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Foil rolling: a new interpretation of experimental results
Journal of Materials Processing Technology 121 (2002) 50±59
Foil rolling: a new interpretation of experimental results H. Keifea, Jingyu Shib, D.L.S. McElwainb,*, T.A.M. Langlandsb a
b
Outokumpu Copper Process Automation, S-72188 Vasteras, Sweden Centre in Statistical Science and Industrial Mathematics, Queensland University of Technology, Gardens Point Campus, 2 George Street, GPO Box 2434, Brisbane, Qld 4001, Australia Accepted 12 September 2001
Abstract This paper explores a new interpretation of experiments on foil rolling. The assumption that the roll remains convex is relaxed so that the strip pro®le may become concave, or thicken in the roll gap. However, we conjecture that the concave pro®le is associated with phenomena which occur after the rolls have stopped. We argue that the yield criterion must be satis®ed in a nonconventional manner if such a phenomenon is caused plastically. Finite element analysis on an extrusion problem appears to con®rm this conjecture. # 2002 Elsevier Science B.V. All rights reserved. Keywords: Foil rolling; Finite element; Modelling
1. Introduction There are several models for foil rolling which assume that the roll ¯attens in the roll gap (see [1,2]). In these models, it is assumed that the strip undergoes two reduction steps, one at the entry and the other at the exit, between which there is a zone where the strip thickness is uniform and the rolls are ¯at. However, a few experiments on thin strip rolling, whose aim has been to measure the contact line, show that the middle ¯at zone assumed in the models does not exist or exists only in a small region. Instead, the strip pro®le appears to be concave in the centre of the roll gap (see [3]). The suggested explanation for this discrepancy is that the experimental conditions are not entirely relevant to rolling since the rolls are stopped for some time before the pressure is released, so that there is a possibility that the strip deforms by creep in a pure compression state. The idea that the strip might thicken during the rolling was suggested by Orowan [4]. He stated that a genuine temporary thickening of the sheet as it passes through neutral zone is ``neither impossible nor improbable''. The paper by Sutcliffe and Rayner [3] describes experiments on the rolling of plasticine strips using elastomer rolls. Both thick and thin strips of plasticine are rolled and the rolls are stopped suddenly and the pro®les of the rolled strips are * Corresponding author. Tel.: 61-7-3864-5185; fax: 61-7-3864-2310. E-mail address: [email protected] (D.L.S. McElwain).
measured. The experimental observations are compared with the theoretical predictions from Fleck et al. [2] model of foil rolling. The comparison is not at all good for the thinnest strips. The strip thickening in the central part of the roll gap is also observed in a copper foil rolling experiment, which is presented in this paper. Pawelski et al. [5] carried out a simulation of cold rolling by imposing a static compressive stress on a strip with a roll-shaped die. They reported a double-indentation strip pro®leÐthe die surface becoming concave in the middle of the contact region. There may be no direct correlation between these static experiments and foil rolling. However, it seems, on the basis of these experiments, worthwhile exploring the possibility that the rolls actually become concave during the rolling of the strip. Grimble et al. [6] studied the possibility of the elastic thickening at the central of the roll gap during the rolling process. Here, we attempt to explore the possibility that the thickening is caused by plastic deformation. The models by Fleck and colleagues [1,2] assume that the central ¯at zone must exist because it is believed that the strip cannot expand perpendicular to the direction of highest pressure, i.e., towards the roll surfaces, and expansion of the strip is in contradiction to the plastic ¯ow rule. In this paper, it will be argued that it may be possible for an expansion zone to exist in rolling without violating a ¯ow rule. However, the relation between the relative surface velocities of the roll and strip must be consistent with the direction of the friction stress.
0924-0136/02/$ ± see front matter # 2002 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 4 - 0 1 3 6 ( 0 1 ) 0 1 1 5 8 - X
H. Keife et al. / Journal of Materials Processing Technology 121 (2002) 50±59
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2. Experiments on copper 2.1. Foil rolling Annealed copper foil was rolled from 0.12 mm down to 0.055 mm in thickness. The foil strip was 22.5 mm wide. The rolls had been made with a local step in diameter to prevent roll-to-roll contact outside the foil (see Fig. 1). The roll diameter was approximately 100 mm. The foil was inserted between the rolls and the roll pressure was applied. Next, the rolls were rotated slowly, stopped and the pressure released. The contact pro®le of the foil was measured with a gauge meter with an accuracy of 10 nm. Two different roll forces were applied, 225 and 740 kN. In order to see any effect of creep, two different holding times (0 and 15 s) were used for the case of a force of 740 kN. Fig. 2 shows the strip pro®le as the semi-thickness t, obtained with roll force 225 kN when the pressure was released immediately when the rotation stopped. Fig. 3 illustrates the two pro®les for the higher roll force (740 kN): one for the case where the pressure was released immediately
Fig. 1. Simple illustration of the setup for the rolling and forging experiments.
when the rotation stopped, and the other for the case where the pressure was held for 15 s after the rotation stopped. The difference between these two cases should give some indication of the creep effect. It can be seen from Fig. 3 that the difference is small, so the creep effect in these experiments is small.
Fig. 2. Profile of the strip surface in roll gap after rolling with roll force of 225 kN.
Fig. 3. Profiles of the strip surface in roll gap after rolling with roll force of 740 kN and two pressure releasing times: 0 and 15 s.
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H. Keife et al. / Journal of Materials Processing Technology 121 (2002) 50±59
Fig. 4. Profile of the strip surface after forging with a force of 740 kN.
2.2. Forging The experiment for a forging process was carried out in the same way as the rolling experiment described above except that there was no rotation of the rolls. The roll force was 740 kN. Fig. 4 shows the pro®le of the strip after the roll is lifted. This ®gure shows the same pattern as that reported in the paper by Pawelski et al. [5]. 3. Yield criterion and strip thickening Let sx denote the horizontal stress in the strip and sz the vertical stress. Most metals are assumed to obey the Tresca yield condition: jsx
sz j Ys ;
where we have assumed that the principal stresses coincide with the horizontal and vertical axes and Ys is the biaxial compression yield stress under plane strain conditions. In the case of rolling, we can further assume that sz p, approximately, with p being the vertical pressure. The horizontal stress sx could be tensile (positive, as in the case where the strip is pulled at the two ends) or compressive (negative). If in the case where the strip is compressed in the horizontal direction, p > sx , then the yield condition can be written as s x p Ys :
(1)
The reason to restrict the expansion of the thickness of strip in the roll gap is that the expansion violates the plastic ¯ow rule, which uses this yield criterion. However, the Tresca yield condition may also be satis®ed if the compressive stress sx < 0 is such that sx p
Ys :
(2)
In this case, since jsx j > p, it is acceptable for the strip to thicken. A similar argument holds when the von Mises yield condition p is assumed, although Ys should be replaced by 2Ys = 3 for plane strain cases.
Fig. 5. Model of extrusion with an expansion zone.
To demonstrate that the yield criterion (2) is associated with the strip thickening, we start by examining a hypothetical forming operation including an expansion zone B as shown in Fig. 5. In this extrusion process, when the deformation is fully developed, it is possible for the material to increase in thickness at the second plastic zone in a manner similar to that which is observed in the experiments on rolling. The necessary condition for the expansion to take place in the zone B is a change in stress state at two places, the beginning and end of zone B. This can be expressed as a change in the yield criterion from (1) to (2) at the beginning and then from (2) to (1) at the end. Using the slab method of analysis, which assumes the homogeneous compression deformation in the vertical direction (the compressive deformation does not vary across the thickness of the strip), the equilibrium equation is t x
dsx dt sx p dx dx
mp 0;
(3)
where t(x) is the strip semi-thickness, x the horizontal distance through the roll gap and m the coef®cient of friction. By solving this equation for the geometry shown in Fig. 5, together with the yield criteria (1) in zones A and C and (2) in zone B (see Appendix A for the details) we can plot the pressure and horizontal stress distribution in the three zones as shown in Fig. 6. The von Mises yield p criterion is used in the calculation with yield stress 2Ys = 3 115 MPa and the coef®cient of the friction is m 0:05. The initial semithickness of the strip is 6 mm. The tool is ®xed and the x-coordinate is measured from the beginning of the ®rst reduction zone, which ends at x 25 mm with the semithickness of the strip being 5 mm. The expansion zone B
H. Keife et al. / Journal of Materials Processing Technology 121 (2002) 50±59
Fig. 6. Variations of the pressure p and compressive horizontal stress
ends at x 50 mm with the semi-thickness of the strip being 5.5 mm. The second reduction zone ends, i.e., the exit at x 100 mm with 2.5 mm of strip semi-thickness. The horizontal stress is assumed to be continuous at the interfaces between the three zones and it is zero at the exit. It is clear from Fig. 6 that a discontinuity in the pressure takes place at the ends of the expansion zone. In order to check these results, a FEM analysis was carried out using ABAQUS/Explicit with the same geometry, material properties and friction conditions as for the slab method of analysis. At the start, front end of the strip is at the beginning of the ®rst reduction zone and then the strip is pushed at the other end with a speed of 100 mm/s. After the front end of the strip passes through the beginning of the expansion zone but before it reaches the second reduction surface, the strip
53
sx in the bite of the extrusion from analytic solution of the slab method.
surface between the beginning of the expansion zone and the front end of the strip becomes horizontal. This creates a gap in the expansion zone and the ®rst part of the second reduction zone. After the front end hits the second reduction surface, the material starts to build up in the gap due to the friction on the contact surface. After a while, the gap is fully ®lled and the pressure is built up in the expansion zone. When t 1:35 s, about 35 mm of the undeformed strip has passed through the exit and it is believed that the deformation in the three zones becomes fully developed. Fig. 7 shows the deformed strip at this time and it can be seen that shearing occurs on the contact surface. The variations of pressure and compressive horizontal stress sx on the contact surface in the reduction±expansion±reduction zone are illustrated in Fig. 8. It can be seen that the pressure and the horizontal stress
Fig. 7. Deformed strip at time t 1:35 s from ABAQUS/Explicit analysis of the extrusion. The arrows on the top surface indicate the reduction and expansion zones.
Fig. 8. Variations of the pressure p and compressive horizontal stress
sx in the bit of the extrusion from ABAQUS/Explicit.
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H. Keife et al. / Journal of Materials Processing Technology 121 (2002) 50±59
vary in the same way as found in the slab analysis of the extrusion. The ABAQUS/Explicit predicts higher stresses than the slab method. This may result from the fact that the ABAQUS/Explicit does not assume homogeneous deformation across the thickness and the principal stress directions are not vertical and horizontal, as assumed in the slab method. The sharp jump of the pressure at the corners is also found if the corners are smoothed. 4. New interpretation for the rolling experiments Having established that strip thickening is possible and is accompanied by a change of stress state in the extrusion process, we turn our attention to the rolling process and suggest a new interpretation for the strip thickening in the roll gap as measured in the experiments. There are two stages in the rolling experiments presented here by Sutcliffe and Rayner [3]: (1) a rolling stage when the rolls rotate; (2) a forging (compression) stage when the rotation is stopped with the roll force being held for a short time. (Although it is claimed that the rotation is stopped suddenly and the pressure is released immediately, there is still some time effect. To provide an accurate measurement for the rolling process only, it should be carried out during the rolling process. However, so far such an experiment has not been performed.) We conjecture that there are two possible interpretations for the strip thickening in the central part of the roll gap in the rolling experiments: (1) the strip thickening occurs plastically during both the rolling and forging stages; (2) no plastic thickening occurs during the rolling stage, but it occurs during the subsequent forging stage. Generally speaking, the roll force needed during the rolling stage cannot be supported by the same contact pro®le after the rotation is stopped (since the conditions have been changed). This will cause a stress redistribution and leads to a forging process. As shown in Fig. 4 and Pawelski et al.'s results [5], a concave strip pro®le is possible for a forging process. Therefore, the nonuniform pro®le might occur during the forging stage. We conjecture, from the experiments mentioned above, that during the stage when the strip thickening occurs, there are six zones in the roll gap (compare Ref. [1]), as shown in Fig. 9. Zone A: elastic compression. This zone starts from the entry and ends when the strip yields sx p Ys . The strip slips backward relative to the roll.
Zone B: plastic reduction zone. This starts from the point where the strip starts to yield and ends when the strip profile becomes horizontal. The condition sx p Ys holds throughout this zone. The strip slips backward relative to the roll. Zone C: elastic zone. This zone starts when the strip profile becomes flat. Here sx increases faster than p such that sx p < Ys . Eventually sx > p and the yield criterion (2), i.e., sx p Ys is satisfied at which the zone ends. The strip slips backward relative to the roll in this zone. Zone D: plastic expansion zone. This zone starts when the yield criterion (2) is satisfied and sx p Ys holds throughout the zone. The strip increases in thickness in this zone. This zone ends when the horizontal stress equals to that from zone E (discussed next). The strip slips backward relative to the roll in the zone and the neutral point could be at the end point. Zone E: plastic reduction zone. The condition sx p Ys holds throughout the zone. It starts when the horizontal stress equals to that from the zone D and ends when the strip profile becomes horizontal. The neutral point could exist in this zone so that part of the strip slips backward and the other part slips forward relative to the roll in the zone. Zone F: elastic recovery zone. This zone starts the strip profile and becomes horizontal and ends when the pressure and the horizontal stress drop to zero. The strip slips forward relative to the roll in the zone. With this type of assumed pro®le of the contact surface, we can predict the pressure, horizontal stress and the deformed strip pro®le with the slab method. The deformation of the strip is governed by the equation t x
dsx dt sx p mp 0: dx dx
(4)
Here the minus sign holds in the forward slip zones (E and F) where the strip moves faster than the roll surface, and the positive sign holds in the backward slip zones (A, B, C and D) where the strip moves slower than the roll. The deformation of the elastic rolls is assumed to be governed by the matrix foundation model proposed by Fleck and Johnson [1] and modi®ed by Luo and Keife [7]. This model predicts the following relationship between the pressure distribution p(x) on the contact surface and the strip semi-thickness t(x) by adapting Hertz' theory to
Fig. 9. Proposed zones in the roll gap during rolling process or forging process. Zones: AÐelastic (compression); BÐplastic (reduction); CÐelastic; DÐ plastic (expansion); EÐplastic (reduction); FÐelastic (recovery).
H. Keife et al. / Journal of Materials Processing Technology 121 (2002) 50±59
rolling:
s 0 2 " x t0 p pp x; t p0 1 0 a0 R2 x02
5. Velocity considerations #
t00 1 ; p R2 a00 2 (5)
where t0 b 1
cos a t cos a x sin a;
q x0 b sin a x cos a t sin a; b t0 R2 a20 ; q t0 t2 : a00 12 a0 a2 2 t0 t2 2 ; a tan 1 a0 a2
Here p0 is the maximum pressure according to the Hertz' model, R being the radius of the undeformed roll, a2 and a0 being the values of x at the exit and the entry, respectively, where the semi-thicknesses of the strip are t2 and t0, respectively. We need extra information to complete the problem for the three unknowns t(x), sx(x) and p(x). This information is provided by the mechanical properties of the strip: the generalised Hooke's law in the elastic zones and yield conditions in the plastic zones, so that ln
t t0
1
ln
t1 t
1
ln
t2 t
1
Es Es Es
n2s n2s n2s
2Ys sx p p 3 sx p
2Ys p 3
p
ns 1 ns sx Es
in zone A;
(6)
p
ns 1 ns sx Es
in zone C;
(7)
p
ns 1 ns sx Es
in zone F;
(8)
in zones B and E; in zone D:
55
(9) (10)
Here t1 is the semi-thickness at the interface between the zones B and C, Es and ns are Young's modulus and Poisson ratio of the strip material, respectively. The computation is carried out in the following way. Firstly, an iteration scheme is employed to determine the three parameters, p0, a0 and a2. The procedure is the same as that used by Luo and Keife [7]. Secondly, Eq. (4) together with the expression (5) and the corresponding one of (6±10), is integrated forward for each zone from A, B, C and to D, with the boundary conditions at the entry and continuity conditions at their interfaces. The calculation in the zone D is extended further to x a2 . Thirdly, Eq. (4) together with the expression (5) and the corresponding one of (6±10) is integrated backward from zone F to E and the calculation in zone E continues until the compressive horizontal stress sx meets the value calculated from zone D in the forward calculation. This point is taken as the interface between zone D and E.
The slab method relates the strip pro®le and the stresses, including the friction stress on the contact surface, whose direction depends on the relative movement between the roll and the strip. Therefore, it is necessary to investigate the relative velocity on the contact surface. We ®rst consider the velocities on the roll surface and on the strip surface in the rolling process. Assume that the tangential velocity at the points on the roll surface is vr and does not vary with the position of the points. The tangential velocity at a generic point on the strip surface is denoted by vs and the horizontal component is given by vs;x vs cos a, with a being the angle between the horizontal direction and the tangent to the contact surface. At the neutral point where the strip moves with the same speed as the roll, we have vsn vr and vsn;x vsn cos an vr cos an . The mass ¯ow of the strip should be conserved, so we have tvs;x tn vsn;x , with tn as the semi-thickness of the strip at the neutral point. By making use of the above relations and assuming that the angle a is de®ned everywhere on the surface, we have tvs cos a tn vr cos an . Thus, we can derive the following expression for the slip ratio: q 1 dt=dx2 t n vs tn cos an q : (11) vr t cos a t 1 dt=dx2n q Here, we have used the relation cos a 1= 1 dt=dx2 : Now, we consider the velocity ®elds in the forging (or compression) process by a die. The velocity of the die (the roll) is denoted by vr and it is downwards. Assume that there is a point x xn on the surface of the strip where the velocity is also downward only, i.e., a neutral point. Then the tangential velocity vs at any point x on the strip surface should be such that tvs cos a x xn vr , i.e., the volume of the material ¯owing out of the section at x is equal to the volume squeezed downward. Thus, the slip ratio is given by s 2 vs x xn dt : (12) 1 dx vr t After the strip deformation is found from the model in the Section 4 with the assumed friction stress direction, the slip ratio should be checked to ensure that the assumed friction stress is consistent. 6. Results from the new interpretation When the plastic expansion zone D exists during the rolling stage, the material of the strip at the exit side of this zone moves slower than the material at the entry side of the zone in order to preserve the material continuity. The speed of the points on the roll surface is relatively uniform. Thus, the neutral point cannot be in this zone or at the interface between this zone and zone E. The speed of the
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H. Keife et al. / Journal of Materials Processing Technology 121 (2002) 50±59
Fig. 10. Variations of the slip ratio for two rolling process models: with an expansion zone (line with square) and without an expansion zone (line with diamond).
material of the strip in zone E increases along the rolling direction and at some point, it matches the speed of the roll surface, so the neutral point could be in this zone (here we assume that there are no tensions at the entry and exit). Furthermore, if it is assumed that the neutral point is at the interface between zones D and E and the sign of the friction stress term mp in the governing equation (3) is taken to be positive, i.e., assuming the roll moving faster than the strip, then the slip ratio calculated from (11) is greater than 1 in most of this zone, as shown in Fig. 10. This indicates that the strip moves faster than the roll. This contradiction partially con®rms the previous conclusion that the neutral point cannot exist at the interface between zone D and E during the rolling stage. However, its precise position in zone E is not known and we have not found a method for its determination within the present elastic foundation model for the roll. It might be possible to determine its position using some other model, such as the in¯uence function model suggested by Fleck et al. [2]. This needs further work and we will not progress
further in this direction here. Instead, we investigate the other possible interpretation for the measured pro®le to be concave: the expansion zone does not exist during the rolling stage, it is created during the subsequent forging stage. We assume that during the rolling stage, the strip pro®le in the roll gap consists of ®ve zones: elastic compression backward slip, plastic reduction backward slip, contained plastic ¯at neutral, plastic reduction forward slip and elastic recovery forward slip zones. The six-zone division, including an expansion zone, outlined in Section 4 holds during the subsequent forging stage. The deformed strip pro®le, variations of the pressure p and the compressive horizontal stress sx and the slip ratio in the roll gap during the rolling and forging stages are computed with the following material and geometrical parameters: the Young's modulus and Poisson ratio of the roll are Er 210 000 MPa and nr 0:3, respectively. The Young's modulus and Poisson ratio of the strip are Es 140 000 MPa and ns 0:3, respectively. The plane p strain compression yield stress of the strip is Ys 400= 3 MPa. The coef®cient
Fig. 11. Variation of the slip ratio for forging process with an expansion zone.
H. Keife et al. / Journal of Materials Processing Technology 121 (2002) 50±59
57
Fig. 12. Profiles of the strip during the rolling process (without expansion zone: line with diamond) and the forging process (with expansion zone: line with square).
Fig. 13. Variations of the pressure p and compressive horizontal stress
Fig. 14. Variations of the pressure p and compressive horizontal stress
sx in the roll gap during the rolling process (without expansion zone).
sx in the roll gap during the forging process (with expansion zone).
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H. Keife et al. / Journal of Materials Processing Technology 121 (2002) 50±59
of friction at the contact is 0.05. The semi-thickness of the strip at the entry and the exit are t0 0:025 mm and t2 0:015 mm, respectively, during the rolling stage. The roll force is calculated for the rolling process and then it is applied to the forging process, which is assumed to make the pro®le concave. The variation of the slip ratio vs =vr during the rolling stage, is illustrated in Fig. 10 and Fig. 11, shows the variation of the slip ratio during the forging stage. The strip pro®les during the rolling and forging processes are shown in Fig. 12. The pro®les support the same roll force. It can be seen that the forged pro®le is lower than the rolling pro®le. The variations of the pressure p and the compressive horizontal stress sx during the rolling and forging stages are shown in Figs. 13 and 14, respectively. During the rolling stage, p > sx in whole roll gap, while sx > p in part of the roll gap during the forging process. In the calculation for the forging process, it is assumed that the compressive horizontal stress is continuous at the interfaces between the various zones and different yield criterion is used in zones D and E. This leads to the discontinuity in the pressure and the strip pro®le at the interface between zone D and E, as seen in Figs. 12 and 14. Such a discontinuity in pressure is also observed in the case of ®nite element ABAQUS/Explicit analysis of the extrusion studied in Section 3 (see Fig. 8). 7. Discussion Results of some experiments on pro®les of thin strip after rolling and forging are shown. These results, together with some other results in the literature, show that strip pro®le may become concave in the roll gap. It is conjectured from these experiments that the compressive horizontal stress in the expansion zone is so much larger than the vertical pressure that the yield criterion is also satis®ed, but in a nonconventional manner. With this state of stress, the expansion of the strip in the vertical direction does not violate the plastic ¯ow rule. The horizontal stress in the reduction zones could be tensile or compressive. If it is compressive, then its magnitude is so much smaller than the pressure that the yield criterion is satis®ed. This kind of changeover of stress state is con®rmed by ®nite element package ABAQUS/Explicit analysis of an extrusion process. Therefore, the more restrictive assumption that the two reduction zones must be connected by a ¯at zone used in the previous foil rolling models [1,2] can be relaxed. Based on these observations, it is proposed that a plastic expansion zone exists in the roll gap in those experiments presented here and in the literature and that this is produced either during both the rolling and forging stages, or during the forging stage only. To achieve the changeover of the stress state, an elastic zone should exist between the plastic reduction and expansion zones. Some cases are analysed by using the proposed contact pro®le and modi®ed elastic foundation model for the rolls.
Acknowledgements This work is partially supported by the Australian Research Council under the Large Grant and SPIRT schemes. A QUT Visit Fellowship to Dr. H. Keife is also appreciated. We also like to thank Drs. S. Domanti and J. Edwards for their valuable discussions. Appendix A We derive in this appendix the analytic expressions for the solution of the governing equation (3) with the yield criterion (1) and (2) in the different zones shown in Fig. 5. Let the surface of the tool in the zone A be denoted by t a1 x b1, where a1 and b1 are constants. Then substituting this together with the yield criterion (1) into the equilibrium equation (3), we have the governing differential equation for sx, a1 x b1
dsx msx Ym a1 dx
m 0;
(A.1)
p where Ym is Ys for the Tresca yield criterion and 2Ys = 3 for the von Mises yield criterion. Let the surface of the tool in the zone B be denoted by t a2 x b2 , where a2 and b2 are constants. With the yield criterion (2) and this expression for the surface, we can derive the governing differential equation for the stress sx in this zone, a2 x b2
dsx msx Ym a2 m 0: dx
(A.2)
Denoting the surface of the tool in the zone C by t a3 x b3 , where a3 and b3 are constants, and using the yield criterion (1), we derive the governing differential equation for the stress sx as a3 x b3
dsx msx Ym a3 dx
m 0:
(A.3)
These three equations have a common form ax b
dsx msx c 0; dx
(A.4)
and this equation has a general solution c ; sx C0 ax b m=a m
(A.5)
where C0 is an arbitrary integral constant. Expressing the general solutions for the three zones, we have sx C1 a1 x b1
m=a1
Ym a1 m
m
in zone A; (A.6)
sx C2 a2 x b2
m=a2
Ym a2 m m
in zone B; (A.7)
H. Keife et al. / Journal of Materials Processing Technology 121 (2002) 50±59
sx C3 a3 x b3
m=a3
Ym a3 m
m
in zone C: (A.8)
The constants C1, C2 and C3 can be determined from the conditions at the exit and the interfaces between the zones. The pressure in the different zones is determined from the corresponding yield criteria.
References [1] N.A. Fleck, K.L. Johnson, Towards a new theory of cold rolling thin foil, Int. J. Mech. Sci. 29 (1987) 507±524.
59
[2] N.A. Fleck, K.L. Johnson, M.E. Mear, L.C. Zhang, Cold rolling of foil, Proc. Inst. Mech. Eng. 206 (1992) 119±131. [3] M.P.F. Sutcliffe, P.J. Rayner, Experimental measurements of load and strip profile in thin strip rolling, Int. J. Mech. Sci. 40 (1998) 887±899. [4] E. Orowan, The calculation of roll pressure in hot and cold flat rolling, Proc. Inst. Mech. Eng. 150 (1943) 140±167. [5] O. Pawelski, W. Rasp, L. Loffler, A. Kramer, Compressive testing of flat specimens with dies similar to rolls for simulation of cold rolling (Stauchversuche an flachproben mit walzenahnichen werkzeugen zur nachahmung des kaltwalzens), Blech Rohre Profile 40 (1993) 379±383. [6] M.J. Grimble, M.A. Fuller, G.F. Bryant, A non-circular arc roll force model for cold rolling, Int. J. Numer. Meth. Eng. 12 (1978) 643±663. [7] C. Luo, H. Keife, The influence of process parameters upon the cold rolling of aluminium foils, in: Proceedings of the First International Conference on Tribology in Manufacturing Processes, Gifu, 1997, pp. 364±371.
Foil rolling: a new interpretation of experimental results H. Keifea, Jingyu Shib, D.L.S. McElwainb,*, T.A.M. Langlandsb a
b
Outokumpu Copper Process Automation, S-72188 Vasteras, Sweden Centre in Statistical Science and Industrial Mathematics, Queensland University of Technology, Gardens Point Campus, 2 George Street, GPO Box 2434, Brisbane, Qld 4001, Australia Accepted 12 September 2001
Abstract This paper explores a new interpretation of experiments on foil rolling. The assumption that the roll remains convex is relaxed so that the strip pro®le may become concave, or thicken in the roll gap. However, we conjecture that the concave pro®le is associated with phenomena which occur after the rolls have stopped. We argue that the yield criterion must be satis®ed in a nonconventional manner if such a phenomenon is caused plastically. Finite element analysis on an extrusion problem appears to con®rm this conjecture. # 2002 Elsevier Science B.V. All rights reserved. Keywords: Foil rolling; Finite element; Modelling
1. Introduction There are several models for foil rolling which assume that the roll ¯attens in the roll gap (see [1,2]). In these models, it is assumed that the strip undergoes two reduction steps, one at the entry and the other at the exit, between which there is a zone where the strip thickness is uniform and the rolls are ¯at. However, a few experiments on thin strip rolling, whose aim has been to measure the contact line, show that the middle ¯at zone assumed in the models does not exist or exists only in a small region. Instead, the strip pro®le appears to be concave in the centre of the roll gap (see [3]). The suggested explanation for this discrepancy is that the experimental conditions are not entirely relevant to rolling since the rolls are stopped for some time before the pressure is released, so that there is a possibility that the strip deforms by creep in a pure compression state. The idea that the strip might thicken during the rolling was suggested by Orowan [4]. He stated that a genuine temporary thickening of the sheet as it passes through neutral zone is ``neither impossible nor improbable''. The paper by Sutcliffe and Rayner [3] describes experiments on the rolling of plasticine strips using elastomer rolls. Both thick and thin strips of plasticine are rolled and the rolls are stopped suddenly and the pro®les of the rolled strips are * Corresponding author. Tel.: 61-7-3864-5185; fax: 61-7-3864-2310. E-mail address: [email protected] (D.L.S. McElwain).
measured. The experimental observations are compared with the theoretical predictions from Fleck et al. [2] model of foil rolling. The comparison is not at all good for the thinnest strips. The strip thickening in the central part of the roll gap is also observed in a copper foil rolling experiment, which is presented in this paper. Pawelski et al. [5] carried out a simulation of cold rolling by imposing a static compressive stress on a strip with a roll-shaped die. They reported a double-indentation strip pro®leÐthe die surface becoming concave in the middle of the contact region. There may be no direct correlation between these static experiments and foil rolling. However, it seems, on the basis of these experiments, worthwhile exploring the possibility that the rolls actually become concave during the rolling of the strip. Grimble et al. [6] studied the possibility of the elastic thickening at the central of the roll gap during the rolling process. Here, we attempt to explore the possibility that the thickening is caused by plastic deformation. The models by Fleck and colleagues [1,2] assume that the central ¯at zone must exist because it is believed that the strip cannot expand perpendicular to the direction of highest pressure, i.e., towards the roll surfaces, and expansion of the strip is in contradiction to the plastic ¯ow rule. In this paper, it will be argued that it may be possible for an expansion zone to exist in rolling without violating a ¯ow rule. However, the relation between the relative surface velocities of the roll and strip must be consistent with the direction of the friction stress.
0924-0136/02/$ ± see front matter # 2002 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 4 - 0 1 3 6 ( 0 1 ) 0 1 1 5 8 - X
H. Keife et al. / Journal of Materials Processing Technology 121 (2002) 50±59
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2. Experiments on copper 2.1. Foil rolling Annealed copper foil was rolled from 0.12 mm down to 0.055 mm in thickness. The foil strip was 22.5 mm wide. The rolls had been made with a local step in diameter to prevent roll-to-roll contact outside the foil (see Fig. 1). The roll diameter was approximately 100 mm. The foil was inserted between the rolls and the roll pressure was applied. Next, the rolls were rotated slowly, stopped and the pressure released. The contact pro®le of the foil was measured with a gauge meter with an accuracy of 10 nm. Two different roll forces were applied, 225 and 740 kN. In order to see any effect of creep, two different holding times (0 and 15 s) were used for the case of a force of 740 kN. Fig. 2 shows the strip pro®le as the semi-thickness t, obtained with roll force 225 kN when the pressure was released immediately when the rotation stopped. Fig. 3 illustrates the two pro®les for the higher roll force (740 kN): one for the case where the pressure was released immediately
Fig. 1. Simple illustration of the setup for the rolling and forging experiments.
when the rotation stopped, and the other for the case where the pressure was held for 15 s after the rotation stopped. The difference between these two cases should give some indication of the creep effect. It can be seen from Fig. 3 that the difference is small, so the creep effect in these experiments is small.
Fig. 2. Profile of the strip surface in roll gap after rolling with roll force of 225 kN.
Fig. 3. Profiles of the strip surface in roll gap after rolling with roll force of 740 kN and two pressure releasing times: 0 and 15 s.
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H. Keife et al. / Journal of Materials Processing Technology 121 (2002) 50±59
Fig. 4. Profile of the strip surface after forging with a force of 740 kN.
2.2. Forging The experiment for a forging process was carried out in the same way as the rolling experiment described above except that there was no rotation of the rolls. The roll force was 740 kN. Fig. 4 shows the pro®le of the strip after the roll is lifted. This ®gure shows the same pattern as that reported in the paper by Pawelski et al. [5]. 3. Yield criterion and strip thickening Let sx denote the horizontal stress in the strip and sz the vertical stress. Most metals are assumed to obey the Tresca yield condition: jsx
sz j Ys ;
where we have assumed that the principal stresses coincide with the horizontal and vertical axes and Ys is the biaxial compression yield stress under plane strain conditions. In the case of rolling, we can further assume that sz p, approximately, with p being the vertical pressure. The horizontal stress sx could be tensile (positive, as in the case where the strip is pulled at the two ends) or compressive (negative). If in the case where the strip is compressed in the horizontal direction, p > sx , then the yield condition can be written as s x p Ys :
(1)
The reason to restrict the expansion of the thickness of strip in the roll gap is that the expansion violates the plastic ¯ow rule, which uses this yield criterion. However, the Tresca yield condition may also be satis®ed if the compressive stress sx < 0 is such that sx p
Ys :
(2)
In this case, since jsx j > p, it is acceptable for the strip to thicken. A similar argument holds when the von Mises yield condition p is assumed, although Ys should be replaced by 2Ys = 3 for plane strain cases.
Fig. 5. Model of extrusion with an expansion zone.
To demonstrate that the yield criterion (2) is associated with the strip thickening, we start by examining a hypothetical forming operation including an expansion zone B as shown in Fig. 5. In this extrusion process, when the deformation is fully developed, it is possible for the material to increase in thickness at the second plastic zone in a manner similar to that which is observed in the experiments on rolling. The necessary condition for the expansion to take place in the zone B is a change in stress state at two places, the beginning and end of zone B. This can be expressed as a change in the yield criterion from (1) to (2) at the beginning and then from (2) to (1) at the end. Using the slab method of analysis, which assumes the homogeneous compression deformation in the vertical direction (the compressive deformation does not vary across the thickness of the strip), the equilibrium equation is t x
dsx dt sx p dx dx
mp 0;
(3)
where t(x) is the strip semi-thickness, x the horizontal distance through the roll gap and m the coef®cient of friction. By solving this equation for the geometry shown in Fig. 5, together with the yield criteria (1) in zones A and C and (2) in zone B (see Appendix A for the details) we can plot the pressure and horizontal stress distribution in the three zones as shown in Fig. 6. The von Mises yield p criterion is used in the calculation with yield stress 2Ys = 3 115 MPa and the coef®cient of the friction is m 0:05. The initial semithickness of the strip is 6 mm. The tool is ®xed and the x-coordinate is measured from the beginning of the ®rst reduction zone, which ends at x 25 mm with the semithickness of the strip being 5 mm. The expansion zone B
H. Keife et al. / Journal of Materials Processing Technology 121 (2002) 50±59
Fig. 6. Variations of the pressure p and compressive horizontal stress
ends at x 50 mm with the semi-thickness of the strip being 5.5 mm. The second reduction zone ends, i.e., the exit at x 100 mm with 2.5 mm of strip semi-thickness. The horizontal stress is assumed to be continuous at the interfaces between the three zones and it is zero at the exit. It is clear from Fig. 6 that a discontinuity in the pressure takes place at the ends of the expansion zone. In order to check these results, a FEM analysis was carried out using ABAQUS/Explicit with the same geometry, material properties and friction conditions as for the slab method of analysis. At the start, front end of the strip is at the beginning of the ®rst reduction zone and then the strip is pushed at the other end with a speed of 100 mm/s. After the front end of the strip passes through the beginning of the expansion zone but before it reaches the second reduction surface, the strip
53
sx in the bite of the extrusion from analytic solution of the slab method.
surface between the beginning of the expansion zone and the front end of the strip becomes horizontal. This creates a gap in the expansion zone and the ®rst part of the second reduction zone. After the front end hits the second reduction surface, the material starts to build up in the gap due to the friction on the contact surface. After a while, the gap is fully ®lled and the pressure is built up in the expansion zone. When t 1:35 s, about 35 mm of the undeformed strip has passed through the exit and it is believed that the deformation in the three zones becomes fully developed. Fig. 7 shows the deformed strip at this time and it can be seen that shearing occurs on the contact surface. The variations of pressure and compressive horizontal stress sx on the contact surface in the reduction±expansion±reduction zone are illustrated in Fig. 8. It can be seen that the pressure and the horizontal stress
Fig. 7. Deformed strip at time t 1:35 s from ABAQUS/Explicit analysis of the extrusion. The arrows on the top surface indicate the reduction and expansion zones.
Fig. 8. Variations of the pressure p and compressive horizontal stress
sx in the bit of the extrusion from ABAQUS/Explicit.
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H. Keife et al. / Journal of Materials Processing Technology 121 (2002) 50±59
vary in the same way as found in the slab analysis of the extrusion. The ABAQUS/Explicit predicts higher stresses than the slab method. This may result from the fact that the ABAQUS/Explicit does not assume homogeneous deformation across the thickness and the principal stress directions are not vertical and horizontal, as assumed in the slab method. The sharp jump of the pressure at the corners is also found if the corners are smoothed. 4. New interpretation for the rolling experiments Having established that strip thickening is possible and is accompanied by a change of stress state in the extrusion process, we turn our attention to the rolling process and suggest a new interpretation for the strip thickening in the roll gap as measured in the experiments. There are two stages in the rolling experiments presented here by Sutcliffe and Rayner [3]: (1) a rolling stage when the rolls rotate; (2) a forging (compression) stage when the rotation is stopped with the roll force being held for a short time. (Although it is claimed that the rotation is stopped suddenly and the pressure is released immediately, there is still some time effect. To provide an accurate measurement for the rolling process only, it should be carried out during the rolling process. However, so far such an experiment has not been performed.) We conjecture that there are two possible interpretations for the strip thickening in the central part of the roll gap in the rolling experiments: (1) the strip thickening occurs plastically during both the rolling and forging stages; (2) no plastic thickening occurs during the rolling stage, but it occurs during the subsequent forging stage. Generally speaking, the roll force needed during the rolling stage cannot be supported by the same contact pro®le after the rotation is stopped (since the conditions have been changed). This will cause a stress redistribution and leads to a forging process. As shown in Fig. 4 and Pawelski et al.'s results [5], a concave strip pro®le is possible for a forging process. Therefore, the nonuniform pro®le might occur during the forging stage. We conjecture, from the experiments mentioned above, that during the stage when the strip thickening occurs, there are six zones in the roll gap (compare Ref. [1]), as shown in Fig. 9. Zone A: elastic compression. This zone starts from the entry and ends when the strip yields sx p Ys . The strip slips backward relative to the roll.
Zone B: plastic reduction zone. This starts from the point where the strip starts to yield and ends when the strip profile becomes horizontal. The condition sx p Ys holds throughout this zone. The strip slips backward relative to the roll. Zone C: elastic zone. This zone starts when the strip profile becomes flat. Here sx increases faster than p such that sx p < Ys . Eventually sx > p and the yield criterion (2), i.e., sx p Ys is satisfied at which the zone ends. The strip slips backward relative to the roll in this zone. Zone D: plastic expansion zone. This zone starts when the yield criterion (2) is satisfied and sx p Ys holds throughout the zone. The strip increases in thickness in this zone. This zone ends when the horizontal stress equals to that from zone E (discussed next). The strip slips backward relative to the roll in the zone and the neutral point could be at the end point. Zone E: plastic reduction zone. The condition sx p Ys holds throughout the zone. It starts when the horizontal stress equals to that from the zone D and ends when the strip profile becomes horizontal. The neutral point could exist in this zone so that part of the strip slips backward and the other part slips forward relative to the roll in the zone. Zone F: elastic recovery zone. This zone starts the strip profile and becomes horizontal and ends when the pressure and the horizontal stress drop to zero. The strip slips forward relative to the roll in the zone. With this type of assumed pro®le of the contact surface, we can predict the pressure, horizontal stress and the deformed strip pro®le with the slab method. The deformation of the strip is governed by the equation t x
dsx dt sx p mp 0: dx dx
(4)
Here the minus sign holds in the forward slip zones (E and F) where the strip moves faster than the roll surface, and the positive sign holds in the backward slip zones (A, B, C and D) where the strip moves slower than the roll. The deformation of the elastic rolls is assumed to be governed by the matrix foundation model proposed by Fleck and Johnson [1] and modi®ed by Luo and Keife [7]. This model predicts the following relationship between the pressure distribution p(x) on the contact surface and the strip semi-thickness t(x) by adapting Hertz' theory to
Fig. 9. Proposed zones in the roll gap during rolling process or forging process. Zones: AÐelastic (compression); BÐplastic (reduction); CÐelastic; DÐ plastic (expansion); EÐplastic (reduction); FÐelastic (recovery).
H. Keife et al. / Journal of Materials Processing Technology 121 (2002) 50±59
rolling:
s 0 2 " x t0 p pp x; t p0 1 0 a0 R2 x02
5. Velocity considerations #
t00 1 ; p R2 a00 2 (5)
where t0 b 1
cos a t cos a x sin a;
q x0 b sin a x cos a t sin a; b t0 R2 a20 ; q t0 t2 : a00 12 a0 a2 2 t0 t2 2 ; a tan 1 a0 a2
Here p0 is the maximum pressure according to the Hertz' model, R being the radius of the undeformed roll, a2 and a0 being the values of x at the exit and the entry, respectively, where the semi-thicknesses of the strip are t2 and t0, respectively. We need extra information to complete the problem for the three unknowns t(x), sx(x) and p(x). This information is provided by the mechanical properties of the strip: the generalised Hooke's law in the elastic zones and yield conditions in the plastic zones, so that ln
t t0
1
ln
t1 t
1
ln
t2 t
1
Es Es Es
n2s n2s n2s
2Ys sx p p 3 sx p
2Ys p 3
p
ns 1 ns sx Es
in zone A;
(6)
p
ns 1 ns sx Es
in zone C;
(7)
p
ns 1 ns sx Es
in zone F;
(8)
in zones B and E; in zone D:
55
(9) (10)
Here t1 is the semi-thickness at the interface between the zones B and C, Es and ns are Young's modulus and Poisson ratio of the strip material, respectively. The computation is carried out in the following way. Firstly, an iteration scheme is employed to determine the three parameters, p0, a0 and a2. The procedure is the same as that used by Luo and Keife [7]. Secondly, Eq. (4) together with the expression (5) and the corresponding one of (6±10), is integrated forward for each zone from A, B, C and to D, with the boundary conditions at the entry and continuity conditions at their interfaces. The calculation in the zone D is extended further to x a2 . Thirdly, Eq. (4) together with the expression (5) and the corresponding one of (6±10) is integrated backward from zone F to E and the calculation in zone E continues until the compressive horizontal stress sx meets the value calculated from zone D in the forward calculation. This point is taken as the interface between zone D and E.
The slab method relates the strip pro®le and the stresses, including the friction stress on the contact surface, whose direction depends on the relative movement between the roll and the strip. Therefore, it is necessary to investigate the relative velocity on the contact surface. We ®rst consider the velocities on the roll surface and on the strip surface in the rolling process. Assume that the tangential velocity at the points on the roll surface is vr and does not vary with the position of the points. The tangential velocity at a generic point on the strip surface is denoted by vs and the horizontal component is given by vs;x vs cos a, with a being the angle between the horizontal direction and the tangent to the contact surface. At the neutral point where the strip moves with the same speed as the roll, we have vsn vr and vsn;x vsn cos an vr cos an . The mass ¯ow of the strip should be conserved, so we have tvs;x tn vsn;x , with tn as the semi-thickness of the strip at the neutral point. By making use of the above relations and assuming that the angle a is de®ned everywhere on the surface, we have tvs cos a tn vr cos an . Thus, we can derive the following expression for the slip ratio: q 1 dt=dx2 t n vs tn cos an q : (11) vr t cos a t 1 dt=dx2n q Here, we have used the relation cos a 1= 1 dt=dx2 : Now, we consider the velocity ®elds in the forging (or compression) process by a die. The velocity of the die (the roll) is denoted by vr and it is downwards. Assume that there is a point x xn on the surface of the strip where the velocity is also downward only, i.e., a neutral point. Then the tangential velocity vs at any point x on the strip surface should be such that tvs cos a x xn vr , i.e., the volume of the material ¯owing out of the section at x is equal to the volume squeezed downward. Thus, the slip ratio is given by s 2 vs x xn dt : (12) 1 dx vr t After the strip deformation is found from the model in the Section 4 with the assumed friction stress direction, the slip ratio should be checked to ensure that the assumed friction stress is consistent. 6. Results from the new interpretation When the plastic expansion zone D exists during the rolling stage, the material of the strip at the exit side of this zone moves slower than the material at the entry side of the zone in order to preserve the material continuity. The speed of the points on the roll surface is relatively uniform. Thus, the neutral point cannot be in this zone or at the interface between this zone and zone E. The speed of the
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H. Keife et al. / Journal of Materials Processing Technology 121 (2002) 50±59
Fig. 10. Variations of the slip ratio for two rolling process models: with an expansion zone (line with square) and without an expansion zone (line with diamond).
material of the strip in zone E increases along the rolling direction and at some point, it matches the speed of the roll surface, so the neutral point could be in this zone (here we assume that there are no tensions at the entry and exit). Furthermore, if it is assumed that the neutral point is at the interface between zones D and E and the sign of the friction stress term mp in the governing equation (3) is taken to be positive, i.e., assuming the roll moving faster than the strip, then the slip ratio calculated from (11) is greater than 1 in most of this zone, as shown in Fig. 10. This indicates that the strip moves faster than the roll. This contradiction partially con®rms the previous conclusion that the neutral point cannot exist at the interface between zone D and E during the rolling stage. However, its precise position in zone E is not known and we have not found a method for its determination within the present elastic foundation model for the roll. It might be possible to determine its position using some other model, such as the in¯uence function model suggested by Fleck et al. [2]. This needs further work and we will not progress
further in this direction here. Instead, we investigate the other possible interpretation for the measured pro®le to be concave: the expansion zone does not exist during the rolling stage, it is created during the subsequent forging stage. We assume that during the rolling stage, the strip pro®le in the roll gap consists of ®ve zones: elastic compression backward slip, plastic reduction backward slip, contained plastic ¯at neutral, plastic reduction forward slip and elastic recovery forward slip zones. The six-zone division, including an expansion zone, outlined in Section 4 holds during the subsequent forging stage. The deformed strip pro®le, variations of the pressure p and the compressive horizontal stress sx and the slip ratio in the roll gap during the rolling and forging stages are computed with the following material and geometrical parameters: the Young's modulus and Poisson ratio of the roll are Er 210 000 MPa and nr 0:3, respectively. The Young's modulus and Poisson ratio of the strip are Es 140 000 MPa and ns 0:3, respectively. The plane p strain compression yield stress of the strip is Ys 400= 3 MPa. The coef®cient
Fig. 11. Variation of the slip ratio for forging process with an expansion zone.
H. Keife et al. / Journal of Materials Processing Technology 121 (2002) 50±59
57
Fig. 12. Profiles of the strip during the rolling process (without expansion zone: line with diamond) and the forging process (with expansion zone: line with square).
Fig. 13. Variations of the pressure p and compressive horizontal stress
Fig. 14. Variations of the pressure p and compressive horizontal stress
sx in the roll gap during the rolling process (without expansion zone).
sx in the roll gap during the forging process (with expansion zone).
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H. Keife et al. / Journal of Materials Processing Technology 121 (2002) 50±59
of friction at the contact is 0.05. The semi-thickness of the strip at the entry and the exit are t0 0:025 mm and t2 0:015 mm, respectively, during the rolling stage. The roll force is calculated for the rolling process and then it is applied to the forging process, which is assumed to make the pro®le concave. The variation of the slip ratio vs =vr during the rolling stage, is illustrated in Fig. 10 and Fig. 11, shows the variation of the slip ratio during the forging stage. The strip pro®les during the rolling and forging processes are shown in Fig. 12. The pro®les support the same roll force. It can be seen that the forged pro®le is lower than the rolling pro®le. The variations of the pressure p and the compressive horizontal stress sx during the rolling and forging stages are shown in Figs. 13 and 14, respectively. During the rolling stage, p > sx in whole roll gap, while sx > p in part of the roll gap during the forging process. In the calculation for the forging process, it is assumed that the compressive horizontal stress is continuous at the interfaces between the various zones and different yield criterion is used in zones D and E. This leads to the discontinuity in the pressure and the strip pro®le at the interface between zone D and E, as seen in Figs. 12 and 14. Such a discontinuity in pressure is also observed in the case of ®nite element ABAQUS/Explicit analysis of the extrusion studied in Section 3 (see Fig. 8). 7. Discussion Results of some experiments on pro®les of thin strip after rolling and forging are shown. These results, together with some other results in the literature, show that strip pro®le may become concave in the roll gap. It is conjectured from these experiments that the compressive horizontal stress in the expansion zone is so much larger than the vertical pressure that the yield criterion is also satis®ed, but in a nonconventional manner. With this state of stress, the expansion of the strip in the vertical direction does not violate the plastic ¯ow rule. The horizontal stress in the reduction zones could be tensile or compressive. If it is compressive, then its magnitude is so much smaller than the pressure that the yield criterion is satis®ed. This kind of changeover of stress state is con®rmed by ®nite element package ABAQUS/Explicit analysis of an extrusion process. Therefore, the more restrictive assumption that the two reduction zones must be connected by a ¯at zone used in the previous foil rolling models [1,2] can be relaxed. Based on these observations, it is proposed that a plastic expansion zone exists in the roll gap in those experiments presented here and in the literature and that this is produced either during both the rolling and forging stages, or during the forging stage only. To achieve the changeover of the stress state, an elastic zone should exist between the plastic reduction and expansion zones. Some cases are analysed by using the proposed contact pro®le and modi®ed elastic foundation model for the rolls.
Acknowledgements This work is partially supported by the Australian Research Council under the Large Grant and SPIRT schemes. A QUT Visit Fellowship to Dr. H. Keife is also appreciated. We also like to thank Drs. S. Domanti and J. Edwards for their valuable discussions. Appendix A We derive in this appendix the analytic expressions for the solution of the governing equation (3) with the yield criterion (1) and (2) in the different zones shown in Fig. 5. Let the surface of the tool in the zone A be denoted by t a1 x b1, where a1 and b1 are constants. Then substituting this together with the yield criterion (1) into the equilibrium equation (3), we have the governing differential equation for sx, a1 x b1
dsx msx Ym a1 dx
m 0;
(A.1)
p where Ym is Ys for the Tresca yield criterion and 2Ys = 3 for the von Mises yield criterion. Let the surface of the tool in the zone B be denoted by t a2 x b2 , where a2 and b2 are constants. With the yield criterion (2) and this expression for the surface, we can derive the governing differential equation for the stress sx in this zone, a2 x b2
dsx msx Ym a2 m 0: dx
(A.2)
Denoting the surface of the tool in the zone C by t a3 x b3 , where a3 and b3 are constants, and using the yield criterion (1), we derive the governing differential equation for the stress sx as a3 x b3
dsx msx Ym a3 dx
m 0:
(A.3)
These three equations have a common form ax b
dsx msx c 0; dx
(A.4)
and this equation has a general solution c ; sx C0 ax b m=a m
(A.5)
where C0 is an arbitrary integral constant. Expressing the general solutions for the three zones, we have sx C1 a1 x b1
m=a1
Ym a1 m
m
in zone A; (A.6)
sx C2 a2 x b2
m=a2
Ym a2 m m
in zone B; (A.7)
H. Keife et al. / Journal of Materials Processing Technology 121 (2002) 50±59
sx C3 a3 x b3
m=a3
Ym a3 m
m
in zone C: (A.8)
The constants C1, C2 and C3 can be determined from the conditions at the exit and the interfaces between the zones. The pressure in the different zones is determined from the corresponding yield criteria.
References [1] N.A. Fleck, K.L. Johnson, Towards a new theory of cold rolling thin foil, Int. J. Mech. Sci. 29 (1987) 507±524.
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[2] N.A. Fleck, K.L. Johnson, M.E. Mear, L.C. Zhang, Cold rolling of foil, Proc. Inst. Mech. Eng. 206 (1992) 119±131. [3] M.P.F. Sutcliffe, P.J. Rayner, Experimental measurements of load and strip profile in thin strip rolling, Int. J. Mech. Sci. 40 (1998) 887±899. [4] E. Orowan, The calculation of roll pressure in hot and cold flat rolling, Proc. Inst. Mech. Eng. 150 (1943) 140±167. [5] O. Pawelski, W. Rasp, L. Loffler, A. Kramer, Compressive testing of flat specimens with dies similar to rolls for simulation of cold rolling (Stauchversuche an flachproben mit walzenahnichen werkzeugen zur nachahmung des kaltwalzens), Blech Rohre Profile 40 (1993) 379±383. [6] M.J. Grimble, M.A. Fuller, G.F. Bryant, A non-circular arc roll force model for cold rolling, Int. J. Numer. Meth. Eng. 12 (1978) 643±663. [7] C. Luo, H. Keife, The influence of process parameters upon the cold rolling of aluminium foils, in: Proceedings of the First International Conference on Tribology in Manufacturing Processes, Gifu, 1997, pp. 364±371.