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A robust model for rolling of thin strip and foil
International Journal of Mechanical Sciences 43 (2001) 1405}1419
A robust model for rolling of thin strip and foil H.R. Le, M.P.F. Sutcli!e* Department of Engineering, University of Cambridge, Trumpington Street, Cambridge CB2 1PZ, UK Received 27 November 1999; received in revised form 10 August 2000
Abstract A new analysis for cold rolling of thin strip and foil is developed. This model follows the approach of Fleck et al. [8], but relaxes their assumption of a central #at neutral zone. Instead of following their inverse method to obtain the pressure distribution in this neutral zone, an explicit equation for the contact pressure variation is obtained from the sticking condition in this region. This signi"cantly simpli"es the solution method, leading to a much more robust algorithm. Moreover the method treats the cases either where the roll retains its circular arc or where there is very signi"cant roll deformation in the same way, greatly simplifying the method of obtaining solutions. This will facilitate the incorporation of other e!ects such as the friction models currently being developed. Results are in line with the theory of Fleck et al. [8]. The e!ect of entry and exit tensions on the non-dimensional load and forward slip is investigated. It is found that the e!ect of equal entry and exit tensions is equivalent to reducing the yield stress of the strip by this tension stress. 2001 Elsevier Science Ltd. All rights reserved.
1. Introduction Modelling of thin strip and foil rolling is of great interest to industry due to the large tonnage of such material consumed each year. Modern set-up and control algorithms for rolling mills rely on robust and accurate mathematical models of the roll bite which can predict key parameters such as load, torque and forward slip as a function of the rolling parameters. A schematic of the metal rolling process is shown in Fig. 1. The strip is rolled between two rolls with an initial radius of R so that its thickness is reduced from t to t . Conventional theories by von Karman [1], Orowan [2], and Bland and Ford [3] are based on a circular roll shape as shown in Fig. 1(a). These models either neglect roll deformation or allow for this by assuming an increased roll radius, for example using the well-known Hitchcock formula [4]. This model is improved by Jortner et al. [5] to allow
* Corresponding author. Tel.: #44-1223-332996; fax: #44-1223-332662. E-mail address: [email protected] (M.P.F. Sutcli!e). 0020-7403/01/$ - see front matter 2001 Elsevier Science Ltd. All rights reserved. PII: S 0 0 2 0 - 7 4 0 3 ( 0 0 ) 0 0 0 9 2 - 8
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Nomenclature b(B) c(C) D GH E (E ) 1 0 EH(EH ) 1 0 M(M K ) p(P) q(Q) r R t t A ¹ < (< ) 1 0 =(= K ) x, z X > (> ) 1 C Z ( ) 1 0 (;) , ( ) 1 0 () , , ( , )
(non-dimensional) normal displacement of the roll surface, B"bEH/R> 0 C (non-dimensional) spacing between elements, C"cEH /R> 0 C in#uence coe$cients for the roll displacement Young's modulus of strip (roll) plane strain Young's modulus of strip (roll) (non-dimensional) roll torque per unit width of strip, M K "MEH/R> 0 C (non-dimensional) normal contact pressure, P"p/> C (non-dimensional) shear stress, Q"qEH /> 0 C reduction in strip thickness radius of undeformed work roll strip thickness strip thickness for an undeformed roll non-dimensional strip thickness, ¹"tEH/R> 0 C velocity of un-strained strip (roll) in x direction (non-dimensional) roll load per unit width of strip, = K "=EH /R> 0 C Cartesian co-ordinates in the rolling and thickness direction, respectively non-dimensional co-ordinate in the rolling direction, X"xEH /R> 0 C (e!ective) plane strain yield stress of strip, > "> !0.5( # ) C 1 forward slip, Z"(< !< )/< 1 0 0 tensile strain of strip (roll) (non-dimensional) friction coe$cient, ;"EH /> 0 C Poisson ratio of strip (roll) creep ratio, "(< !< )/< 1 0 0 (non-dimensional) tensile stress in the rolling direction (non-dimensional) unwind and rewind stress, respectively
for a non-circular roll shape. Unfortunately he maintains the assumption that there is relative slip between the roll and the strip throughout the bite, except at the &neutral point' where the relative slip between the roll and strip changes sign. These theories are successful in thick strip rolling, but unsatisfactory when applied to thin strip where there is signi"cant roll elasticity. An investigation by Johnson and Bentall [6] suggests that the plastic reduction of thin strip between two rolls occurs in two zones separated by an extensive no-slip neutral zone. In this region the frictional traction falls below that for slipping friction. Based on this suggestion, a new theory of cold rolling thin foil is developed by Fleck and Johnson [7], Fig. 1(b), in which the neutral zone is assumed to be #at and the contact pressure is approximated by the Hertzian elliptical shape, onto which is added a perturbation caused by the plastic strain in the strip. This is improved in the model of Fleck et al. [8]. Here, deformation of the rolls is treated by modelling these as elastic half-spaces. The contact length is split into a series of zones, according to whether the strip is plastic or elastic and whether there is slip between the roll and strip. For the slipping regions, equilibrium
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Fig. 1. Schematic of metal rolling process (a) with a circular and (b) with non-circular roll bite (after [8]).
of the strip is used to "nd the variation of pressure with rolling direction. For the no-slip neutral zone, which is taken as #at, a matrix equation is assembled which relates the elastic roll deformation to the normal pressure. This is inverted to "nd the pressure distribution in this region. To meet the continuity conditions at the boundaries between each of the zones, the positions of these boundaries are found using a Newton}Raphson scheme. Theoretical predictions for the roll shape in the bite using this model are in reasonable agreement with experiments with model materials by Sutcli!e and Rayner [9]. This model is extended by Yuen and co-workers [10,11] to include strain hardening of the strip. The computation time is in the order of seconds for minor roll #attening but much slower and unstable for more severe roll deformation. Domanti et al. [12] applied the model to thin strip and temper rolling, modelling roll elasticity using the in#uence functions for circular rolls described by Jortner [5]. They reported that each calculation requires a few seconds, although it is unclear how severe the roll #attening is for these calculation times. A new algorithm is proposed by Domanti et al. [13], using a theory by Greenwood [14] to solve for the contact pressure in the #at region. The solutions are very close to those obtained by Fleck et al. [8], but the execution time is much shorter. This algorithm is particularly useful for the most di$cult cases where there is a large #at region. Although the models described in the previous paragraph have gained widespread support, both from industrialists and academics, they su!er from two major drawbacks. Firstly the principle of separating the bite into several zones, for which the boundaries have to be solved, is numerically unstable and time-consuming. Secondly the regime of roll deformation (e.g. whether there is a central #at region) has to be identi"ed before starting the calculation so that appropriate numerical techniques can be applied. These de"ciencies need to be overcome before friction modelling, which plays a key role in foil rolling, can be successfully coupled into the problem. Zhang [15] released the assumption of a #at no-reduction region, relating the shear stress to the roll shape and contact pressure in this region. This allows direct solution of the shear stress and contact pressure in the central region, giving much improved e$ciency. The execution time is about 15 min for a given rigid body displacement of rolls, although the overall execution time will be signi"cantly longer for the program to converge to a desired exit gauge. An alternative strategy which overcomes the numerical di$culties is described by Gratacos et al. [16], who de"ne a friction law which simulates sticking friction in the neutral zone and slipping friction elsewhere. This strategy has also been used in a recent model which couples tribological and mechanical
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models of foil rolling [17]. The approach described in this paper takes its inspiration for modelling of the neutral zone from this method, although the details of the formulation are quite di!erent. Elsewhere the model follows the theory by Fleck et al. [8]. The theory and numerical scheme are described in Sections 2 and 3, while results are presented in Section 4.
2. Theory Most of the theory is taken directly from the work by Fleck et al. [8]. This is summarised below for completeness. The new element for modelling of the neutral zone is described in detail in Section 2.5. 2.1. Equilibrium of slab element Consider a slab element of the strip, as shown in Fig. 1(a). Equilibrium gives t
dt d #(#p) #2q"0, dx dx
(1)
where x is the rolling distance, t is the strip thickness, is the tensile stress in the rolling direction, averaged through the thickness of the strip, p is the interface pressure and q is the shear stress. 2.2. Roll shape The variation of strip thickness t through the bite is given by t"t #2b(x) (2) A where t "t !(x!x)/R is the strip thickness variation with an initial strip thickness A ? t corresponding to the circular arc for the undeformed roll, as illustrated in Fig. 1(b). Here x is the ? value of x at the entry to the bite. b(x) is the elastic displacement of the roll in the vertical z direction at a point x due to the applied normal pressure distribution p, measured relative to the displacement at the entry position. Using the half-space solution described by Johnson [18], b(x) is given by
VB s#x 2 ? ds, p(s)ln (3) b(x)"! H s!x E 0 \V? where x is the value of x at the exit of the bite and EH "E /(1! ) is the plane strain Young's ? 0 0 0 modulus of the roll, with the Poisson's ratio of the roll. 0 2.3. Elastic slip at entry and exit The vertical elastic strain of the strip is given by
1 p# 1 (z)"! 1 1! EH 1 1
(4)
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with EH and the plane strain elastic modulus and Poisson's ratio of the strip. Since there are no 1 1 plastic strains in these regions, the rate of change of strip thickness associated with the change in elastic strain in the rolling direction, given by di!erentiation of Eq. (4), is equal to the roll slope dt/dx, giving d EH dt dp # 1 "! 1 . t dx dx 1! dx 1 Combining Eqs. (1) and (5) we have
(5)
2q (#p) dt dp EH dt "! 1 # 1 # 1 . (6) t dx dx t dx 1! t 1! 1 1 In practice we can neglect the last term which is small compared to the "rst term on the right-hand side of the equation, to get 2q dp EH dt
! 1 # 1 . (7) dx t dx 1! t 1 Assuming a constant Coulomb friction coe$cient , the shear stress in the slipping elastic entry and exit regions is given by q"$p,
(8)
where the positive sign is used for the backward slip at entry and the negative sign for the forward slip at exit. Alternative formulations incorporating a limiting shear stress or a friction factor approach could straightforwardly be used in the model. 2.4. Plastic slip In the plastic region, a Tresca yield criterion is assumed. p#"> . 1 Substituting this equation into Eq. (1) gives 2q dp > dt " 1 # . t dx t dx
(9)
(10)
In the plastic slip regions, the Coulomb friction law, Eq. (8), is again used. 2.5. Sticking region In this section we derive new expressions relating the roll shape and the shear stress in the sticking zone. Equations for the change in roll and strip strains in the sticking zone can be combined with #ow continuity for the strip through this zone to derive expressions relating the pressure gradient and shear stress in this zone to the roll slope here. Results are summarised below and described in detail in Appendix A. In the plastic no-slip region, the pressure gradient and shear
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stress are given by dp C EH dt "! 1 , dx t dx
(11)
C EH dt , q"! 1 2 dx
(12)
where C "((2!4 )/(1! )!(1!2 )/(1! )EH/EH )\. 1 1 0 0 1 0 For aluminium strip and steel rolls, EH EH , so that C "1.05. 1 0 The pressure gradient and the shear stress in the elastic unloading region are given by dp C EH dt "! 1 , dx t dx
(13)
C C EH dt , q"! 1 dx 2
(14)
where
1!2 1 # 0 C " 1! 1 1! 1! 1! 1 1 0
EH 1 EH 0
\
and 1!2 EH 0 1. C " 1 # 1! 1! EH 1 0 0 For aluminium strip and steel rolls, C "1.36, C "0.62. Comparing these equations with Eqs. (11) and (12) for the plastic no-slip region, we "nd that they di!er only by having slightly di!erent coe$cients. Since the coe$cient C in Eq. (13) is close to C in Eq. (11) and the pressure gradient is small in the sticking region, Eq. (11) is used throughout the sticking region for simplicity. Eqs. (12) and (14) show that the pressure gradient is proportional to the roll slope in the sticking region. In fact, this is exactly the form of dependence arbitrarily assumed by Sutcli!e and Montmitonnet [17] in their model of foil rolling! The approach described above is similar in concept to that of Zhang [15], but uses a more appropriate expression for the shear stress variation in this region. The roll load per unit width is given by
VB
VB
="
p dx.
(15)
V The roll torque per unit width is given by M"
?
V
?
xp dx#0.5( t ! t ).
(16)
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3. Numerical implementation As numerical problems associated with solving the foil rolling problem present a signi"cant barrier to e!ective implementation, in this section we present a step-by-step guide to the solution method used. A simpli"ed #ow diagram is included as Fig. 2. We cast the equations in nondimensional forms, with the independent variables normalised as below tEH EH p xEH X" 0 , ¹" 0 , ;" 0 , P" , " , " , > > R> > > R> C C C C C C where > is an e!ective yield stress, > "> !0.5( # ) and and are the entry and exit C C 1 stresses, respectively. The rationale behind using this e!ective yield stress will become clear when considering the results, Section 4. Other parameters are normalised as below qEH =EH MEH 0, M 0 . " , Q" 0 , = K " K " > > R> R> C C C C
Fig. 2. A #ow chart of the numerical scheme.
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The solution now proceeds as follows: (1) We "rst assume a circular roll arc. An estimate must be made of a roll arc length which will exceed the actual contact arc, to be used in subsequent roll elasticity calculations (item 9). If the actual contact arc exceeds the estimated value, the range is expanded by extrapolation of the roll shape. On the other hand, if the roll arc length is overestimated, no signi"cant waste of execution time occurs because the contact pressure is solved only in the contact region. (2) The entry point X is determined from the roll shape, the roll separation and the inlet ? thickness ¹ . The program starts with an estimate for the neutral point X , based on L a previous iteration where appropriate. (3) Integrate the pressure and tension stress variation through the bite, with the given deformed roll shape and neutral position, as described in steps 4 to 6 below. The integration starts with the inlet boundary conditions P"0, " . (4) In the inlet elastic slip region simultaneously integrate the dimensionless forms of Eqs. (6) and (7), using a standard variable-step size second/third-order Runge}Kutta scheme [19] d¹ 2;P d "!(#P) ! , ¹dX ¹ dX
(17)
EH d¹ 2;P dP
! 1 # 1 . (18) dX > ¹dX 1! ¹ 1 C The end of the elastic slip region is reached when the yield criteria is satis"ed, P#"> /> 1 C at X"X . @ (5) In the central region there is either no-slip or plastic slip. With the approximation that the strip is plastic in the no-slip region, the pressure gradient and corresponding shear stress are given by dimensionless forms of Eqs. (11) and (12) as dP C EH d¹ "! 1 , (19) dX > ¹dX C C EH d¹ Q"! 1 . (20) 2> dX C When the magnitude of Q according to Eq. (20) exceeds the limiting value P, then slip occurs, Q"$;P, and the pressure gradient is given by the dimensionless form of Eq. (10) dP > d¹ 2Q " 1 # . (21) dX > ¹dX ¹ C The pressure can thus be integrated using Eqs. (19) and (21) until the end of the exit plastic slip region X , which is taken as the position where the roll slope is zero, with 2"0. A 6 (6) In the exit elastic slip region Eqs. (17) and (18), suitably modi"ed to account for the reversed direction of slip, are again integrated from X . The exit position X is determined when the A B normal pressure P falls to zero. A measure of the error in the estimate of the neutral point position is given by the di!erence in the calculated exit tension stress and the required Q B value , i.e. " ! . Q B
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(7) The neutral position is found using a standard solver which uses a combination of bisection, secant and inverse quadratic interpolation [19] to adjust the neutral position, running through steps 2}6 for each value of X , until is less than an appropriate tolerance. This L Q typically takes 10 iterations. At the end of this step we have solved the pressure distribution for a given roll shape. (8) Since the contact pressure is very sensitive to the roll shape for very thin strip, the contact pressure is updated using a relaxation factor e. PL>"eP#(1!e)PL.
(22)
(9) Recall (item 1) that a roll arc has been identi"ed for the elasticity calculations which encompasses the contact arc. The roll elastic deformation B(X) is calculated at N nodes located at equal intervals C along this arc using the in#uence coe$cients given by Fleck et al. [8] for an elastic half-space
where
H, B(X )" (D !D )PL>, G GH H H H C D "! [(k#1) ln(k#1)#(k!1) ln(k!1)!2k ln k]#D and k"i!j. GH 2 (23)
The variation in strip thickness ¹ through the bite (and corresponding roll shape) is then given by ¹"¹L#2B, (24) where ¹ is the undeformed roll gap shape. In general the roll gap shape ¹L will not give the required exit gauge. Hence, it is necessary to apply some rigid body displacement of the rolls. This is done by moving the deformed and undeformed roll gap shapes ¹"¹! , (25) R ¹ "¹ ! , (26) R where "¹L!¹ is the error in the exit gauge. R B The roll shape is then updated using the same relaxation factor e as for the pressure updating, Eq. (22), according to the following expression: ¹L>"e¹#(1!e)¹L.
(27)
Typically, values of e in the range 0.1}1.0 are used, with the smaller values for the more severe roll deformation cases. (10) The whole process, from steps (2)}(9), is then repeated until the roll shape converges, as estimated by the criterion max(abs(¹!¹L)( , where a tolerance of "0.01 ¹ is typically used.
(28)
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Fig. 3. The roll shape and contact pressure for R"135 mm, r"50%, "0.03, "0, "0, > "230 MPa, 1 EH "230 GPa, EH"77 GPa. 0 1
(11) The non-dimensional roll load per unit width is given by
6B
6B
= K "
P dX. 6? The non-dimensional roll torque per unit width is given by M K "
(29)
XP dX#0.5( ¹ ! ¹ ). (30) 6? A #ow diagram to illustrate the numerical procedure is given in Fig. 2. It is found that the program is stable given an appropriate relaxation factor. The execution time for cases with minor roll deformation is of the order of seconds. Even for the most severe case presented in Fig. 3 with t "0.03 mm, the execution time is about 20 min on a workstation. The program in its current form is suitable for o!-line optimization. Further work is needed to improve the computation speed for on-line control program. Using an analytical solution as the start guess of the pressure distribution in the #at region (cf. [13]) would greatly reduce the execution time. 4. Theoretical results Results are presented in two sections. The results in Section 4.1 con"rm that the model agrees with that of Fleck et al. [8]. In Section 4.2 new results are presented for the forward slip and for the e!ects of end tensions. 4.1. Roll shape, contact pressure, load and torque The roll shape and contact pressure for a range of foil entry gauges are shown in Fig. 3, using parameters typical of aluminium foil rolling (except that coiling tensions are taken as zero). For
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Fig. 4. Comparison of the current solution with the model by Fleck et al. [8], "0, "0, EH /EH"3. 0 1
a strip inlet thickness of 0.20 mm, the roll shape is approximately circular. When the inlet thickness is reduced to 0.10 mm, the contact pressure goes up resulting in some roll deformation, while at an inlet gauge of 0.03 mm there is a substantial sticking region in the middle of the bite which is nearly #at. These results reproduce the e!ects observed by Fleck et al. [8] for thin foil rolling, including a #at central region, a nearly Hertzian pressure distribution and a pressure spike just after the neutral zone. Although elastic slip regions at entry and exit have been included in the analysis, these do not appear to play a signi"cant role. Following the suggestion of Fleck et al. [8], the non-dimensional groups of load and torque, = K ;\ r\ and M K ;\ r\ are plotted as a function of thickness group 0.5¹ ;\ r\ in Fig. 4. Results are in good agreement with those of Fleck et al. [8]. The new method for calculating the shear stresses in the no-slip region follows the same general approach as Zhang [15], who also considers the creep strains in this region. He deduces the shear stress gradient from the e!ect that shear stresses have on the roll longitudinal strains, using an approximate mattress model [18]. Since the in#uence of shear stresses on roll strains is small, and given the inaccuracy of the mattress model as applied to rolling [8], his method is not able to reproduce the analysis of Fleck et al. [8] in the way that the current model can. 4.2. The ewect of tensions on roll load and forward slip The e!ect of unwind and rewind tensions on the load and forward slip are shown in Fig. 5. Fig. 5(a) shows that the dimensionless load is not signi"cantly changed when equal unwind and rewind tensions are applied, as compared with the case without end tensions. However the actual load will be lower when tensions are applied, as the e!ective yield stress > is reduced by the mean C of the tension stresses. The dimensionless load is only slightly changed where unequal tensions are chosen. This is consistent with conventional circular arc models, where an equal increase in unwind and rewind tensions is equivalent to a corresponding drop in yield stress, reducing the contact
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Fig. 5. The e!ect of entry and exit tensions on the master curve for roll load and forward slip, EH /EH"3. 0 1
Fig. 6. The e!ect of entry and exit tensions on contact pressure and roll shape, t "0.03 mm, r"50%, EH /EH"3. 0 1
pressure throughout the bite by an equal amount. Although it is not obvious that equal tensions can be treated using an e!ective yield stress for thin strip rolling, the result can be understood by examining those parts of the governing equations where the yield stress or e!ective yield stress occur, i.e. in Eqs. (18)}(21). Due to the small slopes in thin strip conditions, the relevant terms in these equations are negligible. Fig. 5(b) shows that, while the forward slip does not change where equal unwind and rewind tensions are applied, unequal tensions do alter the value of forward slip signi"cantly. Unwind and rewind tensions change the forward slip by approximately equal and opposite amounts. The contact pressure and roll shape for a pass with signi"cant roll deformation are illustrated in Fig. 6, showing four cases with di!erent combination of tensions, other conditions being the same. When equal tensions are applied, the contact pressure drops at both entry and exit, so that the roll shape and the forward slip are little changed. With an increased unwind tension, the contact pressure in
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the entry region falls, so that the neutral point and hence neutral zone moves towards the exit to maintain the balance of horizontal stresses. The converse applies for increased rewind tensions. 5. Conclusions A new model is presented for cold rolling of thin strip and foil. In general the model adopts the modelling approach of Fleck et al. [8]. However, instead of assuming a #at central region and solving the pressure distribution in the central neutral zone by inverting the elasticity solution for this region, small elastic and plastic strains in this central region are considered. This leads to an explicit solution for the pressure distribution in the central region, resulting in a much simpler, faster and more robust numerical algorithm. These advantages will facilitate the incorporation of additional complications such as a more sophisticated friction model. The load and torque predicted by this model are in good agreement with those of Fleck et al. [8]. Parametric studies show that the e!ect of equal tensions is equivalent to reducing the yield stress by the value of the tension stress. The model predicts an increase in forward slip with increasing exit tension or decreasing entry tension, an e!ect well known from industrial practice. Acknowledgements The support of EPSRC, Alcan International Ltd. and ALSTOM Drives and Controls Ltd is gratefully acknowledged. The advice and assistance of Drs. K. Waterson and D. Miller at Alcan, and Dr. P. Reeve and Mr. C. Fryer at ALSTOM Drives and Controls Ltd is much appreciated. Appendix A. The contact pressure and shear stress in the sticking region When there is sticking region (as shown in Fig. 1b), the strip and the roll must move at the same speed longitudinally in that region [18]. < !< 0 ", (A.1) (x)! (x)" 1 0 1 < 0 where is the creep ratio, < is the peripheral speed of the undeformed roll and < is the entry 0 1 speed of the strip. The longitudinal strain of the roll is given by the half-space solution [18]
2 VC q 1!2 p 0 ! ds. (A.2) (x)"! 0 x!s 1! EH EH 0 \V? 0 0 Assuming no plastic straining across the width of the strip, the longitudinal and vertical strains in the strip are given by summing the elastic and plastic strains
1 (x)" # 1 p #N , 1 1 EH 1! 1 1 1 (z)"! p# 1 !N . 1 1 EH 1! 1 1
(A.3) (A.4)
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Therefore 1 1!2 1 [!p]. (A.5) (x)"! (z)# 1 1 EH 1! 1 1 Substituting Eqs. (A.2) and (A.5) into Eq. (A.1), neglecting the e!ect of shear stress q in Eq. (A.2), gives the sticking condition 1!2 p 1 1!2 1 [!p]! (z)"!. 0 # 1 H 1! E EH 1! 1 0 0 1 Di!erentiating Eq. (A.6) gives
d (z) EH dt 1!2 EH dp 1!2 d dp 1 0 1 # ! "EH 1 " 1 . 1 dx 1! dx dx t dx 1! EH dx 1 0 0 If the yield criterion is satis"ed, we have dp d # "0. dx dx
(A.6)
(A.7)
(A.8)
Combining Eqs. (A.7) and (A.8) gives the pressure gradient in the sticking region dp d C EH dt "! "! 1 dx dx t dx
(A.9)
in which C "((2!4 )/(1! )!(1!2 )/(1! )EH/EH )\. 1 1 0 0 1 0 For aluminium strip, EH EH , so that C "1.05. 1 0 Substituting Eq. (A.9) into the equilibrium equation gives the shear stress in the plastic sticking region. C EH dt . (A.10) q"! 1 2 dx If elastic unloading occurs, the yield criteria is no longer satis"ed, but N "0. Di!erentiating 1 Eq. (A.4) gives dp d EH dt # 1 "! 1 . (A.11) dx 1! dx t dx 1 The pressure gradient in the elastic unloading region are given by combining Eqs. (A.7) and (A.11) C EH dt dp "! 1 , dx t dx
(A.12)
where
1!2 EH 1 # 0 1 C " 1! 1 1! 1! 1! EH 1 1 0 0 For aluminium strip, C "1.36.
\
.
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Substituting Eq. (A.12) into Eq. (A.11) gives d/dx. Combining this with Eq. (1) gives the shear stress C C EH dt q"! 1 dx 2
(A.13)
in which C " /(1! )#(1!2 )/(1! )EH/EH . For aluminium strip, C "0.62. 1 1 0 0 1 0 References [1] von Karmann T. Beitrag zur theorie des Walzvorganges. Zeitschrift fur Angewandte Mathematic und Mechanik 1925;5:139. [2] Orowan E. Graphical calculation of roll pressure with the assumptions of homogeneous compression and slipping friction. Proceedings of the Institution of Mechanical Engineers 1943;150:141. [3] Bland DR, Ford H. The calculation of roll force and torque in cold strip rolling with tensions. Proceedings of the Institution of Mechanical Engineers 1948;159:144. [4] Hitchcock JH. Roll neck bearings, Report of ASME Research committee, 1935. [5] Jortner D, Osterle JF, Zorowski CF. An analysis of cold rolling. International Journal of Mechanical Sciences 1960;2:179}94. [6] Johnson KL, Bentall RH. The onset of yield in the cold rolling of thin strip. Journal of the Mechanics and Physics of Solids 1969;17:253. [7] Fleck NA, Johnson KL. Towards a new theory of cold rolling thin foil. International Journal of Mechanical Sciences 1987;29(7):507}24. [8] Fleck NA, Johnson KL, Mear ME, Zhang LC. Cold rolling of foil. Proceedings of the Institution of Mechanical Engineers 1992;206:119}31. [9] Sutcli!e MPF, Rayner PJ. Experimental measurements of load and strip pro"le in thin strip rolling. International Journal of Mechanical Sciences 1998;40(9):887}9. [10] Yuen WYD, Dixon A, Nguyen DN. The modelling of the mechanics of the deformation in #at rolling. Journal of Mechanics Processing Technology 1996;60:87}94. [11] Dixon AE, Yuen WYD. A computationally fast method to model thin strip rolling. Proceedings Computational Techniques and Application Conference, 1995. p. 239}46. [12] Domanti SA, Edwards WJ, Thomas PJ, Chefneux IL. Application of foil rolling models to thin steel strip and temper rolling. Sixth International Rolling Conference, Dussedorf, 1994. p. 422}9. [13] Domanti SA, Edwards WJ, Thomas PJ. A model for rolling and thin strip rolling. AISE Annual Convention. Ohio, USA: Cleveland, 1994. [14] Greenwood JA. An extension of the Grubin theory of elastohydrodynamic lubrication. Journal of Physics D 1972;5:2195}211. [15] Zhang LC. A simple approach for cold rolling of foil, computational mechanics from concepts to computations. Southampton: AA Balkema Publishers, 1993. p. 283}6. [16] Gratacos P, Montmitonnet P, Fromholz P, Chenot JL. A plane-strain elastic "nite-element model for cold rolling of thin strip. International Journal of Mechanical Sciences 1992;34(3):195}210. [17] Sutcli!e MPF, Montmitonnet P. A coupled tribological and mechanical model for thin foil rolling in the mixed lubrication regime. Proceedings of the Conference on Modelling of Metal Rolling Processes 3, London, December 1999, Institute of Materials, 1999. p. 187}96. [18] Johnson KL. Contact mechanics. Cambridge: Cambridge University Press, 1985. [19] Matlab, The Mathworks Inc., 1996.
A robust model for rolling of thin strip and foil H.R. Le, M.P.F. Sutcli!e* Department of Engineering, University of Cambridge, Trumpington Street, Cambridge CB2 1PZ, UK Received 27 November 1999; received in revised form 10 August 2000
Abstract A new analysis for cold rolling of thin strip and foil is developed. This model follows the approach of Fleck et al. [8], but relaxes their assumption of a central #at neutral zone. Instead of following their inverse method to obtain the pressure distribution in this neutral zone, an explicit equation for the contact pressure variation is obtained from the sticking condition in this region. This signi"cantly simpli"es the solution method, leading to a much more robust algorithm. Moreover the method treats the cases either where the roll retains its circular arc or where there is very signi"cant roll deformation in the same way, greatly simplifying the method of obtaining solutions. This will facilitate the incorporation of other e!ects such as the friction models currently being developed. Results are in line with the theory of Fleck et al. [8]. The e!ect of entry and exit tensions on the non-dimensional load and forward slip is investigated. It is found that the e!ect of equal entry and exit tensions is equivalent to reducing the yield stress of the strip by this tension stress. 2001 Elsevier Science Ltd. All rights reserved.
1. Introduction Modelling of thin strip and foil rolling is of great interest to industry due to the large tonnage of such material consumed each year. Modern set-up and control algorithms for rolling mills rely on robust and accurate mathematical models of the roll bite which can predict key parameters such as load, torque and forward slip as a function of the rolling parameters. A schematic of the metal rolling process is shown in Fig. 1. The strip is rolled between two rolls with an initial radius of R so that its thickness is reduced from t to t . Conventional theories by von Karman [1], Orowan [2], and Bland and Ford [3] are based on a circular roll shape as shown in Fig. 1(a). These models either neglect roll deformation or allow for this by assuming an increased roll radius, for example using the well-known Hitchcock formula [4]. This model is improved by Jortner et al. [5] to allow
* Corresponding author. Tel.: #44-1223-332996; fax: #44-1223-332662. E-mail address: [email protected] (M.P.F. Sutcli!e). 0020-7403/01/$ - see front matter 2001 Elsevier Science Ltd. All rights reserved. PII: S 0 0 2 0 - 7 4 0 3 ( 0 0 ) 0 0 0 9 2 - 8
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Nomenclature b(B) c(C) D GH E (E ) 1 0 EH(EH ) 1 0 M(M K ) p(P) q(Q) r R t t A ¹ < (< ) 1 0 =(= K ) x, z X > (> ) 1 C Z ( ) 1 0 (;) , ( ) 1 0 () , , ( , )
(non-dimensional) normal displacement of the roll surface, B"bEH/R> 0 C (non-dimensional) spacing between elements, C"cEH /R> 0 C in#uence coe$cients for the roll displacement Young's modulus of strip (roll) plane strain Young's modulus of strip (roll) (non-dimensional) roll torque per unit width of strip, M K "MEH/R> 0 C (non-dimensional) normal contact pressure, P"p/> C (non-dimensional) shear stress, Q"qEH /> 0 C reduction in strip thickness radius of undeformed work roll strip thickness strip thickness for an undeformed roll non-dimensional strip thickness, ¹"tEH/R> 0 C velocity of un-strained strip (roll) in x direction (non-dimensional) roll load per unit width of strip, = K "=EH /R> 0 C Cartesian co-ordinates in the rolling and thickness direction, respectively non-dimensional co-ordinate in the rolling direction, X"xEH /R> 0 C (e!ective) plane strain yield stress of strip, > "> !0.5( # ) C 1 forward slip, Z"(< !< )/< 1 0 0 tensile strain of strip (roll) (non-dimensional) friction coe$cient, ;"EH /> 0 C Poisson ratio of strip (roll) creep ratio, "(< !< )/< 1 0 0 (non-dimensional) tensile stress in the rolling direction (non-dimensional) unwind and rewind stress, respectively
for a non-circular roll shape. Unfortunately he maintains the assumption that there is relative slip between the roll and the strip throughout the bite, except at the &neutral point' where the relative slip between the roll and strip changes sign. These theories are successful in thick strip rolling, but unsatisfactory when applied to thin strip where there is signi"cant roll elasticity. An investigation by Johnson and Bentall [6] suggests that the plastic reduction of thin strip between two rolls occurs in two zones separated by an extensive no-slip neutral zone. In this region the frictional traction falls below that for slipping friction. Based on this suggestion, a new theory of cold rolling thin foil is developed by Fleck and Johnson [7], Fig. 1(b), in which the neutral zone is assumed to be #at and the contact pressure is approximated by the Hertzian elliptical shape, onto which is added a perturbation caused by the plastic strain in the strip. This is improved in the model of Fleck et al. [8]. Here, deformation of the rolls is treated by modelling these as elastic half-spaces. The contact length is split into a series of zones, according to whether the strip is plastic or elastic and whether there is slip between the roll and strip. For the slipping regions, equilibrium
H.R. Le, M.P.F. Sutcliwe / International Journal of Mechanical Sciences 43 (2001) 1405}1419
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Fig. 1. Schematic of metal rolling process (a) with a circular and (b) with non-circular roll bite (after [8]).
of the strip is used to "nd the variation of pressure with rolling direction. For the no-slip neutral zone, which is taken as #at, a matrix equation is assembled which relates the elastic roll deformation to the normal pressure. This is inverted to "nd the pressure distribution in this region. To meet the continuity conditions at the boundaries between each of the zones, the positions of these boundaries are found using a Newton}Raphson scheme. Theoretical predictions for the roll shape in the bite using this model are in reasonable agreement with experiments with model materials by Sutcli!e and Rayner [9]. This model is extended by Yuen and co-workers [10,11] to include strain hardening of the strip. The computation time is in the order of seconds for minor roll #attening but much slower and unstable for more severe roll deformation. Domanti et al. [12] applied the model to thin strip and temper rolling, modelling roll elasticity using the in#uence functions for circular rolls described by Jortner [5]. They reported that each calculation requires a few seconds, although it is unclear how severe the roll #attening is for these calculation times. A new algorithm is proposed by Domanti et al. [13], using a theory by Greenwood [14] to solve for the contact pressure in the #at region. The solutions are very close to those obtained by Fleck et al. [8], but the execution time is much shorter. This algorithm is particularly useful for the most di$cult cases where there is a large #at region. Although the models described in the previous paragraph have gained widespread support, both from industrialists and academics, they su!er from two major drawbacks. Firstly the principle of separating the bite into several zones, for which the boundaries have to be solved, is numerically unstable and time-consuming. Secondly the regime of roll deformation (e.g. whether there is a central #at region) has to be identi"ed before starting the calculation so that appropriate numerical techniques can be applied. These de"ciencies need to be overcome before friction modelling, which plays a key role in foil rolling, can be successfully coupled into the problem. Zhang [15] released the assumption of a #at no-reduction region, relating the shear stress to the roll shape and contact pressure in this region. This allows direct solution of the shear stress and contact pressure in the central region, giving much improved e$ciency. The execution time is about 15 min for a given rigid body displacement of rolls, although the overall execution time will be signi"cantly longer for the program to converge to a desired exit gauge. An alternative strategy which overcomes the numerical di$culties is described by Gratacos et al. [16], who de"ne a friction law which simulates sticking friction in the neutral zone and slipping friction elsewhere. This strategy has also been used in a recent model which couples tribological and mechanical
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models of foil rolling [17]. The approach described in this paper takes its inspiration for modelling of the neutral zone from this method, although the details of the formulation are quite di!erent. Elsewhere the model follows the theory by Fleck et al. [8]. The theory and numerical scheme are described in Sections 2 and 3, while results are presented in Section 4.
2. Theory Most of the theory is taken directly from the work by Fleck et al. [8]. This is summarised below for completeness. The new element for modelling of the neutral zone is described in detail in Section 2.5. 2.1. Equilibrium of slab element Consider a slab element of the strip, as shown in Fig. 1(a). Equilibrium gives t
dt d #(#p) #2q"0, dx dx
(1)
where x is the rolling distance, t is the strip thickness, is the tensile stress in the rolling direction, averaged through the thickness of the strip, p is the interface pressure and q is the shear stress. 2.2. Roll shape The variation of strip thickness t through the bite is given by t"t #2b(x) (2) A where t "t !(x!x)/R is the strip thickness variation with an initial strip thickness A ? t corresponding to the circular arc for the undeformed roll, as illustrated in Fig. 1(b). Here x is the ? value of x at the entry to the bite. b(x) is the elastic displacement of the roll in the vertical z direction at a point x due to the applied normal pressure distribution p, measured relative to the displacement at the entry position. Using the half-space solution described by Johnson [18], b(x) is given by
VB s#x 2 ? ds, p(s)ln (3) b(x)"! H s!x E 0 \V? where x is the value of x at the exit of the bite and EH "E /(1! ) is the plane strain Young's ? 0 0 0 modulus of the roll, with the Poisson's ratio of the roll. 0 2.3. Elastic slip at entry and exit The vertical elastic strain of the strip is given by
1 p# 1 (z)"! 1 1! EH 1 1
(4)
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with EH and the plane strain elastic modulus and Poisson's ratio of the strip. Since there are no 1 1 plastic strains in these regions, the rate of change of strip thickness associated with the change in elastic strain in the rolling direction, given by di!erentiation of Eq. (4), is equal to the roll slope dt/dx, giving d EH dt dp # 1 "! 1 . t dx dx 1! dx 1 Combining Eqs. (1) and (5) we have
(5)
2q (#p) dt dp EH dt "! 1 # 1 # 1 . (6) t dx dx t dx 1! t 1! 1 1 In practice we can neglect the last term which is small compared to the "rst term on the right-hand side of the equation, to get 2q dp EH dt
! 1 # 1 . (7) dx t dx 1! t 1 Assuming a constant Coulomb friction coe$cient , the shear stress in the slipping elastic entry and exit regions is given by q"$p,
(8)
where the positive sign is used for the backward slip at entry and the negative sign for the forward slip at exit. Alternative formulations incorporating a limiting shear stress or a friction factor approach could straightforwardly be used in the model. 2.4. Plastic slip In the plastic region, a Tresca yield criterion is assumed. p#"> . 1 Substituting this equation into Eq. (1) gives 2q dp > dt " 1 # . t dx t dx
(9)
(10)
In the plastic slip regions, the Coulomb friction law, Eq. (8), is again used. 2.5. Sticking region In this section we derive new expressions relating the roll shape and the shear stress in the sticking zone. Equations for the change in roll and strip strains in the sticking zone can be combined with #ow continuity for the strip through this zone to derive expressions relating the pressure gradient and shear stress in this zone to the roll slope here. Results are summarised below and described in detail in Appendix A. In the plastic no-slip region, the pressure gradient and shear
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stress are given by dp C EH dt "! 1 , dx t dx
(11)
C EH dt , q"! 1 2 dx
(12)
where C "((2!4 )/(1! )!(1!2 )/(1! )EH/EH )\. 1 1 0 0 1 0 For aluminium strip and steel rolls, EH EH , so that C "1.05. 1 0 The pressure gradient and the shear stress in the elastic unloading region are given by dp C EH dt "! 1 , dx t dx
(13)
C C EH dt , q"! 1 dx 2
(14)
where
1!2 1 # 0 C " 1! 1 1! 1! 1! 1 1 0
EH 1 EH 0
\
and 1!2 EH 0 1. C " 1 # 1! 1! EH 1 0 0 For aluminium strip and steel rolls, C "1.36, C "0.62. Comparing these equations with Eqs. (11) and (12) for the plastic no-slip region, we "nd that they di!er only by having slightly di!erent coe$cients. Since the coe$cient C in Eq. (13) is close to C in Eq. (11) and the pressure gradient is small in the sticking region, Eq. (11) is used throughout the sticking region for simplicity. Eqs. (12) and (14) show that the pressure gradient is proportional to the roll slope in the sticking region. In fact, this is exactly the form of dependence arbitrarily assumed by Sutcli!e and Montmitonnet [17] in their model of foil rolling! The approach described above is similar in concept to that of Zhang [15], but uses a more appropriate expression for the shear stress variation in this region. The roll load per unit width is given by
VB
VB
="
p dx.
(15)
V The roll torque per unit width is given by M"
?
V
?
xp dx#0.5( t ! t ).
(16)
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3. Numerical implementation As numerical problems associated with solving the foil rolling problem present a signi"cant barrier to e!ective implementation, in this section we present a step-by-step guide to the solution method used. A simpli"ed #ow diagram is included as Fig. 2. We cast the equations in nondimensional forms, with the independent variables normalised as below tEH EH p xEH X" 0 , ¹" 0 , ;" 0 , P" , " , " , > > R> > > R> C C C C C C where > is an e!ective yield stress, > "> !0.5( # ) and and are the entry and exit C C 1 stresses, respectively. The rationale behind using this e!ective yield stress will become clear when considering the results, Section 4. Other parameters are normalised as below qEH =EH MEH 0, M 0 . " , Q" 0 , = K " K " > > R> R> C C C C
Fig. 2. A #ow chart of the numerical scheme.
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The solution now proceeds as follows: (1) We "rst assume a circular roll arc. An estimate must be made of a roll arc length which will exceed the actual contact arc, to be used in subsequent roll elasticity calculations (item 9). If the actual contact arc exceeds the estimated value, the range is expanded by extrapolation of the roll shape. On the other hand, if the roll arc length is overestimated, no signi"cant waste of execution time occurs because the contact pressure is solved only in the contact region. (2) The entry point X is determined from the roll shape, the roll separation and the inlet ? thickness ¹ . The program starts with an estimate for the neutral point X , based on L a previous iteration where appropriate. (3) Integrate the pressure and tension stress variation through the bite, with the given deformed roll shape and neutral position, as described in steps 4 to 6 below. The integration starts with the inlet boundary conditions P"0, " . (4) In the inlet elastic slip region simultaneously integrate the dimensionless forms of Eqs. (6) and (7), using a standard variable-step size second/third-order Runge}Kutta scheme [19] d¹ 2;P d "!(#P) ! , ¹dX ¹ dX
(17)
EH d¹ 2;P dP
! 1 # 1 . (18) dX > ¹dX 1! ¹ 1 C The end of the elastic slip region is reached when the yield criteria is satis"ed, P#"> /> 1 C at X"X . @ (5) In the central region there is either no-slip or plastic slip. With the approximation that the strip is plastic in the no-slip region, the pressure gradient and corresponding shear stress are given by dimensionless forms of Eqs. (11) and (12) as dP C EH d¹ "! 1 , (19) dX > ¹dX C C EH d¹ Q"! 1 . (20) 2> dX C When the magnitude of Q according to Eq. (20) exceeds the limiting value P, then slip occurs, Q"$;P, and the pressure gradient is given by the dimensionless form of Eq. (10) dP > d¹ 2Q " 1 # . (21) dX > ¹dX ¹ C The pressure can thus be integrated using Eqs. (19) and (21) until the end of the exit plastic slip region X , which is taken as the position where the roll slope is zero, with 2"0. A 6 (6) In the exit elastic slip region Eqs. (17) and (18), suitably modi"ed to account for the reversed direction of slip, are again integrated from X . The exit position X is determined when the A B normal pressure P falls to zero. A measure of the error in the estimate of the neutral point position is given by the di!erence in the calculated exit tension stress and the required Q B value , i.e. " ! . Q B
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(7) The neutral position is found using a standard solver which uses a combination of bisection, secant and inverse quadratic interpolation [19] to adjust the neutral position, running through steps 2}6 for each value of X , until is less than an appropriate tolerance. This L Q typically takes 10 iterations. At the end of this step we have solved the pressure distribution for a given roll shape. (8) Since the contact pressure is very sensitive to the roll shape for very thin strip, the contact pressure is updated using a relaxation factor e. PL>"eP#(1!e)PL.
(22)
(9) Recall (item 1) that a roll arc has been identi"ed for the elasticity calculations which encompasses the contact arc. The roll elastic deformation B(X) is calculated at N nodes located at equal intervals C along this arc using the in#uence coe$cients given by Fleck et al. [8] for an elastic half-space
where
H, B(X )" (D !D )PL>, G GH H H H C D "! [(k#1) ln(k#1)#(k!1) ln(k!1)!2k ln k]#D and k"i!j. GH 2 (23)
The variation in strip thickness ¹ through the bite (and corresponding roll shape) is then given by ¹"¹L#2B, (24) where ¹ is the undeformed roll gap shape. In general the roll gap shape ¹L will not give the required exit gauge. Hence, it is necessary to apply some rigid body displacement of the rolls. This is done by moving the deformed and undeformed roll gap shapes ¹"¹! , (25) R ¹ "¹ ! , (26) R where "¹L!¹ is the error in the exit gauge. R B The roll shape is then updated using the same relaxation factor e as for the pressure updating, Eq. (22), according to the following expression: ¹L>"e¹#(1!e)¹L.
(27)
Typically, values of e in the range 0.1}1.0 are used, with the smaller values for the more severe roll deformation cases. (10) The whole process, from steps (2)}(9), is then repeated until the roll shape converges, as estimated by the criterion max(abs(¹!¹L)( , where a tolerance of "0.01 ¹ is typically used.
(28)
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Fig. 3. The roll shape and contact pressure for R"135 mm, r"50%, "0.03, "0, "0, > "230 MPa, 1 EH "230 GPa, EH"77 GPa. 0 1
(11) The non-dimensional roll load per unit width is given by
6B
6B
= K "
P dX. 6? The non-dimensional roll torque per unit width is given by M K "
(29)
XP dX#0.5( ¹ ! ¹ ). (30) 6? A #ow diagram to illustrate the numerical procedure is given in Fig. 2. It is found that the program is stable given an appropriate relaxation factor. The execution time for cases with minor roll deformation is of the order of seconds. Even for the most severe case presented in Fig. 3 with t "0.03 mm, the execution time is about 20 min on a workstation. The program in its current form is suitable for o!-line optimization. Further work is needed to improve the computation speed for on-line control program. Using an analytical solution as the start guess of the pressure distribution in the #at region (cf. [13]) would greatly reduce the execution time. 4. Theoretical results Results are presented in two sections. The results in Section 4.1 con"rm that the model agrees with that of Fleck et al. [8]. In Section 4.2 new results are presented for the forward slip and for the e!ects of end tensions. 4.1. Roll shape, contact pressure, load and torque The roll shape and contact pressure for a range of foil entry gauges are shown in Fig. 3, using parameters typical of aluminium foil rolling (except that coiling tensions are taken as zero). For
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Fig. 4. Comparison of the current solution with the model by Fleck et al. [8], "0, "0, EH /EH"3. 0 1
a strip inlet thickness of 0.20 mm, the roll shape is approximately circular. When the inlet thickness is reduced to 0.10 mm, the contact pressure goes up resulting in some roll deformation, while at an inlet gauge of 0.03 mm there is a substantial sticking region in the middle of the bite which is nearly #at. These results reproduce the e!ects observed by Fleck et al. [8] for thin foil rolling, including a #at central region, a nearly Hertzian pressure distribution and a pressure spike just after the neutral zone. Although elastic slip regions at entry and exit have been included in the analysis, these do not appear to play a signi"cant role. Following the suggestion of Fleck et al. [8], the non-dimensional groups of load and torque, = K ;\ r\ and M K ;\ r\ are plotted as a function of thickness group 0.5¹ ;\ r\ in Fig. 4. Results are in good agreement with those of Fleck et al. [8]. The new method for calculating the shear stresses in the no-slip region follows the same general approach as Zhang [15], who also considers the creep strains in this region. He deduces the shear stress gradient from the e!ect that shear stresses have on the roll longitudinal strains, using an approximate mattress model [18]. Since the in#uence of shear stresses on roll strains is small, and given the inaccuracy of the mattress model as applied to rolling [8], his method is not able to reproduce the analysis of Fleck et al. [8] in the way that the current model can. 4.2. The ewect of tensions on roll load and forward slip The e!ect of unwind and rewind tensions on the load and forward slip are shown in Fig. 5. Fig. 5(a) shows that the dimensionless load is not signi"cantly changed when equal unwind and rewind tensions are applied, as compared with the case without end tensions. However the actual load will be lower when tensions are applied, as the e!ective yield stress > is reduced by the mean C of the tension stresses. The dimensionless load is only slightly changed where unequal tensions are chosen. This is consistent with conventional circular arc models, where an equal increase in unwind and rewind tensions is equivalent to a corresponding drop in yield stress, reducing the contact
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Fig. 5. The e!ect of entry and exit tensions on the master curve for roll load and forward slip, EH /EH"3. 0 1
Fig. 6. The e!ect of entry and exit tensions on contact pressure and roll shape, t "0.03 mm, r"50%, EH /EH"3. 0 1
pressure throughout the bite by an equal amount. Although it is not obvious that equal tensions can be treated using an e!ective yield stress for thin strip rolling, the result can be understood by examining those parts of the governing equations where the yield stress or e!ective yield stress occur, i.e. in Eqs. (18)}(21). Due to the small slopes in thin strip conditions, the relevant terms in these equations are negligible. Fig. 5(b) shows that, while the forward slip does not change where equal unwind and rewind tensions are applied, unequal tensions do alter the value of forward slip signi"cantly. Unwind and rewind tensions change the forward slip by approximately equal and opposite amounts. The contact pressure and roll shape for a pass with signi"cant roll deformation are illustrated in Fig. 6, showing four cases with di!erent combination of tensions, other conditions being the same. When equal tensions are applied, the contact pressure drops at both entry and exit, so that the roll shape and the forward slip are little changed. With an increased unwind tension, the contact pressure in
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the entry region falls, so that the neutral point and hence neutral zone moves towards the exit to maintain the balance of horizontal stresses. The converse applies for increased rewind tensions. 5. Conclusions A new model is presented for cold rolling of thin strip and foil. In general the model adopts the modelling approach of Fleck et al. [8]. However, instead of assuming a #at central region and solving the pressure distribution in the central neutral zone by inverting the elasticity solution for this region, small elastic and plastic strains in this central region are considered. This leads to an explicit solution for the pressure distribution in the central region, resulting in a much simpler, faster and more robust numerical algorithm. These advantages will facilitate the incorporation of additional complications such as a more sophisticated friction model. The load and torque predicted by this model are in good agreement with those of Fleck et al. [8]. Parametric studies show that the e!ect of equal tensions is equivalent to reducing the yield stress by the value of the tension stress. The model predicts an increase in forward slip with increasing exit tension or decreasing entry tension, an e!ect well known from industrial practice. Acknowledgements The support of EPSRC, Alcan International Ltd. and ALSTOM Drives and Controls Ltd is gratefully acknowledged. The advice and assistance of Drs. K. Waterson and D. Miller at Alcan, and Dr. P. Reeve and Mr. C. Fryer at ALSTOM Drives and Controls Ltd is much appreciated. Appendix A. The contact pressure and shear stress in the sticking region When there is sticking region (as shown in Fig. 1b), the strip and the roll must move at the same speed longitudinally in that region [18]. < !< 0 ", (A.1) (x)! (x)" 1 0 1 < 0 where is the creep ratio, < is the peripheral speed of the undeformed roll and < is the entry 0 1 speed of the strip. The longitudinal strain of the roll is given by the half-space solution [18]
2 VC q 1!2 p 0 ! ds. (A.2) (x)"! 0 x!s 1! EH EH 0 \V? 0 0 Assuming no plastic straining across the width of the strip, the longitudinal and vertical strains in the strip are given by summing the elastic and plastic strains
1 (x)" # 1 p #N , 1 1 EH 1! 1 1 1 (z)"! p# 1 !N . 1 1 EH 1! 1 1
(A.3) (A.4)
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Therefore 1 1!2 1 [!p]. (A.5) (x)"! (z)# 1 1 EH 1! 1 1 Substituting Eqs. (A.2) and (A.5) into Eq. (A.1), neglecting the e!ect of shear stress q in Eq. (A.2), gives the sticking condition 1!2 p 1 1!2 1 [!p]! (z)"!. 0 # 1 H 1! E EH 1! 1 0 0 1 Di!erentiating Eq. (A.6) gives
d (z) EH dt 1!2 EH dp 1!2 d dp 1 0 1 # ! "EH 1 " 1 . 1 dx 1! dx dx t dx 1! EH dx 1 0 0 If the yield criterion is satis"ed, we have dp d # "0. dx dx
(A.6)
(A.7)
(A.8)
Combining Eqs. (A.7) and (A.8) gives the pressure gradient in the sticking region dp d C EH dt "! "! 1 dx dx t dx
(A.9)
in which C "((2!4 )/(1! )!(1!2 )/(1! )EH/EH )\. 1 1 0 0 1 0 For aluminium strip, EH EH , so that C "1.05. 1 0 Substituting Eq. (A.9) into the equilibrium equation gives the shear stress in the plastic sticking region. C EH dt . (A.10) q"! 1 2 dx If elastic unloading occurs, the yield criteria is no longer satis"ed, but N "0. Di!erentiating 1 Eq. (A.4) gives dp d EH dt # 1 "! 1 . (A.11) dx 1! dx t dx 1 The pressure gradient in the elastic unloading region are given by combining Eqs. (A.7) and (A.11) C EH dt dp "! 1 , dx t dx
(A.12)
where
1!2 EH 1 # 0 1 C " 1! 1 1! 1! 1! EH 1 1 0 0 For aluminium strip, C "1.36.
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H.R. Le, M.P.F. Sutcliwe / International Journal of Mechanical Sciences 43 (2001) 1405}1419
1419
Substituting Eq. (A.12) into Eq. (A.11) gives d/dx. Combining this with Eq. (1) gives the shear stress C C EH dt q"! 1 dx 2
(A.13)
in which C " /(1! )#(1!2 )/(1! )EH/EH . For aluminium strip, C "0.62. 1 1 0 0 1 0 References [1] von Karmann T. Beitrag zur theorie des Walzvorganges. Zeitschrift fur Angewandte Mathematic und Mechanik 1925;5:139. [2] Orowan E. Graphical calculation of roll pressure with the assumptions of homogeneous compression and slipping friction. Proceedings of the Institution of Mechanical Engineers 1943;150:141. [3] Bland DR, Ford H. The calculation of roll force and torque in cold strip rolling with tensions. Proceedings of the Institution of Mechanical Engineers 1948;159:144. [4] Hitchcock JH. Roll neck bearings, Report of ASME Research committee, 1935. [5] Jortner D, Osterle JF, Zorowski CF. An analysis of cold rolling. International Journal of Mechanical Sciences 1960;2:179}94. [6] Johnson KL, Bentall RH. The onset of yield in the cold rolling of thin strip. Journal of the Mechanics and Physics of Solids 1969;17:253. [7] Fleck NA, Johnson KL. Towards a new theory of cold rolling thin foil. International Journal of Mechanical Sciences 1987;29(7):507}24. [8] Fleck NA, Johnson KL, Mear ME, Zhang LC. Cold rolling of foil. Proceedings of the Institution of Mechanical Engineers 1992;206:119}31. [9] Sutcli!e MPF, Rayner PJ. Experimental measurements of load and strip pro"le in thin strip rolling. International Journal of Mechanical Sciences 1998;40(9):887}9. [10] Yuen WYD, Dixon A, Nguyen DN. The modelling of the mechanics of the deformation in #at rolling. Journal of Mechanics Processing Technology 1996;60:87}94. [11] Dixon AE, Yuen WYD. A computationally fast method to model thin strip rolling. Proceedings Computational Techniques and Application Conference, 1995. p. 239}46. [12] Domanti SA, Edwards WJ, Thomas PJ, Chefneux IL. Application of foil rolling models to thin steel strip and temper rolling. Sixth International Rolling Conference, Dussedorf, 1994. p. 422}9. [13] Domanti SA, Edwards WJ, Thomas PJ. A model for rolling and thin strip rolling. AISE Annual Convention. Ohio, USA: Cleveland, 1994. [14] Greenwood JA. An extension of the Grubin theory of elastohydrodynamic lubrication. Journal of Physics D 1972;5:2195}211. [15] Zhang LC. A simple approach for cold rolling of foil, computational mechanics from concepts to computations. Southampton: AA Balkema Publishers, 1993. p. 283}6. [16] Gratacos P, Montmitonnet P, Fromholz P, Chenot JL. A plane-strain elastic "nite-element model for cold rolling of thin strip. International Journal of Mechanical Sciences 1992;34(3):195}210. [17] Sutcli!e MPF, Montmitonnet P. A coupled tribological and mechanical model for thin foil rolling in the mixed lubrication regime. Proceedings of the Conference on Modelling of Metal Rolling Processes 3, London, December 1999, Institute of Materials, 1999. p. 187}96. [18] Johnson KL. Contact mechanics. Cambridge: Cambridge University Press, 1985. [19] Matlab, The Mathworks Inc., 1996.