inverse tangent function can closely approximate the Coulomb friction at the sliding zone (which is, g(jDuj)Z1 when jDujO0), as well as the Coulomb friction at the sticking zone (which is 0%g(jDuj)%1 when jDujZ0), provided the constant a is sufficiently small. The detailed finite element formulation may be seen in Refs. [6,13].
Fig. 1. Modified Coulomb friction model, stZKmsng(jDuj).
2.2. FE model for analysis of heat transfer in the strip and in the roll The governing equation for steady-state heat transfer in the strip as well as that in the roll is given by rcui T;i Z ðkT; Þ;i C Q
(12)
where r is the density; ui represent the components of the velocity vector and Q represents the heat dissipated by plastic deformation. Note that QZ s 3_ in the strip and QZ0 in the roll, when the roll deformation is purely elastic. During rolling, both the roll and the strip may be subject to various thermal boundary conditions, as follows T Z T;
on GT
kT;i ni Z Khw ðT K Tw Þ;
(13) on Gh
kT;i ni Z Ks3ðT 4 K Te4 Þ K he ðT K Te Þ; kT;i ni Z q;
on Gq
(14) on Ge
(15) (16)
where GT, Gh, Ge, Gq represent the temperature prescribed surface, the surface where convection heat transfer occurs, the surface exposed to the environment, the heat flux prescribed surface, respectively. Detailed formulation may be found in Refs. [6,13]. In the strip and also in the roll, a large amount of heat is transported to the downstream by convection due to high processing speed. It is well known that when the convection term governs heat transfer more significantly than the diffusion term, solutions based on the standard Galerkin formulation are often corrupted by spurious node to node oscillations. To remove such a numerical instability, proper modification of the standard Galerkin formulation is necessary. The detailed description may be also seen in Refs. [6,13].
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3. Integrated FE model for the analysis of the roll–strip system An integrated FE model for the analysis of the thermo-mechanical behavior of the roll–strip system at a single mill stand consisted of three basic FE models—a model for the analysis of thermo-viscoplastic deformation of the strip (Model A), a model for the analysis of steady-state thermal behavior of the strip (Model B), and a model for the analysis of steady-state thermal behavior of the work roll (Model C). Interaction among the thermal and mechanical behaviors of the strip and the thermal behavior of the work roll may be summarized as follows: the heat transfer and plastic deformation occurring in the strip affect each other, since the flow stress is temperature-dependent while plastic deformation generates heat; and the thermal behavior of the strip is coupled to that of the work roll due to roll–strip contact. In the present investigation, the interaction was taken into account by adopting an iterative solution scheme, as shown in Fig. 2. The heat transfer coefficient at the roll–strip interface used in conjunction with the present FE model was an empirical equation derived by Hlady et al [5] hlub Z
kZ
k C
Pr s S
kr kS kr C kS
1:7 (17)
(18)
Fig. 2. An integrated FE process model for the prediction of interfacial thermo-mechanical behavior of roll–strip system occurring at a mill stand.
where CZ35!10K3 mm, kr and kS are the thermal conductivity of the roll and that of the strip, respectively, Pr is the mean roll pressure, and s S is the mean flow stress in the skin of the strip at the bite region. A simpler empirical formula for the coefficient of friction at contact surfaces between roll and strip was developed by Lenard and Leon [12] m Z 0:363
Pr s m
K 0:360
(19)
where s m is the mean flow stress at the bite region. Due to the effect of the interface heat transfer coefficient and friction coefficient on the thermo-mechanical behavior, an additional iteration loop was required, as shown in Fig. 2.
4. Simulation strategy and computational conditions Investigated was the thermo-mechanical behavior of the strip occurring in the finishing mill of POSCO No. 3 hot strip mill, Kwangyang works. The finishing mill consisted of seven mill stands (from F1 to F7), with each inter-stand being 5.8 m long. With respect that the tandem mill has an extremely large line length compared to the strip thickness, finite element simulation considering the entire finishing mill as a single analysis domain is impractical in the light of the computational efficiency. An alternative choice would be to divide the finish mill into several sub zones. As shown in Fig. 3a, each zone may be classified into one of the following four types: the first zone, which represents a region located in front of the first mill stand, the last zone, which represents a region located between the last mill stand and somewhere in front of the run-out-table for water cooling, an inter-stand zone, and a mill stand zone occupied by the roll–strip system. As shown in Fig. 3b, simulation may be performed for each zone in sequence, starting from the first zone until the simulation for the last zone. Note that the temperatures predicted at the current zone were employed as the inlet boundary conditions for the next thermal simulation. The thermal and mechanical properties of the roll and strip and a variety of process variables selected for the present investigation were summarized in Tables 1 and 2, respectively. The thermal and mechanical boundary conditions employed in simulation for each zone are shown in Fig. 4. Fig. 5 shows the finite element meshes representing the work roll and the strip in a mill stand zone, and a finite element mesh representing the strip in other zones. Note that the mesh density distributions in the work roll were designed so as to take into account the occurrence of the large temperature gradients at the bite region.
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Fig. 3. (a) Division of a finishing mill into several sub zones, (b) computational procedure.
Table 1 Thermal and mechanical properties of the roll and strip Strip material
JS-SPHC
Chemical composition (%)
Flow stress
CZ0.068 MnZ0.021 SiZ0.001 PZ0.015 SZ0.007
From Shida [22]
Roll material
rc
k Temperature (8C)
(W/mm 8C)
Temperature (8C)
(J/mm3 8C)
0.0 200.0 400.0 600.0 800.0 1000.0
0.0595 0.0532 0.0491 0.0368 0.0285 0.0276
100.0 200.0 400.0 600.0 750.0 900.0
0.003781 0.004079 0.004677 0.005926 0.008685 0.006650
rc
k
HSS
Shell
Ni-Grain
Core Shell Core
Temperature (8C)
(W/mm 8C)
110.0 213.0 398.0 502.0 530.0
0.0224 0.0246 0.0262 0.0272 0.0273 0.027 0.0235 0.027
Temperature (8C)
(J/mm3 8C) 0.004572
0.004248 0.00405 0.004248
k, thermal conductivity; rc, specific heat capacity. When T!700.0 8C, TZ700.0 8C was assumed for the evaluation of the flow stress. The thickness of shell is 50 mm both for HSS and Ni-Grain.
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Table 2 Process conditions Case No.
Stand No.
D (mm)
H0 (mm)
r (%)
VR (m/ min)
RDT (8C)
W (mm)
1
F1 F2 F3 F4 F5 F6 F7 F1 F2 F3 F4 F5 F6 F7
835.0 820.0 781.0 681.0 635.0 640.0 660.0 835.0 796.0 782.0 681.0 626.0 652.0 664.0
46.31 21.68 11.51 6.88 4.31 3.17 2.61 33.98 16.50 8.56 5.17 3.30 2.32 1.84
53.2 46.9 40.2 37.4 26.5 17.7 11.1 51.4 48.1 39.6 36.2 29.7 20.7 13.6
67.0 126.0 212.0 331.0 462.0 580.0 665.0 62.0 120.0 200.0 310.0 441.0 571.0 672.0
1020
1219
1034
1000
2
D, diameter of work roll; H0, strip entry thickness; r, reduction ratio. VR, rolling speed; RDT, roughing mill delivery temperature; W, width. F1–F3 stands, HSS roll; other stands, Ni-Grain roll. Carbon content for case 2 is 0.059%.
5. Results and discussion 5.1. Temperatures and rolling forces through the entire mill stand Illustrated in Fig. 6 were the distributions of strip temperatures at surface and center along the entire finishing mill. The surface temperatures of the strip were dramatically decreased as the strip passed through a mill stand zone, due to contact with the rather chilled work roll. However, the surface temperatures were quickly restored, as the strip passed through an inter-stand zone, due to the profuse heat supply from the relatively hot mass in the interior of the strip. On the contrary, as may be seen from Fig. 6, the temperatures on the center of the strip were increased as the strip passed through a mill stand, due to heat generation by plastic deformation. However, the center temperatures were decreased as the strip passed through an inter-stand zone, due to the heat loss to the skin of the strip. Rolling forces for two different rolling conditions through the entire mill were examined and shown in Fig. 7. A good agreement between the predictions and measurements was found except at the last two stands associated with the lower predictions. This may be attributed to the retained effective strain arising from the incomplete recrystallization happened at the inter-stands before entering into these two stands according to Refs. [14,15]. The coefficients of friction and interface heat transfer coefficients determined for the entire mill stand are depicted in Fig. 8, and the similar distributions were found for each case, which were lower at both front and rear stands but higher at media stands, consistent with the observation in Ref. [12]. This can be addressed from the reality that the interface heat transfer coefficient and coefficient of friction are strong function of the ratio
Fig. 4. Thermal boundary conditions for (a) the work roll in a mill stand zone, for (b) the strip in a mill stand zone, and for (c) the strip in other zones, and mechanical boundary conditions for (d) the strip in a mill stand zone.
between mean pressure and mean surface effective stress ðPr =s S Þ, as well as that between mean pressure and mean effective stress ðPr =s m Þ in the bite region as described in Eqs. (17) and (19), as illustrated in Fig. 9. Also to be noted was that the maximum points both for m and hlub were located at stand 4 for case 1 as well as stand 5 for case 2, which may be dependent on the process conditions. 5.2. The effect of the reduction As illustrated in Fig. 10a, when the reduction ratio is increased, the separating force can be inevitably increased, which may be contributed by the rise of the contact area and the pass strain, and thus, the roll pressures, increased. Hence the real area of contact may approach the apparent area, indicating that more asperities can be flattened at a rate that
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Fig. 7. Comparison between the predicted and measured rolling forces through the entire mill.
Fig. 5. Finite element meshes for (a) the work roll, for (b) the strip in a mill stand zone, for (c) the strip in other zones. The thickness of the strip was exaggerated.
depends on the deformation resistance of the strip. On the other hand, the rise of the reduction ratio will greatly affect the flow stress of strip not only due to the increment of effective strain but also because of the interaction between the temperature increase due to the heat generation by the plastic deformation and the temperature decrease due to the heat loss by the longer contact time with work roll. As a result, since the dramatic increase of the roll pressure due to the rise of the reduction ratio is much higher comparing to the variance of the flow stress affected by the temperatures and pass strain, both the ratio of Pr to s m and that of Pr to s S
Fig. 6. Strip temperatures at the surface and center through the entire mill. Note that the measurements were made only at the surface of strip.
were noticeably increased, as shown in Fig. 10b. Consequently both m and hlub were remarkably increased, as displayed in Fig. 10c, which can also be interpreted according to the adhesion theory of friction. The more asperities, the more of adhesive bonds formed between the asperities of the roll and rolled strip will be present, indicating the increasing resistance to relative motion as more of them would have to be broken, which definitely reflected the experiment observations published in Refs. [8,12,16]. 5.3. The effect of the rolling speed Increasing the rolling speed may certainly increase the strain rate of the rolled material, resulting in the rise of the flow stress of strip; on the contrary, the increase of speed may improve the strip temperatures, resulting in the decrease of the flow stress. As illustrated in Ref. [6], despite such negative effect, the flow strength of the strip is generally increased as the increase of the speed for low carbon steel, which led to the rise of the rolling force,
Fig. 8. m and hlub through the entire mill.
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Fig. 9. Pr =s s and Pr =s m through the entire mill.
Fig. 11. Effect of rolling speed on (a) rolling force, (b) Pr =s S and Pr =s m , and (c) m and hlub.
Fig. 10. Effect of reduction ratio on (a) rolling force, (b) Pr =s S and Pr =s m , and (c) m and hlub.
as shown in Fig. 11a. However, it can be seen in Fig. 11b that a decreasing effect in terms of the ratio of Pr to s m as raising the rolling speed was observed, indicating the same negative effect of speed on coefficient of friction as depicted in Fig. 11c, which agreed with the experimental observation as described in Refs. [9,17]. In addition, it can be demonstrated from the adhesion theory of friction that the formation of adhesive bonds may take time. Therefore, as the speed is increased, the time available for such formation is less, indicating the general decrease of the coefficient of friction with increasing the rolling speed. At the contact surfaces between the roll and strip, the surface temperatures were increased not only due to the less heat loss to the chilled work roll because of the less contact time but also due to the more heat generation by the frictional force because of the higher relative velocity
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have been ideally made at the location with several depths under the surface, then the surface temperatures were predicted by the inverse analysis. As a consequence, a sound evaluation of the effect of the temperatures on m and hlub by the experiment became almost impossible. Therefore, some contradict results regarding to the effect of temperatures on the friction coefficient may be found in the publications, for example, an invariable effect [18], a decreased effect [19], and an increased effect [20] during hot ring compression when the temperatures were increased. It should be noted that such discrepancy may be resulted from the diverse conditions including the status of scale formation and lubrication coherent to the experiments. In the sense of the thermal and mechanical dependence, raising the strip temperatures may greatly decrease the flow stress of strip, leading to the decrease of rolling force, as shown in Fig. 12a. However, the decrease of the roll pressure may reduce the real area of the contact at the interface, indicating the decrease of the frictional force and heat transfer. Consequently, the negative effect of the temperatures on m and hlub as the rise of strip temperatures, as shown in Fig. 12c, can be well expressed in terms of the ratio of Pr to s m and Pr to s S , respectively, as illustrated in Fig. 12b. It should be noted that such negative effect coincided with the evaluation through the hot rolling experiment [10] and industry analysis in Refs. [12,21] for low carbon and ferritic stainless steel.
6. Concluding remarks
Fig. 12. Effect of temperature on (a) rolling force, (b) Pr =s S and Pr =s m , and (c) m and hlub.
between the roll and strip, resulting in the increase of harmonic conductivities between the roll and strip. As a result, though the ratio of Pr to s S was decreased with respect to the increase of the speed, as shown in Fig. 11b, the hlub was raised as the improvement of the speed, as illustrated in Fig. 11c.
A finite element-based approach was presented for the precise prediction of the interfacial thermo-mechanical behavior of the roll and strip through the entire tandem mill in the hot strip rolling. It was demonstrated through the present investigation that the model was effective for the sound prediction of the interfacial interdependence between the thermal and mechanical behaviors of roll–strip system and, consequently for the successful exploration of the detailed aspect of the effect of the process parameters. However, the model addresses only a part of interfacerelated problems, which including the mechanism of scale formation, the effect of scale and lubricant, the effect of temperature, viscosity and pressure on lubrication, etc. They should constitute a part of the future works to be done to achieve the sound product quality.
5.4. The effect of the temperature As aforementioned that the friction and heat transfer at the interface between the roll and strip are strongly interdependent, however, unfortunately, measuring the surface temperatures is still unfeasible both in the laboratory and industry process. Generally, measurements
Acknowledgements The authors are grateful to POSCO for the financial support, with which the present investigation was possible and for permission to publish the results.
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References [1] Malinowski Z, Lenard JG, Davies ME. A study of the heat-transfer coefficient as a function of temperature and pressure. J Mater Proc Technol 1994;41(2):125–42. [2] Chen BK, Thompson PF, Choi SK. Temperature distribution in the rollgap during hot flat rolling. J Mater Proc Technol 1992;30(1):115–30. [3] Murata K, Morise H, Mitsutssuka M, Haito H, Kumatsu T, Shida S. Heat transfer between metals in contact and its application to protection of rolls. Trans ISIJ 1984;24(9):B309. [4] Sellars CM. Computer modeling of hot-working processes. Mater Sci Technol 1985;1(4):325–32. [5] Hlady CO, Brimacombe IJ, Samarasekera IV H, Hawbolt EB. Heat transfer in the hot rolling of metals. Metall Mater Trans B 1995; 26B(10):1019–28. [6] Sun CG, Yun CS, Chung JS, Hwang SM. Investigation of thermomechanical behavior of a work roll and of roll life in hot strip rolling. Metall Mater Trans A 1998;29A(9):2407–24. [7] Bowden FP, Tabor D. The friction and lubrication of solids. Oxford: Clarendon Press; 1950. [8] Karagiozis AN, Lenard JG. The effect of material properties on the coefficient of friction in cold rolling. In: Proceedings of Eurotrib’95 1985. p. 1–7. [9] Zhang S, Lenard JG. Reduction of the roll force during lubricated cold rolling of aluminium strips. J Synth Lubrication 1996;12:303–21. [10] Munther P, Lenard JG. Tribology during hot, flat rolling of steels. Ann CIRP 1995;44(1):213–6. [11] Li YH, Sellars CM. Modeling surface temperatures during hot rolling of steel. In: Proceedings of modeling of metal rolling processes, vol. 3 1997. p. 178–86.
[12] Lenard JG, Leon BN. The coefficient of friction during hot rolling of low carbon steel strips. ASME J Tribol 2002;124(10):840–5. [13] Hwang SM, Sun CG, Ryoo SR, Kwak WJ. An integrated FE process model for precision analysis of thermo-mechanical behaviors of rolls and strip in hot strip rolling. Comput Methods Appl Mech Eng 2002; 191:4015–33. [14] Hodgson PD, Gibbs RK. A mathematical model to predict the mechanical properties hot rolled C–Mn and microalloyed steels. ISIJ Int 1992;32(12):1329–38. [15] Jonas JJ. The hot strip mill as an experimental tool. ISIJ Int 2000; 40(8):731–8. [16] Hum B, Colquhoun HW, Lenard JG. Measurements of friction during hot rolling of aluminum strips. J Mater Proc Technol 1996;60: 331–8. [17] Yuen WYD, Popelianski Y, Prouten M. Variations of friction in the roll bite and their effects on cold strip rolling. Iron Steelmaker 1996; 22(2):33–9. [18] Wang F, Lenard JG. An experimental study of interfacial friction-hot ring compression. J Eng Mater Technol 1992;114(1):13–18. [19] Venugopal S, Srinivasan G, Venkadesan S, Seetharaman V. A note on the determination of the friction factor by means of the reductioncapacity test. J Mech Work Technol 1989;19:261–6. [20] Pawelski O, Rasp W, Hoerster C. The ring compression test as simulation test for the investigation of friction in hot metal forming. Steel Res 1989;60:395–402. [21] Jin W, Piereder D, Lenard JG. A study of the coefficient of friction during hot rolling of a ferritic stainless steel. J Soc Tribol Lubrication Eng 2002;58(11):29–37. [22] Shida S. Empirical formula of flow-stress of carbon steels. J Jpn Soc Technol Plast 1969;10(8):610–7.