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An investigation into the plastic deformation behavior at the roll gap during plate rolling
Journal of Materials Processing Technology 88 (1999) 97 – 104
An investigation into the plastic deformation behavior at the roll gap during plate rolling Y.M. Hwang *, H.H. Hsu Department of Mechanical Engineering, National Sun Yat-Sen Uni6ersity, Kaohsiung, Taiwan 804, ROC Received 22 October 1997
Abstract A mathematical model using the dual-stream function method under cylindrical coordinates has been proposed to investigate the position of neutral point and the plastic deformation region at the roll gap during plate rolling. The profile of width spread is assumed to be a quadratic function of the cylindrical angle. The rigid – plastic boundary at the entrance of the roll gap is considered to be inclined to the rolling direction. The velocity fields derived from the proposed dual-stream function can automatically satisfy the volume constancy and the velocity boundary conditions within the roll gap. Experiments on plate rolling using aluminum sheets meshed by electrochemical etching were carried out. The predicted profiles of width spread and the plastic flow pattern at the roll gap were compared with experimental measurements. From the comparisons between analytical and experimental results, it is clear that this model can offer useful knowledge in designing the pass-schedule of a plate rolling process. © 1999 Elsevier Science S.A. All rights reserved. Keywords: Plate rolling; Dual-stream function method; Neutral point
1. Introduction The rolling process is an efficient and economical approach for the manufacturing of strip or plate metals. When the aspect ratio of the plate (the ratio of its width to thickness) is relatively small compared with sheets or strips, the width spread at the roll gap can not be neglected, i.e. the plastic deformation of plate rolling at the roll gap is that of three-dimensional deformation. Several experimental works [1,2] have been published on this topic. For example, according to a series of experimental data using copper, aluminum and steel as the specimens during cold rolling, Yanagimoto [1] developed an empirical equation for the profile of the width spread. Among the analytical researches in plate rolling, a general approximate formulation based on the virtual work-rate principle was presented by Hill [3] for analyzing the plastic deformation during forging and bar drawing. * Corresponding author. Fax: +886-7-5254299.
A kinematically admissible velocity field based on the concept suggested by Hill [3] was proposed by Oh and Kobayashi [4] to simulate the width spread of the plate at the roll gap. Mori and Osakada [5] simulated the three-dimensional deformation of the plate during plate rolling using the rigid–plastic finite element analysis. Marques and Martins [6] adopted the dual-stream function concept and discretized the plastic region into several tetrahedral elements to discuss the width spread and rolling force in plate rolling. In all of the above models, the rigid–plastic boundary at the entrance of the roll gap was assumed to be a plane and normal to the rolling direction. However, it is indicated in the literature [7,8] that the plastic deformation at the roll gap is usually inhomogeneous. Therefore, the rigid–plastic boundary at the entrance of the roll gap is inclined toward the rolling direction. Moreover, the position of the neutral point, which affects the forward slip and accordingly the design of the pass schedule of a tandem mill rolling, has not been discussed systematically.
0924-0136/99/$ - see front matter © 1999 Elsevier Science S.A. All rights reserved. PII: S 0 9 2 4 - 0 1 3 6 ( 9 8 ) 0 0 3 9 0 - 2
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Y.M. Hwang, H.H. Hsu / Journal of Materials Processing Technology 88 (1999) 97–104
The present authors have proposed a mathematical model using the dual-stream function method to simulate the three-dimensional deformation of the plate at the roll gap [9]. In this paper, the effects of various rolling parameters such as friction factor, aspect ratio and roll radius-to-width ratio upon the position of neutral point and on the size of the plastic deformation region at the roll gap are examined systematically. Furthermore, experiments on plate rolling employing aluminum plates are carried out. Comparisons of streamlines and the rigid – plastic boundary at the entrance of the roll gap between analytical and experimental results have also been made to verify the validity of this model.
2. Mathematical model Throughout the analysis, the following assumptions are employed: (i) the flattening and bending of the rolls are neglected; (ii) the plate is of rigid – plastic material; (iii) the friction between the plate and roll is considered to have a constant friction factor; (iv) the material is isotropic; and (v) bulge deformation on the side surface of the plate is not taken into account. Fig. 1 presents a schematic illustration of a general symmetric plate rolling using cylindrical coordinates. In a three-dimensional analysis, velocity fields can be derived from dual-stream functions which represent two flow patterns within two orthogonal planes [10]. The flow pattern at the roll gap within the r – u plane is assumed to have a uniform circumferential velocity
distribution along any radial cross section and to be represented by a stream function denoted by f as follows: r−r0(u) f(r, u)= Q r1(u)− r0(u)
(1)
r1(u)= ecosu − (ecosu)2 − (e 2 − R 2)
(2)
r0(u)=
r0 cos u
(3)
e= R+tfm + r0
(4)
where r1 and r0 are the boundary functions along the upper roll surface and the central plane of the plate, respectively; Q denotes the volume flow rate at any cross-section; R is the roll radius; tfm is the half thickness of the plate at the exit of the roll gap; and r0 is the distance from the origin O to the central plane at the exit C. The flow pattern in the u–z plane is also assumed to have a uniform circumferential velocity distribution in the width direction and to be represented by c as follows: c(u, z)=
z b(u)
a= arctan
L− Li tim
u a
2
(6) (7)
where b(u), denoting the width profile function of the plate at the roll gap, is assumed to be a quadratic function of u. The radial, circumferential and width velocities, Vr, Vu and Vz, can be derived directly from the cross-product of the gradients of f and c as:
Vr (r, u)=Q − = Vu (u)= Q
1 (f (c r (u (z
Q (r− r0)(r1,u − r0,u ) r0,u + (r1 − r0) r(r1 − r0)b
(f (c Q = (r (z (r1 − r0)b
Vz (r, u, z)= Q −
Fig. 1. Schematic illustration of the mathematical model of plate rolling.
b(u)= bfm + (bim − bfm )
(5)
1 (f (c Qzb,u = r (r (u r(r1 − r0)b 2
(8) (9) (10)
where ( ),u denotes the partial derivative of ( ) with respect to u. The velocity boundary conditions at the exit of the roll gap, where the velocities in the radial direction (r) and the width direction (z) should be zero and the circumferential velocities should be uniform, can be checked by substituting u= 0 into Eqs. (8)–(10).
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99
Fig. 2. Optimized velocity field in plate rolling.
The boundary conditions along the upper roll surface and the central plane can be checked by substituting r =r1 and r= r0 into Eqs. (8) and (9). More details of the derivation are given in Ref. [9]. The velocity field derived from Eqs. (1) and (5) can also be checked to ensure that it satisfies the velocity boundary conditions along the free surface, the central plane, and the rigid – plastic boundary plane at the entrance of the roll gap [9]. According to the velocity – strain rate relations, an arbitrary dual-stream function is obtained that can automatically satisfy the condition of incompressibility. Thus, the velocity field expressed in Eqs. (8)–(10) is a kinematically admissible velocity field.
The constants Q, bfm and Li are left as the pseudoindependent parameters used to minimize the total power of plastic deformation. The equivalent strain rate in the plastic region can be derived from the normal and shear strain rates as: o; eq =
'
2 2 (o; + o; 2uu + o; 2zz + 2o; 2ru + 2o; 2uz + 2o; 2zr) 3 rr
(11)
The internal strain energy rate of deformation dissipated in the plastic region, W: i, can be calculated from the equivalent strain rate and the equivalent flow stress of the deformed material, s, as:
100
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Fig. 3. Effects of friction factor on the position of the neutral point at various reductions.
Fig. 5. Effects of width-to-thickness ratio on the position of the neutral point at various reductions.
&
W: i = so; eq dV
(12)
The shear loss, W: s, dissipated at the rigid–plastic boundaries, and the friction loss, W: f, dissipated at the roll surfaces, are calculated from the velocity fields in the plastic region, respectively, as: W: s = W: s =
1
3 m
3
& &
s(D6)Gs ds
(13)
s(D6)Gf ds
(14)
Gs
Gf
where (Dn)Gs, (Dn)Gf are the velocity differences of the components tangent to the rigid–plastic boundary and the roll surface. The total power of deformation, J, is obtained as follows: J= W: i + W: s + W: f
(15)
The minimization of J, which is expressed by the function of Q, bfm and Li, is performed by the flexible polyhedron search method [11].
3. Numerical calculations and discussions Fig. 4. Effects of friction factor on the relative length of the plastic region at various reductions.
An optimized velocity filed in plate rolling is shown in Fig. 2. The rolling conditions are R= 100 mm,
Y.M. Hwang, H.H. Hsu / Journal of Materials Processing Technology 88 (1999) 97–104
ti = 10 mm, tf = 8 mm, bi =20 mm, m =0.7 and U= 1 mm s − 1. Clearly, the velocity field shown in Fig. 2(b) satisfies the boundary conditions along the free surface and has a uniform circumferential velocity at any width cross-section. The velocity field shown in Fig. 2(c) satisfies the boundary conditions along the roll surface and has a uniform circumferential velocity at any radial cross section, as expected. In other words, the velocity field satisfies all of the velocity boundary conditions. Fig. 3 shows the effects of friction factor, m, on the position of the neutral point, Ln /L, at various reductions. Because larger friction factors will increase the speed of the plate at the exit of the roll gap, the forward slip will increase accordingly and the neutral points will become closer to the entrance of the roll gap. On the other hand, the neutral points are closer to the exit of the roll gap as the thickness reduction increases. The results show the same tendency as that found in other work [4]. Fig. 4 shows the effects of friction factor, m, on the relative length of the plastic region, Li /L, at various reductions. The relative length of the plastic region will influence the streamlines of the plastic flow and the homogeneity of plastic deformation. Because larger friction factors enhance the inhomogeneous velocity distributions in the plastic region, the rigid–
Fig. 6. Effects of width-to-thickness ratio on the relative length of the plastic region at various reductions.
101
Fig. 7. Effects of roll radius-to-thickness ratio on the position of the neutral point at various reductions.
plastic boundary at the entrance of the roll gap becomes more inclined toward the rolling direction. However, the plastic deformation is more homogeneous for larger thickness reductions, thus the rigid– plastic boundary is less inclined. Fig. 5 shows the effects of the width-to-thickness ratio, bi /ti, on the position of neutral point, Ln /L, at various reductions. It is obvious that the position of the neutral point approaches the entrance of the roll gap as bi /ti increases. That is because when bi /ti increases (i.e. closer to a plane-strain state) the speed of the plate at the exit of the roll gap will increase due to the constraint of the plastic flow in the width direction, and thus the neutral point becomes closer to the entrance of the roll gap. Fig. 6 shows the effects of the width-to-thickness ratio, bi /ti, on the relative length of the plastic region, Li /L, at various reductions. Clearly, for larger reductions, the relative length increases with the increase of the width-to-thickness ratio, whereas at lower reductions, the tendency is reversed. On the other hand, the effects of the reduction upon the relative length are more profound at greater width-to-thickness ratios. The opposite tendencies at larger and smaller bi /ti are probably caused by the contact length-towidth ratio, L/bi. In other words, when bi /ti is smaller and L/bi is larger, the plastic flow toward the width
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direction is profound, and accordingly, the inclined angle a (or the relative length of the plastic region) is not significantly affected. Fig. 7 shows the effects of the roll radius-to-thickness ratio, R/ti, on the position of the neutral point, Ln /L, at various reductions. Plate rolling can be approximately regarded as simple compression plus drawing. When R/ti increases, the arc die becomes flatter and the neutral point is closer to the center of the roll surface. Fig. 8 shows the effects of the roll radius-to-thickness ratio, R/ti, on the relative length of the plastic region at various reductions. Obviously, the relative length of the plastic region increases with increasing R/ti, which means that the plastic deformation at larger R/ti is closer to homogeneous deformation
4. Experiments on plate rolling Experiments on plate rolling using a two-high test rolling mill have been performed. The radius of both rolls was 100 mm, and the peripheral speed of both rolls was 16.67 mm s − 1. Specimens employed in the experiments were A6061T6 aluminum. The top and bottom surfaces were first cleaned using a cloth with acetone and, then slightly polished with sand-paper to ensure dry friction during the plate rolling experiments. Before rolling the side surface of the plate was electro-
Fig. 9. Comparison of predicted streamlines with experimental results.
Fig. 10. Comparison of predicted streamlines with experimental results.
Fig. 8. Effects of roll radius-to-thickness ratio on the relative length of the plastic region.
chemically etched to form a 2× 2 mm mesh. A tensile test of the specimens was carried out at an average strain rate of 0.017 s − 1. The strength coefficient K and the strain-hardening exponent n obtained for the 10.41 mm-thick plates were 414.7 MPa and 0.083, respectively, and for the 8.08 mm-thick plates K=516.2 MPa and n= 0.078, respectively. Fig. 9 presents a comparison of the predicted stream-
Y.M. Hwang, H.H. Hsu / Journal of Materials Processing Technology 88 (1999) 97–104
lines with experimental results. A friction factor of m =0.7 is assumed during the theoretical calculations, which is generally close to the condition for dry friction. From Fig. 9, it was found that the plastic flow along the
103
roll surface at the roll gap was faster than that at the central plane of the plate. The distortion of the meshes near to the surface was found to be more profound than that in the central plane. The actual rigid–plastic boundary at the entrance of the roll gap was approximately a convex surface, the predicted rigid–plastic boundary was close to the experimentally-observed boundary. Fig. 10 also shows comparisons of the predicted streamlines and the rigid–plastic boundary at the entrance of the roll gap with the experimental results of Piispanen et al. [12]. The friction factor is assumed to be one during the theoretical calculations because the experiment was hot rolling. The rolling conditions are R/ti = 10.3 and r= 36%. From Fig. 10, it was found that the plastic flow along the roll surface at the roll gap was faster than that at the central plane of the plate. The meshes near the surface of the roll are distorted more profoundly than those near the central symmetrical plane. The theoretical rigid–plastic boundary at the entrance of the roll gap was clearly close to that of the experimental observation. Fig. 11 shows comparisons of the actual width spread of the plate at the roll gap with that of prediction. The initial thickness and the initial width of the plate were ti = 8.08 mm and bi = 8.14 mm, respectively. The reductions in Figs. 11(a) and (b) were r= 25.5% and r=36.1%, respectively. Dashed lines represent the theoretical profiles of the width spread of the plate. Clearly, the shape of the assumed quadratic function for the width spread profile matches well with that of the actual plastic deformation.
5. Conclusions
Fig. 11. Comparison of theoretical width spread at the roll gap with experimental results for : (a) r= 25.5%; (b) r= 36.1%.
The effects of various rolling conditions such as thickness reduction, friction factor, width-to-thickness ratio and roll radius-to-thickness upon the position of the neutral point and the relative plastic region have been discussed systematically. The analytical results are summarized in Table 1, where ‘P’ and ‘o’ indicate that the analytical results increase and decrease with the corresponding rolling condition, respectively. From Table 1, the following can be concluded. (1) The neutral point Ln /L increases with increasing width-to-thickness ratio bi /ti, friction factor m and roll radius-to-thickness ratio R/ti (R is fixed), whereas it decreases with increasing reduction r. (2) The relative length of the plastic region Li /L increases with increasing reduction r and roll radius-tothickness ratio R/ti, whereas it decreases with increasing friction factor m. The analytical results for the flow pattern at the roll gap and the profile of the width spread agree quantitatively with experimental results.
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Table 1 Summary of the analytical results of plate rolling Rolling conditions
Analysis results
Reduction r (%) Friction factor m Width-to-thickness ratio bi /ti Roll radius-to-thickness ratio R/ti (R is fixed)
Position of neutral point Ln /L
Relative length of plastic region Li /L
o P P P
P o P (r\30%), o (rB30%) P
6. Nomenclature
bi, bf bim, bfm J Li Ln R r r0 ti, tf tim, tfm U Ui, Uf Vr, Vu, Vz W: i, W: s, W: f
Subscripts i f Greek letters r0 s
Acknowledgements
width of the plate at the entrance and exit of the roll gap half-width of the plate at the entrance and exit of the roll gap total power relative length of the plastic region relative length of the neutral point roll radius boundary function along the roll surface boundary function for the central plane thickness of the plate at the entrance and exit of the roll gap half-thickness of the plate at the entrance and exit of the roll gap peripheral speed of the roll velocities of the plate at the entrance and exit of the roll gap velocities in the radial, circumferential and width directions strain-energy-rate dissipation due to internal deformation, internal shear loss at the rigid–plastic boundary and friction loss at the surface of the roll before rolling (or at the entrance of the roll gap) after rolling (or at the exit of the roll gap) distance from O to C flow stress
The authors would like to extend their thanks to the National Science Council of the Republic of China under grant no. NSC 87-2212-E-110-009. The advice and financial support of NSC are gratefully acknowledged.
References [1] S. Yanagimoto, An experimental study of width spread in cold rolling, J. Jpn. Soc. Technol. Plast. 2 (1961) 129 –134. (In Japanese). [2] J. Kihara, K. Ishiguro, A mathematical model on the widthspread behavior in the flat rolling of steel based on an experimental investigation, J. Inst. Iron Steel Jpn. 69 (1983) 2016 – 2023. (In Japanese). [3] R. Hill, A general method of analysis for metal-working process, J. Mech. Phys. Solids 11 (1963) 305 – 326. [4] S.I. Oh, S. Kobayashi, An approximate method for a three-dimensional analysis of rolling, Int. J. Mech. Sci. 17 (1975) 293– 305. [5] K. Mori, K. Osakada, Simulation of three-dimensional deformation in rolling by the finite element method, Int. J. Mech. Sci. 26 (1984) 515 – 525. [6] M.J.M. Barata Marques, P.A.F. Martins, The use of dualstream functions in the analysis of three-dimensional metal forming process, Int. J. Mech. Sci. 33 (1991) 313 – 323. [7] W.A. Backofen, Deformation processing, Metall. Trans. 4 (1973) 2679 – 2699. [8] K. Kato, T. Murota, Y. Miura, Morie method using layered plasticine and its application to measurement of three-dimensional strains in flat rolling, J. Jpn. Soc. Technol. Plast. 24 (1983) 471 – 479. (In Japanese). [9] Y.M. Hwang, H.H. Hsu, Analysis of plate rolling using the dual-stream function method and cylindrical coordinates, Int. J. Mech. Sci. 40 (1998) 371 – 385. [10] V. Nagpal, On the solution of three-dimensional metal-forming process, J. Eng. Ind. Trans. ASME 99 (1977) 624– 629. [11] S.S. Rao, Optimization: Theory and Application, 2nd ed., Wiley, New Delhi, India, 1984, pp. 292 – 300. [12] R. Piispanen, R. Eriksson, O. Piispanen, Plastic processes during rolling, II, Ba¨nder Bleche Ro¨hre 8 (1967) 819.
.
An investigation into the plastic deformation behavior at the roll gap during plate rolling Y.M. Hwang *, H.H. Hsu Department of Mechanical Engineering, National Sun Yat-Sen Uni6ersity, Kaohsiung, Taiwan 804, ROC Received 22 October 1997
Abstract A mathematical model using the dual-stream function method under cylindrical coordinates has been proposed to investigate the position of neutral point and the plastic deformation region at the roll gap during plate rolling. The profile of width spread is assumed to be a quadratic function of the cylindrical angle. The rigid – plastic boundary at the entrance of the roll gap is considered to be inclined to the rolling direction. The velocity fields derived from the proposed dual-stream function can automatically satisfy the volume constancy and the velocity boundary conditions within the roll gap. Experiments on plate rolling using aluminum sheets meshed by electrochemical etching were carried out. The predicted profiles of width spread and the plastic flow pattern at the roll gap were compared with experimental measurements. From the comparisons between analytical and experimental results, it is clear that this model can offer useful knowledge in designing the pass-schedule of a plate rolling process. © 1999 Elsevier Science S.A. All rights reserved. Keywords: Plate rolling; Dual-stream function method; Neutral point
1. Introduction The rolling process is an efficient and economical approach for the manufacturing of strip or plate metals. When the aspect ratio of the plate (the ratio of its width to thickness) is relatively small compared with sheets or strips, the width spread at the roll gap can not be neglected, i.e. the plastic deformation of plate rolling at the roll gap is that of three-dimensional deformation. Several experimental works [1,2] have been published on this topic. For example, according to a series of experimental data using copper, aluminum and steel as the specimens during cold rolling, Yanagimoto [1] developed an empirical equation for the profile of the width spread. Among the analytical researches in plate rolling, a general approximate formulation based on the virtual work-rate principle was presented by Hill [3] for analyzing the plastic deformation during forging and bar drawing. * Corresponding author. Fax: +886-7-5254299.
A kinematically admissible velocity field based on the concept suggested by Hill [3] was proposed by Oh and Kobayashi [4] to simulate the width spread of the plate at the roll gap. Mori and Osakada [5] simulated the three-dimensional deformation of the plate during plate rolling using the rigid–plastic finite element analysis. Marques and Martins [6] adopted the dual-stream function concept and discretized the plastic region into several tetrahedral elements to discuss the width spread and rolling force in plate rolling. In all of the above models, the rigid–plastic boundary at the entrance of the roll gap was assumed to be a plane and normal to the rolling direction. However, it is indicated in the literature [7,8] that the plastic deformation at the roll gap is usually inhomogeneous. Therefore, the rigid–plastic boundary at the entrance of the roll gap is inclined toward the rolling direction. Moreover, the position of the neutral point, which affects the forward slip and accordingly the design of the pass schedule of a tandem mill rolling, has not been discussed systematically.
0924-0136/99/$ - see front matter © 1999 Elsevier Science S.A. All rights reserved. PII: S 0 9 2 4 - 0 1 3 6 ( 9 8 ) 0 0 3 9 0 - 2
98
Y.M. Hwang, H.H. Hsu / Journal of Materials Processing Technology 88 (1999) 97–104
The present authors have proposed a mathematical model using the dual-stream function method to simulate the three-dimensional deformation of the plate at the roll gap [9]. In this paper, the effects of various rolling parameters such as friction factor, aspect ratio and roll radius-to-width ratio upon the position of neutral point and on the size of the plastic deformation region at the roll gap are examined systematically. Furthermore, experiments on plate rolling employing aluminum plates are carried out. Comparisons of streamlines and the rigid – plastic boundary at the entrance of the roll gap between analytical and experimental results have also been made to verify the validity of this model.
2. Mathematical model Throughout the analysis, the following assumptions are employed: (i) the flattening and bending of the rolls are neglected; (ii) the plate is of rigid – plastic material; (iii) the friction between the plate and roll is considered to have a constant friction factor; (iv) the material is isotropic; and (v) bulge deformation on the side surface of the plate is not taken into account. Fig. 1 presents a schematic illustration of a general symmetric plate rolling using cylindrical coordinates. In a three-dimensional analysis, velocity fields can be derived from dual-stream functions which represent two flow patterns within two orthogonal planes [10]. The flow pattern at the roll gap within the r – u plane is assumed to have a uniform circumferential velocity
distribution along any radial cross section and to be represented by a stream function denoted by f as follows: r−r0(u) f(r, u)= Q r1(u)− r0(u)
(1)
r1(u)= ecosu − (ecosu)2 − (e 2 − R 2)
(2)
r0(u)=
r0 cos u
(3)
e= R+tfm + r0
(4)
where r1 and r0 are the boundary functions along the upper roll surface and the central plane of the plate, respectively; Q denotes the volume flow rate at any cross-section; R is the roll radius; tfm is the half thickness of the plate at the exit of the roll gap; and r0 is the distance from the origin O to the central plane at the exit C. The flow pattern in the u–z plane is also assumed to have a uniform circumferential velocity distribution in the width direction and to be represented by c as follows: c(u, z)=
z b(u)
a= arctan
L− Li tim
u a
2
(6) (7)
where b(u), denoting the width profile function of the plate at the roll gap, is assumed to be a quadratic function of u. The radial, circumferential and width velocities, Vr, Vu and Vz, can be derived directly from the cross-product of the gradients of f and c as:
Vr (r, u)=Q − = Vu (u)= Q
1 (f (c r (u (z
Q (r− r0)(r1,u − r0,u ) r0,u + (r1 − r0) r(r1 − r0)b
(f (c Q = (r (z (r1 − r0)b
Vz (r, u, z)= Q −
Fig. 1. Schematic illustration of the mathematical model of plate rolling.
b(u)= bfm + (bim − bfm )
(5)
1 (f (c Qzb,u = r (r (u r(r1 − r0)b 2
(8) (9) (10)
where ( ),u denotes the partial derivative of ( ) with respect to u. The velocity boundary conditions at the exit of the roll gap, where the velocities in the radial direction (r) and the width direction (z) should be zero and the circumferential velocities should be uniform, can be checked by substituting u= 0 into Eqs. (8)–(10).
Y.M. Hwang, H.H. Hsu / Journal of Materials Processing Technology 88 (1999) 97–104
99
Fig. 2. Optimized velocity field in plate rolling.
The boundary conditions along the upper roll surface and the central plane can be checked by substituting r =r1 and r= r0 into Eqs. (8) and (9). More details of the derivation are given in Ref. [9]. The velocity field derived from Eqs. (1) and (5) can also be checked to ensure that it satisfies the velocity boundary conditions along the free surface, the central plane, and the rigid – plastic boundary plane at the entrance of the roll gap [9]. According to the velocity – strain rate relations, an arbitrary dual-stream function is obtained that can automatically satisfy the condition of incompressibility. Thus, the velocity field expressed in Eqs. (8)–(10) is a kinematically admissible velocity field.
The constants Q, bfm and Li are left as the pseudoindependent parameters used to minimize the total power of plastic deformation. The equivalent strain rate in the plastic region can be derived from the normal and shear strain rates as: o; eq =
'
2 2 (o; + o; 2uu + o; 2zz + 2o; 2ru + 2o; 2uz + 2o; 2zr) 3 rr
(11)
The internal strain energy rate of deformation dissipated in the plastic region, W: i, can be calculated from the equivalent strain rate and the equivalent flow stress of the deformed material, s, as:
100
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Fig. 3. Effects of friction factor on the position of the neutral point at various reductions.
Fig. 5. Effects of width-to-thickness ratio on the position of the neutral point at various reductions.
&
W: i = so; eq dV
(12)
The shear loss, W: s, dissipated at the rigid–plastic boundaries, and the friction loss, W: f, dissipated at the roll surfaces, are calculated from the velocity fields in the plastic region, respectively, as: W: s = W: s =
1
3 m
3
& &
s(D6)Gs ds
(13)
s(D6)Gf ds
(14)
Gs
Gf
where (Dn)Gs, (Dn)Gf are the velocity differences of the components tangent to the rigid–plastic boundary and the roll surface. The total power of deformation, J, is obtained as follows: J= W: i + W: s + W: f
(15)
The minimization of J, which is expressed by the function of Q, bfm and Li, is performed by the flexible polyhedron search method [11].
3. Numerical calculations and discussions Fig. 4. Effects of friction factor on the relative length of the plastic region at various reductions.
An optimized velocity filed in plate rolling is shown in Fig. 2. The rolling conditions are R= 100 mm,
Y.M. Hwang, H.H. Hsu / Journal of Materials Processing Technology 88 (1999) 97–104
ti = 10 mm, tf = 8 mm, bi =20 mm, m =0.7 and U= 1 mm s − 1. Clearly, the velocity field shown in Fig. 2(b) satisfies the boundary conditions along the free surface and has a uniform circumferential velocity at any width cross-section. The velocity field shown in Fig. 2(c) satisfies the boundary conditions along the roll surface and has a uniform circumferential velocity at any radial cross section, as expected. In other words, the velocity field satisfies all of the velocity boundary conditions. Fig. 3 shows the effects of friction factor, m, on the position of the neutral point, Ln /L, at various reductions. Because larger friction factors will increase the speed of the plate at the exit of the roll gap, the forward slip will increase accordingly and the neutral points will become closer to the entrance of the roll gap. On the other hand, the neutral points are closer to the exit of the roll gap as the thickness reduction increases. The results show the same tendency as that found in other work [4]. Fig. 4 shows the effects of friction factor, m, on the relative length of the plastic region, Li /L, at various reductions. The relative length of the plastic region will influence the streamlines of the plastic flow and the homogeneity of plastic deformation. Because larger friction factors enhance the inhomogeneous velocity distributions in the plastic region, the rigid–
Fig. 6. Effects of width-to-thickness ratio on the relative length of the plastic region at various reductions.
101
Fig. 7. Effects of roll radius-to-thickness ratio on the position of the neutral point at various reductions.
plastic boundary at the entrance of the roll gap becomes more inclined toward the rolling direction. However, the plastic deformation is more homogeneous for larger thickness reductions, thus the rigid– plastic boundary is less inclined. Fig. 5 shows the effects of the width-to-thickness ratio, bi /ti, on the position of neutral point, Ln /L, at various reductions. It is obvious that the position of the neutral point approaches the entrance of the roll gap as bi /ti increases. That is because when bi /ti increases (i.e. closer to a plane-strain state) the speed of the plate at the exit of the roll gap will increase due to the constraint of the plastic flow in the width direction, and thus the neutral point becomes closer to the entrance of the roll gap. Fig. 6 shows the effects of the width-to-thickness ratio, bi /ti, on the relative length of the plastic region, Li /L, at various reductions. Clearly, for larger reductions, the relative length increases with the increase of the width-to-thickness ratio, whereas at lower reductions, the tendency is reversed. On the other hand, the effects of the reduction upon the relative length are more profound at greater width-to-thickness ratios. The opposite tendencies at larger and smaller bi /ti are probably caused by the contact length-towidth ratio, L/bi. In other words, when bi /ti is smaller and L/bi is larger, the plastic flow toward the width
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direction is profound, and accordingly, the inclined angle a (or the relative length of the plastic region) is not significantly affected. Fig. 7 shows the effects of the roll radius-to-thickness ratio, R/ti, on the position of the neutral point, Ln /L, at various reductions. Plate rolling can be approximately regarded as simple compression plus drawing. When R/ti increases, the arc die becomes flatter and the neutral point is closer to the center of the roll surface. Fig. 8 shows the effects of the roll radius-to-thickness ratio, R/ti, on the relative length of the plastic region at various reductions. Obviously, the relative length of the plastic region increases with increasing R/ti, which means that the plastic deformation at larger R/ti is closer to homogeneous deformation
4. Experiments on plate rolling Experiments on plate rolling using a two-high test rolling mill have been performed. The radius of both rolls was 100 mm, and the peripheral speed of both rolls was 16.67 mm s − 1. Specimens employed in the experiments were A6061T6 aluminum. The top and bottom surfaces were first cleaned using a cloth with acetone and, then slightly polished with sand-paper to ensure dry friction during the plate rolling experiments. Before rolling the side surface of the plate was electro-
Fig. 9. Comparison of predicted streamlines with experimental results.
Fig. 10. Comparison of predicted streamlines with experimental results.
Fig. 8. Effects of roll radius-to-thickness ratio on the relative length of the plastic region.
chemically etched to form a 2× 2 mm mesh. A tensile test of the specimens was carried out at an average strain rate of 0.017 s − 1. The strength coefficient K and the strain-hardening exponent n obtained for the 10.41 mm-thick plates were 414.7 MPa and 0.083, respectively, and for the 8.08 mm-thick plates K=516.2 MPa and n= 0.078, respectively. Fig. 9 presents a comparison of the predicted stream-
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lines with experimental results. A friction factor of m =0.7 is assumed during the theoretical calculations, which is generally close to the condition for dry friction. From Fig. 9, it was found that the plastic flow along the
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roll surface at the roll gap was faster than that at the central plane of the plate. The distortion of the meshes near to the surface was found to be more profound than that in the central plane. The actual rigid–plastic boundary at the entrance of the roll gap was approximately a convex surface, the predicted rigid–plastic boundary was close to the experimentally-observed boundary. Fig. 10 also shows comparisons of the predicted streamlines and the rigid–plastic boundary at the entrance of the roll gap with the experimental results of Piispanen et al. [12]. The friction factor is assumed to be one during the theoretical calculations because the experiment was hot rolling. The rolling conditions are R/ti = 10.3 and r= 36%. From Fig. 10, it was found that the plastic flow along the roll surface at the roll gap was faster than that at the central plane of the plate. The meshes near the surface of the roll are distorted more profoundly than those near the central symmetrical plane. The theoretical rigid–plastic boundary at the entrance of the roll gap was clearly close to that of the experimental observation. Fig. 11 shows comparisons of the actual width spread of the plate at the roll gap with that of prediction. The initial thickness and the initial width of the plate were ti = 8.08 mm and bi = 8.14 mm, respectively. The reductions in Figs. 11(a) and (b) were r= 25.5% and r=36.1%, respectively. Dashed lines represent the theoretical profiles of the width spread of the plate. Clearly, the shape of the assumed quadratic function for the width spread profile matches well with that of the actual plastic deformation.
5. Conclusions
Fig. 11. Comparison of theoretical width spread at the roll gap with experimental results for : (a) r= 25.5%; (b) r= 36.1%.
The effects of various rolling conditions such as thickness reduction, friction factor, width-to-thickness ratio and roll radius-to-thickness upon the position of the neutral point and the relative plastic region have been discussed systematically. The analytical results are summarized in Table 1, where ‘P’ and ‘o’ indicate that the analytical results increase and decrease with the corresponding rolling condition, respectively. From Table 1, the following can be concluded. (1) The neutral point Ln /L increases with increasing width-to-thickness ratio bi /ti, friction factor m and roll radius-to-thickness ratio R/ti (R is fixed), whereas it decreases with increasing reduction r. (2) The relative length of the plastic region Li /L increases with increasing reduction r and roll radius-tothickness ratio R/ti, whereas it decreases with increasing friction factor m. The analytical results for the flow pattern at the roll gap and the profile of the width spread agree quantitatively with experimental results.
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Table 1 Summary of the analytical results of plate rolling Rolling conditions
Analysis results
Reduction r (%) Friction factor m Width-to-thickness ratio bi /ti Roll radius-to-thickness ratio R/ti (R is fixed)
Position of neutral point Ln /L
Relative length of plastic region Li /L
o P P P
P o P (r\30%), o (rB30%) P
6. Nomenclature
bi, bf bim, bfm J Li Ln R r r0 ti, tf tim, tfm U Ui, Uf Vr, Vu, Vz W: i, W: s, W: f
Subscripts i f Greek letters r0 s
Acknowledgements
width of the plate at the entrance and exit of the roll gap half-width of the plate at the entrance and exit of the roll gap total power relative length of the plastic region relative length of the neutral point roll radius boundary function along the roll surface boundary function for the central plane thickness of the plate at the entrance and exit of the roll gap half-thickness of the plate at the entrance and exit of the roll gap peripheral speed of the roll velocities of the plate at the entrance and exit of the roll gap velocities in the radial, circumferential and width directions strain-energy-rate dissipation due to internal deformation, internal shear loss at the rigid–plastic boundary and friction loss at the surface of the roll before rolling (or at the entrance of the roll gap) after rolling (or at the exit of the roll gap) distance from O to C flow stress
The authors would like to extend their thanks to the National Science Council of the Republic of China under grant no. NSC 87-2212-E-110-009. The advice and financial support of NSC are gratefully acknowledged.
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