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Behaviour of longitudinal surface cracks in the hot rolling of steel slabs
Journal of Materials Processing Technology 94 (1999) 141±150
Behaviour of longitudinal surface cracks in the hot rolling of steel slabs Esa Ervasti, Ulf StaÊhlberg* Materials Forming, Department of Materials Processing, Royal Institute of Technology, Brinellvagen 23, Stockholm S-10044, Sweden Received 10 February 1998
Abstract The behaviour of a longitudinal V-shaped crack, on the surface of a continuously cast steel slab, is studied during hot rolling. The analysis is carried out by means of the commercial FE-code LS-DYNA3D. Process parameters obtained from industry are used as a reference. The slab of initial width 1000 mm and thickness 220 mm is rolled down to 30 mm. It is assumed that the material can be treated as rigid-perfectly plastic and that the cracks do not propagate. The latter assumption is in agreement with industrial observations for a steel grade similar to that analysed here. The aim of the study is to investigate the possibility of controlling the plastic deformation so that the cracks disappear or so that their deteriorating effects are minimised. The analysis is focused upon the in¯uence of friction, roll radius and rolling schedule on the change in the shape of a crack of initial depth 20 mm and a crack angle of 68. The reliability of the simulations is checked by pilot-plant experiments using aluminium as the model material for steel. The results indicate that it is not possible to prevent the bottom side surfaces of the crack from coming into contact, especially not for small reductions/pass and small roll radii. The in¯uence of friction was found to be marginal. Contact between the crack surfaces is found already at the beginning of the rolling, as the V-shaped crack is being changed to Y-shape. Considering the upper part of the crack, this remained open during the majority of the schedules studied. However for heavy reduction/pass and a large roll radius, this part of the crack is closed also, but not before the ®nal passes. If the bottom side surfaces of the crack are in complete contact, they are prevented from further oxidisation. It is assumed that for such conditions the deteriorating in¯uence of the bottom part of the defect decreases during subsequent rolling. During the elongation of the workpiece, the oxide ¯ake of the Y-crack bottom is broken into splinters with oxide free material in between, making the creation of a high performance weld possible. Provided that this supposition is correct, the best results should be obtained for light reductions/pass at the beginning of the rolling, resulting in an early closure of the crack bottom, followed by heavy reductions/pass enabling also the closure and oxide splintering of the upper part of the folded crack. # 1999 Published by Elsevier Science S.A. All rights reserved. Keywords: Longitudinal surface cracks; Hot rolling; Steel slabs
1. Introduction Longitudinal surface cracks are often found on continuously cast slabs, Fig. 1. Compared to transversal cracks, longitudinal cracks are usually deeper. Often they are located in regions close to the centre line of the slab surface. Almost without exception, they are found to be dif®cult to eliminate during subsequent hot rolling and consequently they will constitute defects on the rolled product [1]. The cracks are formed because of transverse tensile stresses caused by non-homogenous cooling conditions during the *Corresponding author. Tel.: +46-8790-83-84; fax: +46-8790-09-40.
solidi®cation of the material. Very often, the so-called ``hot spots'' are found on the skin of the slab during casting. Other reasons for crack formation could be tensile stresses introduced by high hydrostatic pressures within the liquid part of the strand, peritectic phase transformation and friction in the transverse direction between the skin and the mould [2±8]. Examples of sensitive steel grades are those characterised by approximately 0.12% C and a high content of sulphur compared with manganese. Much research has been carried out in order to improve the casting process for avoiding these cracks, some examples being represented by [2,3,8,9]. No theoretical work has been found dealing with the behaviour of longitudinal cracks during hot rolling.
0924-0136/99/$ ± see front matter # 1999 Published by Elsevier Science S.A. All rights reserved. PII: S 0 9 2 4 - 0 1 3 6 ( 9 9 ) 0 0 0 9 0 - 4
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Fig. 1. (a) Longitudinal cracks on the surface of a continuously cast slab. (b) Cross-section showing a longitudinal crack.
The present analysis is carried out with reference data obtained from industry. The initial slab is of 1000 mm width and 220 mm thickness. The rolling is performed as reverse rolling during the ®rst 7 passes. During the ®rst 4 passes, the slab thickness is reduced by 30 mm/pass, during pass nos. 5 and 6 by 25 mm/pass and ®nally during pass no. 7 by 20 mm down to a thickness of 30 mm. The reversal rolling is followed by continuous wide strip rolling in six stands. The study presented here is limited to the reversing part of the rolling. The behaviour of the cracks as in¯uenced by friction, roll radius and rolling schedule is analysed by means of the commercial FE-code, LS-DYNA3D. The aim of the work is to study whether it is possible to eliminate the cracks or to minimise their deteriorating in¯uence by hot rolling. The reliability of the theoretical results are checked by pilot-plant experiments using aluminium as a model material for steel. 2. FE-simulations To begin with, the input data and process characteristics are presented, after which details regarding the technique of simulation are discussed.
2.1. Input data and process characteristics The initial length of the workpiece is chosen to be 550 mm for all the rolling schedules studied. The initial crack has been assumed to be V-shaped with an angle of 68 and a depth of 20 mm. The rolling speed of the reference mill is 0.8 m/s during the ®rst pass and 2.4 m/s during the remaining passes. In Tables 1 and 2, input data used for different simulations are speci®ed. The material composition and work temperatures of the reference mill are clear from Tables 3 and 4. In addition to the simulations regarding the reference conditions, simulations have been carried out for: (i) two friction coef®cients, both of higher value than that of the reference; (ii) one smaller and one larger roll radius; and (iii) one rolling schedule, built on lighter reductions/pass and one built on heavier reductions/pass. All of the simulations cover a reduction in thickness of 220±30 mm according to Tables 1 and 2. Reference data according to industrial production are marked in italics and the symbol *. From the stress±strain curves presented in [10], it is clear that it is reasonable to treat the material as rigid-perfectly plastic. The strain rate is approximately 1 sÿ1 during the ®rst pass of the reference schedule and increases during the following to a maximum value of approximately 15 sÿ1.
Table 1 Process parameters Simulation no
Roll radius (mm)
Reductions/pass due to reference mill*
1 2 3 4 5 6 7
425* 425* 425* 225 625 425* 425*
X X X X X
Light reductions/pass
X
Heavy reductions/pass
X
0.20* 0.35 0.50 0.20* 0.20* 0.20* 0.20*
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Table 2 Rolling schedules Schedule
H0
H1
H2
H3
H4
H5
H6
H7
H8
H9
H10
H11
H12
H13
Reference Light reductions Heavy reductions
220 220 220
190 205 175
160 190 130
130 175 85
100 160 50
75 145 30
50 130
30 115
100
85
70
55
40
30
Table 3 Chemical composition of the workpiece material (wt.%). C
Si
Cr
Mn
P
S
Cu
0.12
0.20
0.08
0.50
0.01
0.03
0.10
Table 4 Workpiece temperatures after each pass due reference rolling Pass no
1
2
3
4
5
6
7
Workpiece temperature (C8)
1230
1223
1218
1210
1200
1188
1156
2.2. Element structure
Fig. 3. The positions of nodes on the boundary surfaces of the crack.
Shell elements are used for describing the rolls, which are assumed to be rigid. The work material is built up by brick elements. The elements were chosen in such a way that the ratio between their longest and shortest sides at no time exceeded a value of 3. On the contact surface between the workpiece and the roll, the element side of the material was chosen to be shorter than that of the roll in order to secure a stable simulation. The structure of the elements at the beginning of a simulation is clear from Fig. 2. In order to simulate the behaviour of the crack, the earlier described structure was completed with smaller elements
characterised by ®ve nodes in a crack cross-section, Fig. 3. The V-shaped cracks have an extension equal to the length of the workpiece. When simulating the crack behaviour during the subsequent pass, the shape obtained during the earlier passes was utilised. The step-by-step change in crack geometry was determined by establishing the new positions of the nodes. If opposite sides of the crack came into contact, the friction coef®cient along the oxidised surfaces was assumed to be 0.2.
Fig. 2. The rolls are built up by shell elements and the workpiece by brick elements.
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2.3. Practical problems In order to prevent the nodes of the rolls from penetrating into the work material, penalty functions were used. These functions act as springs along the roll/workpiece contact surface and are associated with a time step scale factor (TSSF) in order to avoid a critical length of the time step being exceeded. By combining these two functions in an adequate way, stable contact between the rolls and the workpiece was obtained. The elements of the workpiece were made stiff, Hancock-stiffness, in order to avoid the socalled hourglass structure [11]. The front end of the workpiece was chamfered for achieving a well-de®ned contact surface at the beginning of the rolling. In order to drive the material into the roll gap a small force was utilised, this force being reduced to zero as soon as biting had taken place. 2.4. Calculation times It is dif®cult to carry out simulations that are reliable and at the same time are fast: here is an example. The behaviour of the crack should be determined for steady state conditions and at the same time the calculation time should be kept low. Therefore a rather short workpiece length was chosen, but not so short that the steady state conditions necessary for evaluating the behaviour of the crack in a meaningful way, were violated. Regarding the ®rst pass, a slab length of 550 mm was used. The length obtained after the reduction in thickness was then reduced for keeping the simulations reasonably rapid. This procedure was repeated after each pass. The calculation time is controlled by numerous parameters, but it is possible to vary only a few of them. Amongst these parameters the FE-mesh, Young's modulus, density, TSSF and the dampers are to be mentioned. Young's modulus used in the simulations is higher than the true value. The in¯uence of dynamic effects was assumed to be negligible, with the density being regarded as a parameter that could be used for controlling the calculation time. The number of elements in the vicinity of the crack has a very big in¯uence. More nodes than 5 for describing the crack behaviour should imply the possibility of a more accurate description, however, the calculation time would then increase drastically. 3. Reliability tests The reliability of the theoretical predictions was checked in a pilot-plant mill of roll radius 50 mm with aluminium as the model material. The workpiece had a width of 70 mm. The initial thickness was chosen to H025.9 mm for obtaining similarity with the rolling schedule of the industrial reference mill. By this choice the same R/H01.93 was obtained. A central longitudinal V-shaped surface crack of
Fig. 4. Comparison between theory and pilot-plant Al-experiments.
depth 2 mm was machined on the upper surface of the specimen. In order to facilitate the evaluation of the experiments, the crack angle was chosen to be as large as 308. The workpiece was rolled to the following thicknesses: H122.4 mm, H218.9 mm and H315.4 mm. The workpiece, of temperature 4508C, was rolled at a speed of 0.3 m/s. After each pass the rolling was interrupted and a test piece was cut off perpendicular to the rolling direction, for polishing and photographing. The workpiece was then reheated and rolled once again. The friction coef®cient used in the FE-simulations was chosen to be 0.4 [12] and the material was assumed to be rigid-perfectly plastic. From the comparison between theory and experiments (Fig. 4), it is clear that the predictions obtained from the simulation program are satisfactory.
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Fig. 7. The crack depth decreases slower than the thickness of the workpiece at the end of the rolling schedule. The influence of friction is marginal. Rolling schedule due to reference mill.
Fig. 5. Cross sections A±A, B±B, C±C and D±D showing the shape of the crack during the first pass of the reference mill.
4. Results Five parameters were used in order to describe the crack behaviour: h/H, 0 , 00 , h and w (Fig. 3). A decreasing value of theratio h/H meansthat the crack depth is reduced morerapidly than the thickness of the workpiece. This constitutes a criterion that must be ful®lled for eliminating the crack. The angles 0 and 00 describe the risk of forming a defect, made up by a rolled-in oxide ¯ake:thesmallerthe angles become,thegreater the risk. The crack width w and the crack depth h are utilised for giving a more complete description of the behaviour during the different rolling schedules. Consequently the optimum case should include a rapid decrease of h, at the same time as 0 , 00 and w increase. The bottom of the crack then moves towards the workpiece surface at the same time as the crack is torn apart. 4.1. Crack behaviour during one pass The crack shape has been studied for four positions during the ®rst pass of the reference mill (Fig. 5). They are denoted
as the cross section A±A before entering the roll gap, B±B just before the entrance, C±C within the gap and ®nally D±D after exiting the plastic region. The simulation was carried out for the process data in the italicized parts of Tables 1 and 2, i.e. R425 mm, H0220 mm, H1190 mm and 0.2. From Fig. 5, it is clear that the crack is slightly deformed just before entering the gap, cross section B±B. For this position, the bottom angle 00 has already decreased a little. When passing the gap both crack angles 0 and 00 are reduced in size. The reduction of the bottom angle, 00 , is so heavy that the risk of a remaining defect is obvious (D±D). 4.2. Influence of friction Simulations 1±3 of Table 1, using the process data of the reference mill, are carried out for the friction coef®cients, 0.20, 0.35 and 0.50. From Fig. 6 it is clear that the crack width w decreases less when the friction is high. The depth of the crack decreases approximately linearly with the height of the workpiece and the in¯uence of friction is found to be insigni®cant. A more detailed study (Fig. 7) utilising the parameter h/H, reveals however that the crack depth decreases more slowly than the height of the workpiece during the ®nal passes. Also from this way of presenting the results, it is
Fig. 6. Influence of friction on the reduction of crack width and crack depth. Rolling schedule due to reference mill.
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Fig. 8. Influence of friction on crack behaviour. Rolling schedule due to reference mill.
clear that the in¯uence of friction is negligible regarding the decrease in crack depth. In order to analyse the behaviour of the crack regarding the risk of surface oxide ¯ake formation as in¯uenced by friction, the crack angles 0 and 00 have been studied. As is clear from Fig. 8, the bottom angle 00 equals zero, or is very close to zero, at the end of the rolling. At the beginning of the rolling schedule, passes 1±3, the upper angle 0 decreases rapidly. During the remaining passes, the angle remains almost constant for 0.20 and 0.35. However, for 0.5, the angle increases during the remaining passes. The main conclusion is that it is dif®cult to prevent the bottom part of the crack from forming a defect. The series of crack pro®les that make up the basis for the evaluations presented in Figs. 6±8, is shown in Fig. 9. 4.3. Influence of roll radius In order to study the in¯uence of the roll radius, three radii have been used, R225, 425 and 625 mm, simulations 1, 4 and 5 in Table 1, Fig. 10. R425 mm represents the value of the reference mill. The friction coef®cient has been kept constant and equal to the reference value, 0.2. Also, the reduction in thickness/pass has been chosen from the reference. As mentioned earlier, it is clear that the reduction in crack depth is independent of friction (Fig. 6). Here it is found that this is true also regarding the in¯uence of roll radius (Fig. 10). The depth of the crack decreases linearly with the workpiece height. The greatest decrease in crack width, w, is obtained for the largest roll radius R625 mm and equals zero at the end of the rolling schedule. The results presented so far indicate that the likelihood for the formation of an oxide ¯ake defect is great for a large roll radius. When the parameter h/H is used for a more detailed study (Fig. 11), the result is that the reduction in crack depth follows that of the workpiece thickness during the initial passes. The curve h/Hf(H) is almost horizontal. Towards the end of the schedule, similar to the study regarding
Fig. 9. Series of pictures from FE-simulations. Influence of friction due to the rolling schedule of the reference mill. The pictures show the crack shape after the workpiece has passed the roll gap. Horizontal scale 5:1 and vertical scale 1:1.
different friction conditions, the crack depth decreases more slowly. The in¯uence of the roll radius seems to be marginal. The avoiding of a surface oxide defect at the crack bottom by choosing another roll radius compared to that used in the reference mill does not seem possible (Fig. 12). The diagram is built on the same reductions/pass and the same
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147
Fig. 10. Influence of roll radius on crack width and crack depth. Draft/pass and 0.2 due to reference mill.
Fig. 11. The crack depth decreases slower than the thickness of the workpiece at the end of the rolling schedule. The influence of roll radius is marginal. Draft/pass and 0.2 due to reference mill.
Fig. 13. Series of pictures from FE-simulations. Influence of roll radius. The pictures show the crack shape after the workpiece has passed the roll gap. Horizontal scale 5:1 and vertical scale 1:1. Draft/pass and 0.2 due to reference mill.
Fig. 12. Influence of roll radius on crack behaviour. Draft/pass and 0.2 due to reference mill.
friction coef®cient as those of the reference mill. The upper part of the crack, 0 decreases rapidly and ®nally closes for a large roll radius. This closure is, however, much slower than for the bottom part of the crack, 00 , where all roll radii studied result in crack closure. Regarding the bottom crack behaviour, the roll radius seems to have very little in¯uence. Basic data for the diagrams presented in Figs. 10±12 was extracted from the series of pro®les presented in Fig. 13.
4.4. Influence of rolling schedules The behaviour of the crack has been studied for simulations 1, 6 and 7 (Table 1, Fig. 14). Simulation 1 corresponds to that of the reference, simulation 6 to light reductions/pass and simulation 7 to heavy reductions/pass. The rolling schedules are indicated in Table 2. The schedules were chosen in such a way that it is possible to compare the results after rolling to different workpiece thicknesses. For light reductions/pass, the crack depth h, decreases rapidly, whilst at the same time the decrease in crack width, w, is slow compared to the rolling schedule built on heavy reductions/pass (Fig. 14).
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Fig. 14. Influence of rolling schedule on crack width depth and crack depth. R425 mm and 0.2 due to reference mill. Series 1: light reductions/pass; series 2: reference mill; series 3: heavy reductions/pass.
Fig. 15. The crack depth decreases slower than the thickness of the workpiece at the end of the rolling schedule especially for heavy reductions/pass.
A more detailed study, making use of the parameter h/H, reveals that the decrease in crack depth becomes slower during the ®nal passes of a rolling schedule, especially if the schedule is built on heavy reductions/pass (Fig. 15). When the behaviour of the crack is studied by means of the angles 0 and 00 (Fig. 16), it is once again found that the bottom angle, 00 , is closed for all of the rolling schedules studied, and most rapidly for that built on light reductions/ pass, series 1. The upper angle, 0 , decreases more slowly. However heavy reductions/pass, series 3, result in complete closure during the ®nal passes; this angle is not closed for the two other series. Consequently the bottom angle decreases
Fig. 17. Series of pictures from FE-simulations. Influence of rolling schedules. The pictures show the crack shape after the workpiece has passed the roll gap. Horizontal scale 5:1 and vertical scale 1:1. R425 mm and 0.2 due to reference mill.
more rapidly for light reductions/pass, contradictory to the top angle, which is reduced more rapidly for heavy reductions/pass. In the same way as before the basis for the diagrams presented under this heading is the series of pro®les obtained from LS-DYNA3D (Fig. 17). 5. Discussion Fig. 16. Influence of rolling schedules on crack behaviour. Series 2: R425 mm and 0.2 due to reference mill.
The rapid decrease in crack width, w, for the large roll radius (Fig. 10), and heavy reductions/pass (Fig. 14), is
E. Ervasti, U. StaÊhlberg / Journal of Materials Processing Technology 94 (1999) 141±150
expected because for such conditions the contact length in the rolling direction becomes large compared to the mean value of the workpiece height. These conditions mean that a high hydrostatic pressure, favouring crack closure, is found close to the contact surface. The heaviest decreases in crack depth were found for light reductions/pass (Fig. 14). This is dif®cult to explain, but the result seems reasonable because for such conditions the largest strains are expected close to the roll/workpiece contact surface The curve h/Hf(H) (Fig. 15), reveals that the decrease in crack depth becomes slower during the ®nal passes of a rolling schedule, especially if the schedule is built on heavy reductions/pass. This might have to do with the development of a large rigid, or almost rigid, region beneath the contact surface, during the last passes when the ratio between the contact length and mean workpiece height is large. A possible explanation for the result that the crack bottom angle, 00 , for all of the conditions studied, decreases more rapidly than the upper angle, 0 , might be that the distance from the roll/workpiece contact surface is slightly greater and thus the in¯uence of the contact shear stress, counteracting the inward material ¯ow is smaller. Heat transfer and heat generation due to plastic deformation have not been taken into account. The contact times are, however, short and the volume of material is large so that the reliability of the present results should not be strongly violated because of this assumption. The investigation has been restricted to cracks along the longitudinal central line of the slab surface. Because of the large width of the slabs and consequently the small spread, the results are considered to be representative for a large part of the workpiece. The work presented here, however, cannot be used for predicting the crack behaviour very close to the long sides of the workpiece. Regardless of the choice of rolling schedule, roll radius or friction, it was found that the bottom side surfaces of the crack came into contact, which in turn, means that a surface oxide defect is formed. However, this result can be interpreted in the following way. If the bottom part of the crack is fully welded already at the very beginning of a rolling schedule, the oxidised surface band will be prevented from further oxidising at the same time as it will be heavily extended during the rest of the rolling. This means that the oxide band will be torn to pieces with strongly bonded virgin metal in between and consequently the crack deteriorating in¯uence on the ®nal properties of the product might be negligible. Due to the hypothesis mentioned above, closure of the crack at an early stage of the rolling should be bene®cial. However, according to the results, the upper part of the crack is very dif®cult to close. The only way to do this is to make use of very heavy reductions/pass (Fig. 16). Fig. 12 also reveals that a large roll radius should favour the closure of the upper part of the crack.
149
6. Conclusions Due to the analysis, the following conclusions can be drawn for a V-shaped longitudinal crack of depth 20 mm and angle of 68, found on the surface of a slab of width 1000 mm and thickness 220 mm: 1. The reduction of the crack depth follows that of the slab at the beginning of the rolling schedules. At the end of the rolling, the reduction of the crack depth is smaller than that of the bulk material. 2. It is impossible to avoid the formation of a Y-shaped crack. This occurs no matter what the choice of rolling schedule, roll radius and friction. Thus, the creation of an oxide flake cannot be prevented by optimising the rolling conditions. 3. The damage caused by oxidation can, however, in all probability, be heavily reduced if the rolling is carried out in such a way that the bottom part of the crack is fully welded at the beginning of the rolling schedule. The bottom part of the crack is then prevented from further oxidising. During the subsequent rolling, the oxidised surface band will be extended and torn to pieces with virgin metal in between, ensuring a strong bond. 4. Early bonding of the bottom part of the crack is promoted by: (i) light reductions/pass; and (ii) small roll radii. 5. Early closure of the upper part of the crack is more difficult to obtain. Later closure is possible by utilising heavy reductions/pass and a large roll radii. For the same reason as for the bottom part of the crack, this should be advantageous. 6. The practical recommendations of this work are: (i) light reductions/pass during the first passes in order to close the crack bottom rapidly; and (ii) after the bottom part of the crack has been closed, heavy reductions/pass should be employed in order to close the upper part of the crack also as soon as possible. 7. Nomenclature h H H0 R v w 0 00
depth of crack height of workpiece initial height of workpiece roll radius velocity of rolls width of crack angle defining the upper part of the crack angle defining the bottom part of the crack friction coefficient due to Coulomb
Acknowledgements The authors wish to thank Harry Petersson, SSAB TunnplaÊt AB, for valuable ideas and access to reference process data. One of us (Esa Ervasti) is indebted to the Iron and Steel
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Master's Association for ®nancial support. Thanks are also expressed to Christer Eggertsson, Swedish Institute for Metals Research, for great help with the Al-experiments and to our DYNA-support, Ragnar Nilsson. References [1] E. Hawbolt, F. Weinberg, J. Brimacombe, Influence of hot working on internal cracks in continuously cast steel billets, Metall. Trans. B 10B (1979) 229±236. [2] K. Brimacombe, K. Surimachi, Crack formation in the continuous casting of steel, Metall. Trans. B 8B (1977) 489±504. [3] Y. Maehara, K. Yasumoto, H. Tomono, T. Nagamichi, Y. Ohmori, Surface cracking mechanism of continuously cast low alloy steel slabs, Mat., Sci. and Tech. 6 (1990) 793±806. [4] Y. Maehara, K. Yasumoto, Y. Sugitani, K. Gunji, Effect of carbon on hot ductility of as cast low alloy steels, Trans. ISIJ 25 (1985) 1045± 1052.
[5] F. Vodopivec, M. Torkar, M. Debelak, M. Kmetic, F. Haller, F. Kaucic, Influence of aluminium on solidification structure and initial deformability of continuously cast C±Mn±Si±N± steel, Mat., Sci. and Tech. 4 (1988) 917±925. [6] H. Bruce, Surface defects on continuously cast materials, Swedish Institute for Metals and Research, IM-2791 (1991) (in Swedish). [7] H. Suzuki, S. Nishimura, S. Yamaguchi, Characteristic of hot ductility in steels subjected to the melting and solidification, Trans. ISIJ 22 (1982) 48±56. [8] A. Lieberman, Y. Kan, N. Mironova, M. Luker, O. Lapshin, Transformation of surface defects on continuously cast billets during hot rolling, steel in the USSR, UDC 621.746.628.047, 14 (1984) 428±431. [9] Y. Takemura, O. Tsubakibara, M. Saito, S. Mizogichi, T. Kuwabara, Direct-linked continuous caster-hot rolling mill process, Nippon Steel technical report no. 21, 1983, pp. 189±201. [10] H. Suzuki et al., Studies of flow stress of metals and alloys, Report of Industrial Science, University of Tokyo 18 (3) (1968) 174±240. [11] J. Hallquist, LS-DYNA, Theoretical Manual, LSTC-report, 1991. [12] J.A. Schey, Metal Forming Processes, Friction and Lubrication, Marcel Dekker, New York, 1970.
Behaviour of longitudinal surface cracks in the hot rolling of steel slabs Esa Ervasti, Ulf StaÊhlberg* Materials Forming, Department of Materials Processing, Royal Institute of Technology, Brinellvagen 23, Stockholm S-10044, Sweden Received 10 February 1998
Abstract The behaviour of a longitudinal V-shaped crack, on the surface of a continuously cast steel slab, is studied during hot rolling. The analysis is carried out by means of the commercial FE-code LS-DYNA3D. Process parameters obtained from industry are used as a reference. The slab of initial width 1000 mm and thickness 220 mm is rolled down to 30 mm. It is assumed that the material can be treated as rigid-perfectly plastic and that the cracks do not propagate. The latter assumption is in agreement with industrial observations for a steel grade similar to that analysed here. The aim of the study is to investigate the possibility of controlling the plastic deformation so that the cracks disappear or so that their deteriorating effects are minimised. The analysis is focused upon the in¯uence of friction, roll radius and rolling schedule on the change in the shape of a crack of initial depth 20 mm and a crack angle of 68. The reliability of the simulations is checked by pilot-plant experiments using aluminium as the model material for steel. The results indicate that it is not possible to prevent the bottom side surfaces of the crack from coming into contact, especially not for small reductions/pass and small roll radii. The in¯uence of friction was found to be marginal. Contact between the crack surfaces is found already at the beginning of the rolling, as the V-shaped crack is being changed to Y-shape. Considering the upper part of the crack, this remained open during the majority of the schedules studied. However for heavy reduction/pass and a large roll radius, this part of the crack is closed also, but not before the ®nal passes. If the bottom side surfaces of the crack are in complete contact, they are prevented from further oxidisation. It is assumed that for such conditions the deteriorating in¯uence of the bottom part of the defect decreases during subsequent rolling. During the elongation of the workpiece, the oxide ¯ake of the Y-crack bottom is broken into splinters with oxide free material in between, making the creation of a high performance weld possible. Provided that this supposition is correct, the best results should be obtained for light reductions/pass at the beginning of the rolling, resulting in an early closure of the crack bottom, followed by heavy reductions/pass enabling also the closure and oxide splintering of the upper part of the folded crack. # 1999 Published by Elsevier Science S.A. All rights reserved. Keywords: Longitudinal surface cracks; Hot rolling; Steel slabs
1. Introduction Longitudinal surface cracks are often found on continuously cast slabs, Fig. 1. Compared to transversal cracks, longitudinal cracks are usually deeper. Often they are located in regions close to the centre line of the slab surface. Almost without exception, they are found to be dif®cult to eliminate during subsequent hot rolling and consequently they will constitute defects on the rolled product [1]. The cracks are formed because of transverse tensile stresses caused by non-homogenous cooling conditions during the *Corresponding author. Tel.: +46-8790-83-84; fax: +46-8790-09-40.
solidi®cation of the material. Very often, the so-called ``hot spots'' are found on the skin of the slab during casting. Other reasons for crack formation could be tensile stresses introduced by high hydrostatic pressures within the liquid part of the strand, peritectic phase transformation and friction in the transverse direction between the skin and the mould [2±8]. Examples of sensitive steel grades are those characterised by approximately 0.12% C and a high content of sulphur compared with manganese. Much research has been carried out in order to improve the casting process for avoiding these cracks, some examples being represented by [2,3,8,9]. No theoretical work has been found dealing with the behaviour of longitudinal cracks during hot rolling.
0924-0136/99/$ ± see front matter # 1999 Published by Elsevier Science S.A. All rights reserved. PII: S 0 9 2 4 - 0 1 3 6 ( 9 9 ) 0 0 0 9 0 - 4
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Fig. 1. (a) Longitudinal cracks on the surface of a continuously cast slab. (b) Cross-section showing a longitudinal crack.
The present analysis is carried out with reference data obtained from industry. The initial slab is of 1000 mm width and 220 mm thickness. The rolling is performed as reverse rolling during the ®rst 7 passes. During the ®rst 4 passes, the slab thickness is reduced by 30 mm/pass, during pass nos. 5 and 6 by 25 mm/pass and ®nally during pass no. 7 by 20 mm down to a thickness of 30 mm. The reversal rolling is followed by continuous wide strip rolling in six stands. The study presented here is limited to the reversing part of the rolling. The behaviour of the cracks as in¯uenced by friction, roll radius and rolling schedule is analysed by means of the commercial FE-code, LS-DYNA3D. The aim of the work is to study whether it is possible to eliminate the cracks or to minimise their deteriorating in¯uence by hot rolling. The reliability of the theoretical results are checked by pilot-plant experiments using aluminium as a model material for steel. 2. FE-simulations To begin with, the input data and process characteristics are presented, after which details regarding the technique of simulation are discussed.
2.1. Input data and process characteristics The initial length of the workpiece is chosen to be 550 mm for all the rolling schedules studied. The initial crack has been assumed to be V-shaped with an angle of 68 and a depth of 20 mm. The rolling speed of the reference mill is 0.8 m/s during the ®rst pass and 2.4 m/s during the remaining passes. In Tables 1 and 2, input data used for different simulations are speci®ed. The material composition and work temperatures of the reference mill are clear from Tables 3 and 4. In addition to the simulations regarding the reference conditions, simulations have been carried out for: (i) two friction coef®cients, both of higher value than that of the reference; (ii) one smaller and one larger roll radius; and (iii) one rolling schedule, built on lighter reductions/pass and one built on heavier reductions/pass. All of the simulations cover a reduction in thickness of 220±30 mm according to Tables 1 and 2. Reference data according to industrial production are marked in italics and the symbol *. From the stress±strain curves presented in [10], it is clear that it is reasonable to treat the material as rigid-perfectly plastic. The strain rate is approximately 1 sÿ1 during the ®rst pass of the reference schedule and increases during the following to a maximum value of approximately 15 sÿ1.
Table 1 Process parameters Simulation no
Roll radius (mm)
Reductions/pass due to reference mill*
1 2 3 4 5 6 7
425* 425* 425* 225 625 425* 425*
X X X X X
Light reductions/pass
X
Heavy reductions/pass
X
0.20* 0.35 0.50 0.20* 0.20* 0.20* 0.20*
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Table 2 Rolling schedules Schedule
H0
H1
H2
H3
H4
H5
H6
H7
H8
H9
H10
H11
H12
H13
Reference Light reductions Heavy reductions
220 220 220
190 205 175
160 190 130
130 175 85
100 160 50
75 145 30
50 130
30 115
100
85
70
55
40
30
Table 3 Chemical composition of the workpiece material (wt.%). C
Si
Cr
Mn
P
S
Cu
0.12
0.20
0.08
0.50
0.01
0.03
0.10
Table 4 Workpiece temperatures after each pass due reference rolling Pass no
1
2
3
4
5
6
7
Workpiece temperature (C8)
1230
1223
1218
1210
1200
1188
1156
2.2. Element structure
Fig. 3. The positions of nodes on the boundary surfaces of the crack.
Shell elements are used for describing the rolls, which are assumed to be rigid. The work material is built up by brick elements. The elements were chosen in such a way that the ratio between their longest and shortest sides at no time exceeded a value of 3. On the contact surface between the workpiece and the roll, the element side of the material was chosen to be shorter than that of the roll in order to secure a stable simulation. The structure of the elements at the beginning of a simulation is clear from Fig. 2. In order to simulate the behaviour of the crack, the earlier described structure was completed with smaller elements
characterised by ®ve nodes in a crack cross-section, Fig. 3. The V-shaped cracks have an extension equal to the length of the workpiece. When simulating the crack behaviour during the subsequent pass, the shape obtained during the earlier passes was utilised. The step-by-step change in crack geometry was determined by establishing the new positions of the nodes. If opposite sides of the crack came into contact, the friction coef®cient along the oxidised surfaces was assumed to be 0.2.
Fig. 2. The rolls are built up by shell elements and the workpiece by brick elements.
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2.3. Practical problems In order to prevent the nodes of the rolls from penetrating into the work material, penalty functions were used. These functions act as springs along the roll/workpiece contact surface and are associated with a time step scale factor (TSSF) in order to avoid a critical length of the time step being exceeded. By combining these two functions in an adequate way, stable contact between the rolls and the workpiece was obtained. The elements of the workpiece were made stiff, Hancock-stiffness, in order to avoid the socalled hourglass structure [11]. The front end of the workpiece was chamfered for achieving a well-de®ned contact surface at the beginning of the rolling. In order to drive the material into the roll gap a small force was utilised, this force being reduced to zero as soon as biting had taken place. 2.4. Calculation times It is dif®cult to carry out simulations that are reliable and at the same time are fast: here is an example. The behaviour of the crack should be determined for steady state conditions and at the same time the calculation time should be kept low. Therefore a rather short workpiece length was chosen, but not so short that the steady state conditions necessary for evaluating the behaviour of the crack in a meaningful way, were violated. Regarding the ®rst pass, a slab length of 550 mm was used. The length obtained after the reduction in thickness was then reduced for keeping the simulations reasonably rapid. This procedure was repeated after each pass. The calculation time is controlled by numerous parameters, but it is possible to vary only a few of them. Amongst these parameters the FE-mesh, Young's modulus, density, TSSF and the dampers are to be mentioned. Young's modulus used in the simulations is higher than the true value. The in¯uence of dynamic effects was assumed to be negligible, with the density being regarded as a parameter that could be used for controlling the calculation time. The number of elements in the vicinity of the crack has a very big in¯uence. More nodes than 5 for describing the crack behaviour should imply the possibility of a more accurate description, however, the calculation time would then increase drastically. 3. Reliability tests The reliability of the theoretical predictions was checked in a pilot-plant mill of roll radius 50 mm with aluminium as the model material. The workpiece had a width of 70 mm. The initial thickness was chosen to H025.9 mm for obtaining similarity with the rolling schedule of the industrial reference mill. By this choice the same R/H01.93 was obtained. A central longitudinal V-shaped surface crack of
Fig. 4. Comparison between theory and pilot-plant Al-experiments.
depth 2 mm was machined on the upper surface of the specimen. In order to facilitate the evaluation of the experiments, the crack angle was chosen to be as large as 308. The workpiece was rolled to the following thicknesses: H122.4 mm, H218.9 mm and H315.4 mm. The workpiece, of temperature 4508C, was rolled at a speed of 0.3 m/s. After each pass the rolling was interrupted and a test piece was cut off perpendicular to the rolling direction, for polishing and photographing. The workpiece was then reheated and rolled once again. The friction coef®cient used in the FE-simulations was chosen to be 0.4 [12] and the material was assumed to be rigid-perfectly plastic. From the comparison between theory and experiments (Fig. 4), it is clear that the predictions obtained from the simulation program are satisfactory.
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Fig. 7. The crack depth decreases slower than the thickness of the workpiece at the end of the rolling schedule. The influence of friction is marginal. Rolling schedule due to reference mill.
Fig. 5. Cross sections A±A, B±B, C±C and D±D showing the shape of the crack during the first pass of the reference mill.
4. Results Five parameters were used in order to describe the crack behaviour: h/H, 0 , 00 , h and w (Fig. 3). A decreasing value of theratio h/H meansthat the crack depth is reduced morerapidly than the thickness of the workpiece. This constitutes a criterion that must be ful®lled for eliminating the crack. The angles 0 and 00 describe the risk of forming a defect, made up by a rolled-in oxide ¯ake:thesmallerthe angles become,thegreater the risk. The crack width w and the crack depth h are utilised for giving a more complete description of the behaviour during the different rolling schedules. Consequently the optimum case should include a rapid decrease of h, at the same time as 0 , 00 and w increase. The bottom of the crack then moves towards the workpiece surface at the same time as the crack is torn apart. 4.1. Crack behaviour during one pass The crack shape has been studied for four positions during the ®rst pass of the reference mill (Fig. 5). They are denoted
as the cross section A±A before entering the roll gap, B±B just before the entrance, C±C within the gap and ®nally D±D after exiting the plastic region. The simulation was carried out for the process data in the italicized parts of Tables 1 and 2, i.e. R425 mm, H0220 mm, H1190 mm and 0.2. From Fig. 5, it is clear that the crack is slightly deformed just before entering the gap, cross section B±B. For this position, the bottom angle 00 has already decreased a little. When passing the gap both crack angles 0 and 00 are reduced in size. The reduction of the bottom angle, 00 , is so heavy that the risk of a remaining defect is obvious (D±D). 4.2. Influence of friction Simulations 1±3 of Table 1, using the process data of the reference mill, are carried out for the friction coef®cients, 0.20, 0.35 and 0.50. From Fig. 6 it is clear that the crack width w decreases less when the friction is high. The depth of the crack decreases approximately linearly with the height of the workpiece and the in¯uence of friction is found to be insigni®cant. A more detailed study (Fig. 7) utilising the parameter h/H, reveals however that the crack depth decreases more slowly than the height of the workpiece during the ®nal passes. Also from this way of presenting the results, it is
Fig. 6. Influence of friction on the reduction of crack width and crack depth. Rolling schedule due to reference mill.
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Fig. 8. Influence of friction on crack behaviour. Rolling schedule due to reference mill.
clear that the in¯uence of friction is negligible regarding the decrease in crack depth. In order to analyse the behaviour of the crack regarding the risk of surface oxide ¯ake formation as in¯uenced by friction, the crack angles 0 and 00 have been studied. As is clear from Fig. 8, the bottom angle 00 equals zero, or is very close to zero, at the end of the rolling. At the beginning of the rolling schedule, passes 1±3, the upper angle 0 decreases rapidly. During the remaining passes, the angle remains almost constant for 0.20 and 0.35. However, for 0.5, the angle increases during the remaining passes. The main conclusion is that it is dif®cult to prevent the bottom part of the crack from forming a defect. The series of crack pro®les that make up the basis for the evaluations presented in Figs. 6±8, is shown in Fig. 9. 4.3. Influence of roll radius In order to study the in¯uence of the roll radius, three radii have been used, R225, 425 and 625 mm, simulations 1, 4 and 5 in Table 1, Fig. 10. R425 mm represents the value of the reference mill. The friction coef®cient has been kept constant and equal to the reference value, 0.2. Also, the reduction in thickness/pass has been chosen from the reference. As mentioned earlier, it is clear that the reduction in crack depth is independent of friction (Fig. 6). Here it is found that this is true also regarding the in¯uence of roll radius (Fig. 10). The depth of the crack decreases linearly with the workpiece height. The greatest decrease in crack width, w, is obtained for the largest roll radius R625 mm and equals zero at the end of the rolling schedule. The results presented so far indicate that the likelihood for the formation of an oxide ¯ake defect is great for a large roll radius. When the parameter h/H is used for a more detailed study (Fig. 11), the result is that the reduction in crack depth follows that of the workpiece thickness during the initial passes. The curve h/Hf(H) is almost horizontal. Towards the end of the schedule, similar to the study regarding
Fig. 9. Series of pictures from FE-simulations. Influence of friction due to the rolling schedule of the reference mill. The pictures show the crack shape after the workpiece has passed the roll gap. Horizontal scale 5:1 and vertical scale 1:1.
different friction conditions, the crack depth decreases more slowly. The in¯uence of the roll radius seems to be marginal. The avoiding of a surface oxide defect at the crack bottom by choosing another roll radius compared to that used in the reference mill does not seem possible (Fig. 12). The diagram is built on the same reductions/pass and the same
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Fig. 10. Influence of roll radius on crack width and crack depth. Draft/pass and 0.2 due to reference mill.
Fig. 11. The crack depth decreases slower than the thickness of the workpiece at the end of the rolling schedule. The influence of roll radius is marginal. Draft/pass and 0.2 due to reference mill.
Fig. 13. Series of pictures from FE-simulations. Influence of roll radius. The pictures show the crack shape after the workpiece has passed the roll gap. Horizontal scale 5:1 and vertical scale 1:1. Draft/pass and 0.2 due to reference mill.
Fig. 12. Influence of roll radius on crack behaviour. Draft/pass and 0.2 due to reference mill.
friction coef®cient as those of the reference mill. The upper part of the crack, 0 decreases rapidly and ®nally closes for a large roll radius. This closure is, however, much slower than for the bottom part of the crack, 00 , where all roll radii studied result in crack closure. Regarding the bottom crack behaviour, the roll radius seems to have very little in¯uence. Basic data for the diagrams presented in Figs. 10±12 was extracted from the series of pro®les presented in Fig. 13.
4.4. Influence of rolling schedules The behaviour of the crack has been studied for simulations 1, 6 and 7 (Table 1, Fig. 14). Simulation 1 corresponds to that of the reference, simulation 6 to light reductions/pass and simulation 7 to heavy reductions/pass. The rolling schedules are indicated in Table 2. The schedules were chosen in such a way that it is possible to compare the results after rolling to different workpiece thicknesses. For light reductions/pass, the crack depth h, decreases rapidly, whilst at the same time the decrease in crack width, w, is slow compared to the rolling schedule built on heavy reductions/pass (Fig. 14).
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Fig. 14. Influence of rolling schedule on crack width depth and crack depth. R425 mm and 0.2 due to reference mill. Series 1: light reductions/pass; series 2: reference mill; series 3: heavy reductions/pass.
Fig. 15. The crack depth decreases slower than the thickness of the workpiece at the end of the rolling schedule especially for heavy reductions/pass.
A more detailed study, making use of the parameter h/H, reveals that the decrease in crack depth becomes slower during the ®nal passes of a rolling schedule, especially if the schedule is built on heavy reductions/pass (Fig. 15). When the behaviour of the crack is studied by means of the angles 0 and 00 (Fig. 16), it is once again found that the bottom angle, 00 , is closed for all of the rolling schedules studied, and most rapidly for that built on light reductions/ pass, series 1. The upper angle, 0 , decreases more slowly. However heavy reductions/pass, series 3, result in complete closure during the ®nal passes; this angle is not closed for the two other series. Consequently the bottom angle decreases
Fig. 17. Series of pictures from FE-simulations. Influence of rolling schedules. The pictures show the crack shape after the workpiece has passed the roll gap. Horizontal scale 5:1 and vertical scale 1:1. R425 mm and 0.2 due to reference mill.
more rapidly for light reductions/pass, contradictory to the top angle, which is reduced more rapidly for heavy reductions/pass. In the same way as before the basis for the diagrams presented under this heading is the series of pro®les obtained from LS-DYNA3D (Fig. 17). 5. Discussion Fig. 16. Influence of rolling schedules on crack behaviour. Series 2: R425 mm and 0.2 due to reference mill.
The rapid decrease in crack width, w, for the large roll radius (Fig. 10), and heavy reductions/pass (Fig. 14), is
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expected because for such conditions the contact length in the rolling direction becomes large compared to the mean value of the workpiece height. These conditions mean that a high hydrostatic pressure, favouring crack closure, is found close to the contact surface. The heaviest decreases in crack depth were found for light reductions/pass (Fig. 14). This is dif®cult to explain, but the result seems reasonable because for such conditions the largest strains are expected close to the roll/workpiece contact surface The curve h/Hf(H) (Fig. 15), reveals that the decrease in crack depth becomes slower during the ®nal passes of a rolling schedule, especially if the schedule is built on heavy reductions/pass. This might have to do with the development of a large rigid, or almost rigid, region beneath the contact surface, during the last passes when the ratio between the contact length and mean workpiece height is large. A possible explanation for the result that the crack bottom angle, 00 , for all of the conditions studied, decreases more rapidly than the upper angle, 0 , might be that the distance from the roll/workpiece contact surface is slightly greater and thus the in¯uence of the contact shear stress, counteracting the inward material ¯ow is smaller. Heat transfer and heat generation due to plastic deformation have not been taken into account. The contact times are, however, short and the volume of material is large so that the reliability of the present results should not be strongly violated because of this assumption. The investigation has been restricted to cracks along the longitudinal central line of the slab surface. Because of the large width of the slabs and consequently the small spread, the results are considered to be representative for a large part of the workpiece. The work presented here, however, cannot be used for predicting the crack behaviour very close to the long sides of the workpiece. Regardless of the choice of rolling schedule, roll radius or friction, it was found that the bottom side surfaces of the crack came into contact, which in turn, means that a surface oxide defect is formed. However, this result can be interpreted in the following way. If the bottom part of the crack is fully welded already at the very beginning of a rolling schedule, the oxidised surface band will be prevented from further oxidising at the same time as it will be heavily extended during the rest of the rolling. This means that the oxide band will be torn to pieces with strongly bonded virgin metal in between and consequently the crack deteriorating in¯uence on the ®nal properties of the product might be negligible. Due to the hypothesis mentioned above, closure of the crack at an early stage of the rolling should be bene®cial. However, according to the results, the upper part of the crack is very dif®cult to close. The only way to do this is to make use of very heavy reductions/pass (Fig. 16). Fig. 12 also reveals that a large roll radius should favour the closure of the upper part of the crack.
149
6. Conclusions Due to the analysis, the following conclusions can be drawn for a V-shaped longitudinal crack of depth 20 mm and angle of 68, found on the surface of a slab of width 1000 mm and thickness 220 mm: 1. The reduction of the crack depth follows that of the slab at the beginning of the rolling schedules. At the end of the rolling, the reduction of the crack depth is smaller than that of the bulk material. 2. It is impossible to avoid the formation of a Y-shaped crack. This occurs no matter what the choice of rolling schedule, roll radius and friction. Thus, the creation of an oxide flake cannot be prevented by optimising the rolling conditions. 3. The damage caused by oxidation can, however, in all probability, be heavily reduced if the rolling is carried out in such a way that the bottom part of the crack is fully welded at the beginning of the rolling schedule. The bottom part of the crack is then prevented from further oxidising. During the subsequent rolling, the oxidised surface band will be extended and torn to pieces with virgin metal in between, ensuring a strong bond. 4. Early bonding of the bottom part of the crack is promoted by: (i) light reductions/pass; and (ii) small roll radii. 5. Early closure of the upper part of the crack is more difficult to obtain. Later closure is possible by utilising heavy reductions/pass and a large roll radii. For the same reason as for the bottom part of the crack, this should be advantageous. 6. The practical recommendations of this work are: (i) light reductions/pass during the first passes in order to close the crack bottom rapidly; and (ii) after the bottom part of the crack has been closed, heavy reductions/pass should be employed in order to close the upper part of the crack also as soon as possible. 7. Nomenclature h H H0 R v w 0 00
depth of crack height of workpiece initial height of workpiece roll radius velocity of rolls width of crack angle defining the upper part of the crack angle defining the bottom part of the crack friction coefficient due to Coulomb
Acknowledgements The authors wish to thank Harry Petersson, SSAB TunnplaÊt AB, for valuable ideas and access to reference process data. One of us (Esa Ervasti) is indebted to the Iron and Steel
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Master's Association for ®nancial support. Thanks are also expressed to Christer Eggertsson, Swedish Institute for Metals Research, for great help with the Al-experiments and to our DYNA-support, Ragnar Nilsson. References [1] E. Hawbolt, F. Weinberg, J. Brimacombe, Influence of hot working on internal cracks in continuously cast steel billets, Metall. Trans. B 10B (1979) 229±236. [2] K. Brimacombe, K. Surimachi, Crack formation in the continuous casting of steel, Metall. Trans. B 8B (1977) 489±504. [3] Y. Maehara, K. Yasumoto, H. Tomono, T. Nagamichi, Y. Ohmori, Surface cracking mechanism of continuously cast low alloy steel slabs, Mat., Sci. and Tech. 6 (1990) 793±806. [4] Y. Maehara, K. Yasumoto, Y. Sugitani, K. Gunji, Effect of carbon on hot ductility of as cast low alloy steels, Trans. ISIJ 25 (1985) 1045± 1052.
[5] F. Vodopivec, M. Torkar, M. Debelak, M. Kmetic, F. Haller, F. Kaucic, Influence of aluminium on solidification structure and initial deformability of continuously cast C±Mn±Si±N± steel, Mat., Sci. and Tech. 4 (1988) 917±925. [6] H. Bruce, Surface defects on continuously cast materials, Swedish Institute for Metals and Research, IM-2791 (1991) (in Swedish). [7] H. Suzuki, S. Nishimura, S. Yamaguchi, Characteristic of hot ductility in steels subjected to the melting and solidification, Trans. ISIJ 22 (1982) 48±56. [8] A. Lieberman, Y. Kan, N. Mironova, M. Luker, O. Lapshin, Transformation of surface defects on continuously cast billets during hot rolling, steel in the USSR, UDC 621.746.628.047, 14 (1984) 428±431. [9] Y. Takemura, O. Tsubakibara, M. Saito, S. Mizogichi, T. Kuwabara, Direct-linked continuous caster-hot rolling mill process, Nippon Steel technical report no. 21, 1983, pp. 189±201. [10] H. Suzuki et al., Studies of flow stress of metals and alloys, Report of Industrial Science, University of Tokyo 18 (3) (1968) 174±240. [11] J. Hallquist, LS-DYNA, Theoretical Manual, LSTC-report, 1991. [12] J.A. Schey, Metal Forming Processes, Friction and Lubrication, Marcel Dekker, New York, 1970.