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A new method for predicting the three-dimensional surface texture transfer in the skin pass rolling of metal strips
Wear 426–427 (2019) 1246–1264
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A new method for predicting the three-dimensional surface texture transfer in the skin pass rolling of metal strips
T
Chuhan Wua, Liangchi Zhanga, , Peilei Qub, Shanqing Lib, Zhenglian Jiangb ⁎
a b
School of Mechanical and Manufacturing Engineering, The University of New South Wales, NSW 2052, Australia Baoshan Iron & Steel Co., Ltd., Shanghai 200941, China
ARTICLE INFO
ABSTRACT
Keywords: Random asperity contact Texture transfer Skin pass Finite element analysis
Revealing the mechanisms of texture transfer in the skin pass of metal rolling is challenging due to the complex contact of randomly distributed surface asperities in both the roll and strip surfaces. This paper presents a novel three-dimensional characterisation method to overcome such difficulties. In this method, the plastic deformation of the random asperities was treated by the material redistribution at asperity tips. A time increment scheme was used to draw the rough surfaces of work roll and strip into the rolling bite. Within each time increment, an iterative technique was developed to predict the surface topography of the strip with the integration of the randomly-rough surface of the work roll. A systematic comparison with the finite element analysis (FEA) confirmed that the method developed can reliably predict the surface texture transfer. In addition, it was found that the surface topography of a metal strip can be significantly affected by the texture transfer from the work roll, that a larger reduction ratio results in a higher texture transfer ratio, and that the work roll surface topography is critical to the formation of desired strip surfaces.
1. Introduction Skin pass rolling plays a critical role in determining the surface topography of rolled metal strips. Compared with the cold rolling involving a larger reduction ratio, rolling in a skin pass usually results in a smaller reduction in strip thickness. Revealing the mechanisms of texture transfer process in skin pass rolling has encountered significant challenges because of the complex contact of randomly distributed surface asperities in both the work roll and strip surfaces [1,2]. For instance, the plastic deformation of the microscale surface asperities of a strip at the roll-strip interface is usually significant. However, the surface asperities of a work roll are usually under a much smaller elastic deformation because a work roll surface is much harder than a metal strip [3]. As such, the microscale asperities of the work roll can indent into the strip surface, resulting in a texture transfer from the work roll surface to that of the strip surface. In addition, relative sliding at the roll-strip interface occurs when the strip is drawn into the rolling bite [4,5]. The sliding from the entry of the rolling bite to its exit can bring about a continual topography evolution of the strip surface as well, which further complicates the texture transfer process. To obtain a desired surface topography of a rolled strip, it is necessary to reach an in-depth understanding of the microscale asperity deformation of a strip when it is travelling through the rolling bite in its skin pass. ⁎
In the past decades, experimental studies and theoretical investigations have been conducted to try to reveal the mechanisms of the texture transfer process. Experimental measurement of the surface topography of rolled strips can provide certain information about the resultant surface topography of a strip corresponding to a given surface texture of a work roll. Yet, such experiment is the easiest to perform. Thus, extensive experimental studies have been carried to investigate the effects of rolling conditions on the strip surface finish [1–3,6–10]. However, this approach cannot bring about the understanding of the texture transfer mechanisms in the rolling bite. On the other hand, numerical simulations, e.g., those using the finite element (FE) method, have also been carried out to try to uncover the microscale deformation process of asperities which cannot be realised by experimental observations. To do this, however, a very fine FE mesh is required to describe the microscale surface texture of a work roll, which often results in significant computational problems. To overcome such difficulties, the FE simulations on the texture transfer analysis to date have been using the plane-strain deformation assumption [11–16]. To further reduce the computational cost, roll surface asperities are simplified by circle segments, although circle segments cannot describe correctly the random shapes of surface asperities. The microscale surface features of a work roll are three-dimensional
Corresponding author. E-mail address: [email protected] (L. Zhang).
https://doi.org/10.1016/j.wear.2018.12.020 Received 3 September 2018; Received in revised form 8 December 2018; Accepted 11 December 2018 0043-1648/ © 2018 Elsevier B.V. All rights reserved.
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[10,13,17,18], generated by a surfacing process, e.g., by the pulsed laser dispersing method [18]. Some FE investigations have attempted to simulate the three-dimensional texture transfer in skin pass rolling [19]. Again, the geometry of a surface asperity was assumed to be simply-shaped, evenly-distributed grains [19] because of the limitation on the computational capability. Although statistical methods have been used to investigate the rough surface contact in cold rolling [9,20–22], the application of the statistical approach to investigating the texture transfer in skin pass rolling is a challenge. This is because the surface topography parameters of both the roll and strip surfaces are the input quantities for the statistical characterisation; but these quantities/parameters are changing from the inlet to the outlet of the rolling bite due to the surface asperity deformation. This paper develops a new method for predicting the three-dimensional surface texture transfer in the skin pass rolling of metal strips. The FE analysis will be used to obtain the overall distributions of the contact pressure and strip surface speed in the rolling bite. For an arbitrary contact point at the roll-strip interface, microscale randomlyrough surfaces of the strip and work roll will be used to model the contact. Incorporating with the instant contact pressure and relative sliding at a contact point, the plastic deformation at the random asperity tips will be treated by the material redistribution method. A penalty method will be used to predict the contact and contact pressure. An iterative method will be used in conjunction with the penalty approach to reflect the instant surface topography change in the rolling bite induced by the instant asperity deformation. To model the overall texture transfer process throughout the rolling bite, a time increment approach will be adopted to draw the control surfaces of the strip and the work roll to the rolling bite.
on these asperities, the surface is discretised by uniform cubic bars as shown in Fig. 1(b). For an arbitrary cubic bar, the contact pressure pi on its surface can be calculated by using a penalty method, i.e.,
pi =
0 K (h i , s H
hi , r
(h i , s h i , r d ) 0 d) 0 < (hi, s hi, r d ) (h i , s h i , r d ) > h
h (1)
where i represents the ith surface height, K is the penalty stiffness; hr and hs are the surface heights measured from their reference planes, respectively; H is the strip surface hardness; and ∆h is the critical interference defined by
h=
H K
(2)
The surface hardness H can be related to the yielding strength S of the strip material by [21] (3)
H = 2.8S
In practice, the yielding strength of metal sheet is dependent on the strain where S is a function of strain ε [23,24]. The calculations considering the variations of flow stress with strain are presented in Appendix A. Since the main purpose of this manuscript is to demonstrate the capability of the established approach in predicting the surface texture transfer in skin pass rolling, an elastic-perfectly plastic material of metal sheet, i.e., a constant yielding strength S, was used to simplify the analysis. The real contact area A can be calculated by [22]
A= A
f (h
d ) dh
(4)
where ∆A is the uniform surface area associated with those cubic bars. Function f(h-d) is given by
2. Modelling The rough surface contact at the roll-strip interface in a skin pass rolling is demonstrated by Fig. 1(a). Since the hardness of the roll material is much greater than that of the strip, it is reasonable to assume that the roll surface is rigid. As shown in Fig. 1(b), the rough surfaces will contact intimately when the distance d between the reference planes of the roll and strip surfaces decreases. Therefore, some asperities in the roll surface will indent into the strip surface, causing the deformation of the strip asperities. In rolling, relative sliding also occurs at the roll-strip interface, as shown in Figs. 2(a) to 2(b). These require that both the indentation and sliding should be coupled in the analysis to reflect the instant variation of the strip surface topography, and in turn the instant change of the contact stresses throughout the rolling bite. Because of the surface asperities, the rolling force is carried by the asperities in contact in the rolling bite. To calculate the contact pressure
f h
d =
0 h 1 h
d 0 d>0
(5)
where h = hs – hr. Therefore, the nominal contact pressure P at the rough surface contact can be determined by
P=
A
p (h
d ) f (h A
d ) dh
(6)
where A′ is the nominal contact area associated with the control surface. As the rigid rough surface of the work roll indents into the strip surface, the material on the contact zone flows to the surrounding noncontacting zone [22]. This inevitably results in a material redistribution on the surface heights and brings about a surface topography variation.
Fig. 1. (a) Rough surface contact, (b) asperity deformation. 1247
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Fig. 2. Sliding-induced asperity deformation (a) 2D (b) 3D.
To take this into the account, it assumes that the non-contact zone has a uniform rise [22], and that the plastic flow initiates when the contact pressure at the surface height reaches hardness H. Hence, the uniform rise ∆u at the non-contact zone can be determined by
u=
A A
f1 (h
(1
and Vr, respectively. To complete the analysis, the nominal contact pressure Pt and strip surface speed Vs at point m were determined by using the distributions of pressure and speed obtained above, respectively. A time t, the surface height on the roll surface is hr,t. Thus its location in the rolling bite xr,t can be determined as
d) dh
f (h
d)) dh
d
h h
(7)
f1 h
d =
0 h
h
d h d> h
(9)
x r , t = x r ,0 + Vr t
where
The distribution of the roll surface heights in the rolling bite is governed by Eq. (9). With hr,t, hs,t-∆t and Pt, an iterative calculation of the material redistribution at the strip surface heights could be carried. For a given reference distance d, the updated height hsk, t at the kth iteration can be given by
(8)
It should be noted that some surface asperities in the initially noncontact zone will come into contact due to the deformation-caused uniform rise and bring about new contact areas. Therefore, an iterative process is required to complete the material redistribution. Relative sliding occurs at the roll-strip interface, which changes the strip surface topography. Thus, to reliably predict the texture transfer process, both the asperity indentation and relative sliding need to be integrated. This will be outlined in the following sections.
hsk, t
1
hsk, t = hsk, t
1
t
hk
+ u
t
hsk, t 1 t
(hsk, t 1 t
d
h)
1
d
0
0 < hk
1
d
hk 1
d> h
h (10)
This process was repeated until the new uniform rise ∆u became zero, ensuring that the surface height hsk, t 1 t + u was no longer in contact with the rigid roll surface. With the converged surface height hsk, t , the nominal contact pressure P was determined by Eq. (6). Then, the reference distance d was adjusted according to the difference between P and Pt, i.e.,
3. Implementation As discussed above, the overall distributions of the nominal contact pressure and strip surface speed are important to the texture transfer analysis. A commercially available finite element code, ABAQUS EXPLICIT, was used to obtain the distributions along the rolling bite. In this part of the analysis, however, the effect of surface roughness is negligible. For a point m at the entry of the rolling bite, as shown in Fig. 3, the control surface of the strip was defined at the point when it was drawn into the rolling bite. The rough surface heights were then superimposed to the rigid roll. The roll surface speed, Vr, along the rolling direction was constant because the roll was considered to be rigid. To draw the point m and roll surface to the rolling bite, a time increment method was used. Within each time increment, the control surface at point m and the rigid roll surface move to the rolling bite with surface speeds Vs
dn = d n
1
+
(P n
1
(11)
Pt )
where ω is a relaxation factor. With the updated reference dn, the material redistribution process of surface height hs, t was conducted again by using Eq. (10). This process was repeated until Pt converges to P, i.e.,
=
Pn
Pt Pt
1.0 × 10
4
(12)
Then, the updated surface height hs, t at the time t was used as the input parameter for the texture transfer analysis at t + ∆t. Such incremental process was repeated until point m had completely gone through the rolling bite. With the obtained heights hs, t on the control surface, its distribution and surface roughness were then predicted. The overall calculation process is outlined in Fig. 4. 4. Results and discussion The above model has been well verified by a systematic finite element analysis as detailed in Appendix A at the end of the paper. Base on the discussion and comparison in Appendix A, it can be seen that the developed approach can successfully predict the surface texture transfer in skin pass rolling. However, it is worthwhile mentioning that the FEA is very time consuming because a very fine mesh is usually required to
Fig. 3. Control surface in the analysis model. 1248
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where λx and λy are the wave length along the x (rolling direction) and y directions, respectively. The dimension of the control surface on the metal strip is 24λx × 5λy. To investigate the effect of wave length on the surface topography in the rolled strip, four wave lengths were used in the analysis. To facilitate the comparison, the strip surface before rolling was smooth. Fig. 5 shows the rolled strip surface predicted by the established approach with λx = λy = 20 µm. Fig. 5(a) demonstrates the constructed sinusoidal surface on the rigid work roll with the highlighted wave length. Fig. 5(b) illustrates the overall strip surface topography after rolling. The total number of peaks along x and y directions are 24 and 5, respectively, bounded by the dimension of control surface selected above. Fig. 5(c) gives some details on the distribution of surface heights within a smaller dimension. To compare the rolled surface topography, 2D surface profiles along the cross-section n1, as shown in Fig. 5(a), n2 and n3, as shown in Fig. 5(c), are plotted in Fig. 5(d). By comparing the 2D profiles on the metal strip and that on the rigid roll, it can be seen that only part of the roll surface texture has been transferred to the metal strip. The height of the non-contact region rises due to the material redistribution aforementioned. As the wave length is reduced to 10 µm and 5 µm, the non-contact area decreases (Figs. 6 and 7). The surface texture transfer becomes more complete. When the wave length is further reduced to 2 µm, the non-contact area vanishes, as shown in Fig. 8 that the rolled strip surface mimic the roll surface texture very well. Because of the relative sliding at the roll-strip interface, however, the surface height rise along the cross-section plane n3 is no longer uniform. The above analysis shows that the wave length of the roll surface texture has an obvious effect on the texture transfer process to the strip surface. Under the same rolling force and speed, a smaller wave length results in a more complete texture transfer. 4.2. Rough surface contact Fig. 4. Flow chart.
4.2.1. Generation of rough surface The surfaces of any roll and strip in manufacturing practice are randomly rough. Thus, a rolling contact in production is always a random contact. To investigate such a process, three-dimensional rough surfaces were generated, whose asperity heights followed a Gaussian distribution, i.e.,
reflect the microscale surface pattern. To obtain the results given in Fig. 7(a), for example, the CPU time was about 9 h on a desktop PC with Intel(R) Core(TM) i5–6500 CPU@ 3.20 GHz. Clearly, the CPU time problem will become a more significant barrier for FEA if a three-dimensional of texture transfer is to be analysed. By contrast, the CPU time for the current model was only 20 min using the same desktop PC. Thus using the new method established in this study, a three-dimensional surface topography evolution in skin pass rolling will no longer be a problem. This will be detailed in the following sections.
f (h ) =
In the calculation of Pt and Vs, the half strip thickness at the entry of the strip was taken as 0.5 mm while the radius of the roll was 225 mm. The strip material was elastic-perfectly plastic with the yielding strength of 0.5 GPa. The rolling speed was fixed at 5 m/s and the reduction ratio was 0.6%. In addition, a constant friction coefficient of 0.1 was applied to the roll-strip interface. By using the Pt and Vs obtained above, a three-dimensional texture transfer was conducted with a regular surface pattern on the rigid roll surface. Here, the pattern height ha (mm) is governed by (13)
fy =
=
1 x
(14)
(16)
Rq, r Rq, s1
(17)
where Rq,r is the roll surface roughness and Rq,s1 is the surface roughness on the strip before rolling. The calculation of surface roughness and the ratio of surface texture transfer are detailed in Appendix A. Fig. 11 demonstrates the effect of ζ on the surface topography of the
1 y
1 (h / R q ) 2 2
4.2.2. Roughness ratio effects In skin pass rolling, the roll surface roughness is usually higher than that of the strip. To understand the effect of the roughness difference on the texture transfer process, let us introduce a roughness ratio ζ defined by
where
fx =
exp
To do this, the distribution of the control points in the rough surface was first generated randomly using Eq. (16), as shown in Fig. 9(a) in which the average spacing lengths between the control points along the x and y directions are denoted by ∆x and ∆y, respectively. Those control points were then interpolated by cubic splines, as shown in Fig. 9(b), such that the height of thus generated rough surface also follows the Gaussian distribution. The rough work roll surface was also generated in the same way, as shown in Fig. 10. These surfaces were used for the texture transfer analysis detailed in the sections below.
4.1. Effect of wave length on texture transfer in rolling
ha = 0.003(sin(2 fx x ) + sin(2 f y y ))
1 Rq 2
(15) 1249
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Fig. 5. Three-dimensional surface topography with λx= λy= 20 µm.
Fig. 6. Three-dimensional surface topography with λx= λy= 10 µm.
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Fig. 7. Three-dimensional surface topography with λx= λy= 5 µm.
Fig. 8. Three-dimensional surface topography with λx= λy= 2 µm. 1251
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Fig. 9. (a) Surface heights of the control points (b) generation of rough surface by using cubic spline interpolation.
Fig. 10. (a) rough surface of work roll (b) surface height distribution.
rolled strip. The average spacing length ∆x and ∆y between control points are 20 µm. The reduction ratio and rolling speed are 2.74% and 5 m/s, respectively. The strip surface roughness before rolling, Rq,s1, is 0.78 µm. It can be seen that the strip surface has experienced a dramatic change due to the texture transfer in rolling, although its surface height distribution after rolling also approximates a Gaussian distribution.
With a larger roughness ratio ζ, the surface roughness of the rolled strip is also greater. Fig. 12 presents the strip surface topography after rolling at smaller average spacing length of ∆x = ∆y = 5 µm. Comparing the results in Figs. 11 and 12, it can be seen that a roll of smaller spacing lengths ∆x and ∆y produces smoother strips. 1252
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Fig. 11. Surface prediction with ∆x= ∆y= 20 µm (a) Rq,s= 0.815 µm, ζ = 1, (b) Rq,s= 1.64 µm, ζ = 2 (c) Rq,s= 2.4 µm, ζ = 3.
4.2.3. Reduction ratio effects Fig. 13 shows the effect of reduction ratio on the surface topography of rolled strips, in which Rq,s1 and ζ are fixed at 0.78 µm and 3, respectively. The average spacing lengths ∆x and ∆y are 5 µm and the rolling speed is 5 m/s. It is interesting to note that at a smaller reduction ratio the distribution of the surface asperity heights after rolling is not Gaussian, as shown in Figs. 13(a) to 13(b). However, as the reduction ratio increases, it turns to be Gaussian again approximately, As shown in Fig. 13(c) and Fig. 12(c). Fig. 13(d) shows that when the reduction ratio ψ is below 1%, increasing the reduction ratio will bring about a
rougher rolled strip surface. However, when the reduction ratio further increases, the ψ variation stabilizes because of the bounding by the surface topography of the rigid roll. These results demonstrate that reduction ratio can be used as an effective rolling parameter to produce desirable surface roughness in the skin pass rolling of strips. 4.2.4. Effect of rolling speed Fig. 14 shows the effect of rolling speed on the surface texture transfer, in which ∆x = ∆y = 5 µm, and Rq,s1 and ζ are 0.78 µm and 3, respectively. The range of rolling speeds used in the simulation are from 1253
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Fig. 12. Surface prediction with ∆x= ∆y= 5 µm (a) Rq,s= 0.718 µm, ζ = 1, (b) Rq,s= 0.1.42 µm, ζ = 2 (c) Rq,s= 2.12 µm, ζ = 3.
2.5 to 20 m/s. It can be seen that the rough surface profiles after rolling generally follow Gaussian distributions. In addition, the increase in the rolling speed only leads to small variations on the ratio of surface texture transfer, as show in Fig. 15. This is because the overall distribution of contact pressure is not significantly affected by the rolling speed, as evidenced by Fig. 16. When the relative sliding occurs in rolling, the rigid roll asperity indents into the strip surface and brings about the sliding process in the contact zone. Therefore, the overall sliding distance l, as shown in Fig. 16, can be used to assess the effect of rolling speed. It should be noted that l is dependent on the reduction
ratio rather the rolling speed. Thus, the variation of rolling speed has a negligible effect on the texture transfer process. 5. Conclusions This paper has developed a new method for predicting the threedimensional surface texture transfer in the skin pass rolling of metal strips. In this method, the FE analysis was used to obtain the distributions of the contact pressure and strip surface speed in the rolling bite. In addition, the plastic deformation of the random asperities was 1254
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Fig. 13. (a) Reduction ratio 0.049%, (b) reduction ratio 0.584%, (c) reduction ratio 1.14% (d) Rq,s vs reduction ratio.
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Fig. 14. (a) rolling speed 2 m/s, (b) rolling speed 5 m/s, (c) rolling speed 10 m/s, (d) rolling speed 20 m/s.
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Fig. 15. Effect of rolling speed on the surface roughness of rolled metal strips.
Fig. 16. Effect of rolling speed on the surface texture.
treated by the material redistribution at asperity tips. A time increment scheme was used to draw the rough surfaces of work roll and strip into the rolling bite. Within each time increment, an iterative method was used in conjunction with the penalty approach to reflect the instant surface topography change in the rolling bite induced by the instant asperity deformation. The established approach has overcome the difficulties due to the complex contact of randomly distributed surface asperities in both the roll and strip surfaces. A systematic comparison with the corresponding finite element analysis confirmed that the method developed is much cost-effective and can reliably predict the surface texture transfer processes in the skin pass of metal strip production. The obtained results revealed that the surface topography of a metal strip can be significantly affected by the texture transfer from the work roll and that the work roll surface topography is critical to the formation of desired strip surfaces. A higher ratio of the roll surface roughness
to that of the strip before rolling brings about a rougher strip surface after rolling. In addition, a larger reduction ratio results in a higher texture transfer ratio. As the reduction ratio increases, the texture transfer ratio will increase gradually before reaching a stabilized state bounded by the roll surface topography. Compared with the reduction ratio, the effect of rolling speed on the texture transfer is negligible. These results demonstrate that reduction ratio can be used as an effective rolling parameter to produce desirable surface roughness (and thus texture) in the skin pass rolling of metal strips. Acknowledgement This research forms part of the Baosteel Australia Research and Development Centre (BAJC) portfolio of projects and has received financial support from the Centre through Project BA15001.
Appendix A Because of the microscale surface roughness, it is challenging to experimentally investigate how the dynamic process of texture transfer occurs in the rolling bite. To investigate such processes [11,12,15], FEA has been widely used. Therefore, the feasibility of established model in predicting the texture transfer process will be verified by the relevant FEA in this study.
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A.1 Indentation simulation: FEM vs current model To verify the established model, a systematic finite element analysis (FEA) was conducted to simulate an indentation process between a rigid indenter with a regular surface pattern and a material with a smooth surface, as shown in Fig. A1 (a). The element type used for the former was R2D2, and that for the latter was CPE4R. The dimension of the deformable block was 2.4 mm× 0.5 mm. For simplicity, the FEA was under the planestrain condition. A sinusoidal function was used to construct the regular surface pattern of the rigid indenter. Let the height of the surface pattern be ha (mm), i.e.,
ha = 3 × 10
3
(A-1)
sin(2 fx )
Fig. A1. (a) Dimension of indentation simulation and boundary conditions in FEA, (b) Von-Mises stress at loading and unloading stage with an indentation depth of 4.5 µm, (c) Comparison of contact pressure.
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where
f=
1
(A-2)
where λ is the wave length, which is 0.2 mm in this verification case study. In the FEA, a y-symmetrical condition was applied to the edge BC while an x-symmetrical condition was applied to edges AB and CD. The indentation depth was ∆d. The block under indentation was an elastic-perfectly plastic material with a yielding strength of 0.5 GPa. A constant friction coefficient of 0.1 was applied to the indenter-block interface. The indentation simulation consisted of two steps, i.e., loading and unloading stages. The surface profile of the deformed block after unloading was used to compare with the prediction given by the established method. Fig. A1(b) shows the distributions of von-Mises stress at the end of the loading stage as well as after complete unloading. The maximum indentation depth used was ∆d = 4.5 µm. It can be seen that the regular surface pattern of the rigid indenter brings about a regular pattern of vonMises stress within the deformed material associated with a regular surface pattern. The total indentation force F at the maximum loading was used as the input parameter to our texture transfer model described by using Eqs. (6), (7) and (10)–(12)). For comparison, an identical surface pattern of the indenter was constructed by Eq. (A-1). Then, the total force Fn on the indenter was determined by
Fn = A
p (h
d) f (h
d) dh
(A-3)
The reference distance d was adjusted until Fn converged to F. During the calculation, the penalty stiffness K was set to be 1.0 × 1018 Pa/m. Fig. A1(c) shows the comparison of the distributions of contact pressure given by the FEA and Eq. (1). A dimensionless contact pressures, P/S, was plotted in Fig. A1(c). It can be noted that the prediction given by Eq. (1) agrees well with those given by the FEA although a lower maximum contact pressure was predicted by the current model. This is because, unlike the FEA, the contact pressure in Eq. (1) is limited by the surface hardness H, as shown in Eqs. (2)–(3), and that the elastic deformation is not taken into the consideration in Eq. (1). Fig. A2(a) demonstrates that the surface profiles predicted by Eq. (10) agree with those given by the FEA with different indentation depths. However, an obvious difference between these results can be noted for a lower indentation depth. Again, this is because the FEA considered both the elastic and plastic deformation. As a result, the elastic deformation also contributes to the texture transfer process in the indentation simulation, as evidenced by Fig. A2(b). Under a smaller indentation depth, a non-uniform pile-up can be observed on the non-contact zone due to such elastic deformation, as confirmed by Fig. A2(a). As the indentation depth increases gradually, the contribution of the elastic deformation becomes negligible due to the large plastic deformation, as demonstrated in Fig. A1(b). A better agreement between the surface profiles given by Eq. (10) and that given by the FEA is therefore reached, as shown in Fig. A2(a). With the obtained surface profiles above, the ratio of texture transfer from the rigid indenter to the material can be calculated. Since those profiles are represented by digital form, the surface roughness is calculated by
Rq =
1 N
N
(hi
have )2
(A-4)
i=1
where hi is the surface height on the digital point, have is the average surface height, and N is the total number of digital points. Thus, the texture transfer ratio can be defined by
=
Rq, s (A-5)
Rq, r
where Rq,s is the surface roughness of block after unloading, Rq,r is the surface roughness of rigid indenter. Fig. A2(c) plots texture transfer ratio ψ against the indentation depth. As expected, the ψ increases when the indentation depth increases. Overall, the ψ given by the established approach agrees well with those from the FEA. A.2 Rolling simulation-FEM vs current model To verify the surface topography predicted by Eq. (10) in the skin pass of rolling, an FEA was further conducted by using a rigid roll with a constructed sinusoidal surface profile which was governed by Eq. (A-1). The half strip thickness at the entry of the strip was 0.5 mm while the radius of the roll was 225 mm. The strip material was elastic-perfectly plastic with the yielding strength of 0.5 GPa. The rolling speed was fixed at 5 m/s and a constant friction coefficient of 0.1 was applied to the roll-strip interface. To reduce the computation cost, a y-symmetrical boundary was applied to edge AB (see Fig. A3(a)). Fig. A3(a) shows the prediction by the FEA. It can be seen that some texture on the roll surface has been transferred to the strip surface after rolling, demonstrated by the regular pattern on the strip surface. Because of the roll surface pattern, the distributions of contact pressure P and shear stress T in the rolling bite are no longer smooth. The corresponding distribution of von-Mises stress demonstrates that a regular pattern of residual stresses is caused by the coining of the roll pattern. To calculate the reference reduction ∆Y, a reference plane on the rolled strip surface can be obtained by averaging the surface heights over the selected region, as shown in Fig. A3(a). With the same ∆Y, i.e., 3 µm, a skin pass rolling with a smooth work roll was also carried out, as shown in Fig. A3(b). The overall distributions of contact pressure P and strip surface speed Vs along the rolling bite shown in Fig. A4(a) were the input parameters for the current model, and the analysis of texture transfer followed the procedure in Fig. 4. Fig. A4(b) demonstrates that the strip surface profile given by Eq. (10), as shown in blue line, agrees with that given by the FEA. A3 Effects of strain-dependent yielding strength In practice, the yielding strength of metal sheet is dependent on the strain ε [23,24]. Thus, Eq. (3) can be extended to describe the variations of flow stress with strain. To do this, the yielding strength S is considered to be dependent on strain, i.e., [21] (A-6)
S=f( ) 1259
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Fig. A2. (a) Comparison of surface profile between current model and FEA, (b) Elastic deformation vs plastic deformation for a lower ∆d, (c) Comparison of texture transfer ratio.
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Fig. A3. (a) FEA of skin pass rolling with constructed sinusoidal surface, (b) FEA of skin pass rolling with a smooth work roll.
As long as strain is known, the surface hardness H will be updated accordingly by Eqs. (A-6) and (3). To investigate the effects of strain-dependent yielding strength on the texture transfer, a strain-stress curve experimentally measured by Yoshida et al. [23] as show in Fig. A5, was used as an input parameter in the FEA. The initial yielding instability caused by strain localisation, which is known as the Lüders' bands, were well reflected by such strain-stress curve [23]. At the end of the FEA, the strain distribution in the rolling bite was obtained, and was used to calculate the surface hardness H of point m by Eqs. (3) and (A6). This process was repeated until the control surface completely passed the rolling bite. Fig. A6 demonstrates the predicted surface topography with or without considering the variations of flow stress with strain. For the elasticperfectly plastic material shown in Fig. A6(a), the yielding strength is obtained from the sharp yielding point shown in Fig. A5. Rolling speed and reduction ratio were fixed at 5 m/s and 2.7%, respectively. To simplify the analysis, the strip surface before rolling was regarded to be smooth. The roughness of the work roll, whose height followed a Gaussian distribution, was 0.81 µm. A lower surface roughness was predicted when considering the variation of flow stress with strain. Fig. A7 (a) further demonstrates that the strain-dependent yielding strength has a strong influence on the distribution of surface height after skin pass rolling because the surface hardness H is dependent on the yielding strength. In addition, the initial yielding instability caused by the strain localisation leads to a lower contact pressure at the entry, as shown in Fig. A7(b). However, a higher contact pressure at the exit can be noted due to the work-hardening of sheet metal. Overall, the proposed approach can successfully deal with the variations of yielding strength with strain in analysing the surface texture transfer in skin pass rolling.
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Fig. A4. (a) Distribution of contact pressure and strip surface speed in the rolling bite (b) Comparison of surface profile predicted by the current method and that given by FEA.
Fig. A5. Strain-stress curve [23].
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Fig. A6. Predicted surface topography (a) Elastic-perfectly plastic, 0.76 µm (b) Strain-dependent yielding strength, 0.69 µm.
Fig. A7. (a) Surface height distribution, (b) Contact pressure comparison. 1263
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References
[13] H. Kijima, N. Bay, Influence of tool roughness and lubrication on contact conditions in skin-pass rolling, J. Mater. Process. Technol. 209 (2009) 4835–4841. [14] H. Kijima, Influence of roll radius on contact condition and material deformation in skin-pass rolling of steel strip, J. Mater. Process. Technol. 213 (2013) 1764–1771. [15] H. Kijima, Influence of roll radius on roughness transfer in skin-pass rolling of steel strip, J. Mater. Process. Technol. 214 (2014) 1111–1119. [16] C. Wu, L. Zhang, P. Qu, S. Li, Z. Jiang, A simple approach for analysing the surface texture transfer in cold rolling of metal strips, Int. J. Adv. Manuf. Technol. (2017) 1–12. [17] S. Sheu, J.L.G. Hector, O. Richmond, Tool surface topographies for controlling friction and wear in metal-forming processes, J. Tribol. 120 (1998) 517–527. [18] K. Hilgenberg, K. Steinhoff, Texturing of skin-pass rolls by pulsed laser dispersing, J. Mater. Process. Technol. 225 (2015) 84–92. [19] R. Bünten, K. Steinhoff, W. Rasp, R. Kopp, O. Pawelski, Development of a FEMmodel for the simulation of the transfer of surface structure in cold-rolling processes, J. Mater. Process. Technol. 60 (1996) 369–376. [20] C. Wu, L. Zhang, S. Li, Z. Jiang, P. Qu, A novel multi-scale statistical characterization of interface pressure and friction in metal strip rolling, Int. J. Mech. Sci. 89 (2014) 391–402. [21] J. Hol, V. Meinders, M. de Rooij, A. van den Boogaard, Multi-scale friction modeling for sheet metal forming: the boundary lubrication regime, Tribol. Int. 81 (2015) 112–128. [22] J. Pullen, J. Williamson, On the plastic contact of rough surfaces, Proc. R. Soc. Lond. A: Math. Phys. Eng. Sci.: R. Soc. (1972) 159–173. [23] F. Yoshida, Y. Kaneda, S. Yamamoto, A plasticity model describing yield-point phenomena of steels and its application to FE simulation of temper rolling, Int. J. Plast. 24 (2008) 1792–1818. [24] A. Giarola, P. Pereira, P. Stemler, A. Pertence, H. Campos, M. Aguilar, et al., Strain heterogeneities in the rolling direction of steel sheets submitted to the skin pass: a finite element analysis, J. Mater. Process. Technol. 216 (2015) 234–247.
[1] B. Ma, A. Tieu, C. Lu, Z. Jiang, An experimental investigation of steel surface characteristic transfer by cold rolling, J. Mater. Process. Technol. 125 (2002) 657–663. [2] M.P.F. Sutcliffe, H.R. Le, Measurements of surface roughness in cold metal rolling in the mixed lubrication regime, Tribol. Trans. 43 (2000) 39–44. [3] K. Dick, J.G. Lenard, The effect of roll roughness and lubricant viscosity on the loads on the mill during cold rolling of steel strips, J. Mater. Process. Technol. 168 (2005) 16–24. [4] J.G. Lenard, Measurements of friction in cold flat rolling, J. Mater. Shap. Technol. 9 (1991) 171–180. [5] J.G. Lenard, Friction and forward slip in cold strip rolling, Tribol. Trans. 35 (1992) 423–428. [6] H. Kijima, An experimental investigation on the influence of lubrication on roughness transfer in skin-pass rolling of steel strip, J. Mater. Process. Technol. 225 (2015) 1–8. [7] H. Kijima, Influence of lubrication on roughness crushing in skin-pass rolling of steel strip, J. Mater. Process. Technol. 216 (2015) 1–9. [8] B. Çolak, N. Kurgan, An experimental investigation into roughness transfer in skinpass rolling of steel strips, Int. J. Adv. Manuf. Technol. (2018) 1–10. [9] C. Wu, L. Zhang, S. Li, Z. Jiang, P. Qu, A unified method for characterizing multiple lubrication regimes involving plastic deformation of surface asperities, Tribol. Int. 100 (2016) 70–83. [10] F. Plouraboué, M. Boehm, Multi-scale roughness transfer in cold metal rolling, Tribol. Int. 32 (1999) 45–57. [11] H. Kijima, N. Bay, Skin-pass rolling I—Studies on roughness transfer and elongation under pure normal loading, Int. J. Mach. Tools Manuf. 48 (2008) 1313–1317. [12] H. Kijima, N. Bay, Skin-pass rolling II—Studies of roughness transfer under combined normal and tangential loading, Int. J. Mach. Tools Manuf. 48 (2008) 1308–1312.
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Wear journal homepage: www.elsevier.com/locate/wear
A new method for predicting the three-dimensional surface texture transfer in the skin pass rolling of metal strips
T
Chuhan Wua, Liangchi Zhanga, , Peilei Qub, Shanqing Lib, Zhenglian Jiangb ⁎
a b
School of Mechanical and Manufacturing Engineering, The University of New South Wales, NSW 2052, Australia Baoshan Iron & Steel Co., Ltd., Shanghai 200941, China
ARTICLE INFO
ABSTRACT
Keywords: Random asperity contact Texture transfer Skin pass Finite element analysis
Revealing the mechanisms of texture transfer in the skin pass of metal rolling is challenging due to the complex contact of randomly distributed surface asperities in both the roll and strip surfaces. This paper presents a novel three-dimensional characterisation method to overcome such difficulties. In this method, the plastic deformation of the random asperities was treated by the material redistribution at asperity tips. A time increment scheme was used to draw the rough surfaces of work roll and strip into the rolling bite. Within each time increment, an iterative technique was developed to predict the surface topography of the strip with the integration of the randomly-rough surface of the work roll. A systematic comparison with the finite element analysis (FEA) confirmed that the method developed can reliably predict the surface texture transfer. In addition, it was found that the surface topography of a metal strip can be significantly affected by the texture transfer from the work roll, that a larger reduction ratio results in a higher texture transfer ratio, and that the work roll surface topography is critical to the formation of desired strip surfaces.
1. Introduction Skin pass rolling plays a critical role in determining the surface topography of rolled metal strips. Compared with the cold rolling involving a larger reduction ratio, rolling in a skin pass usually results in a smaller reduction in strip thickness. Revealing the mechanisms of texture transfer process in skin pass rolling has encountered significant challenges because of the complex contact of randomly distributed surface asperities in both the work roll and strip surfaces [1,2]. For instance, the plastic deformation of the microscale surface asperities of a strip at the roll-strip interface is usually significant. However, the surface asperities of a work roll are usually under a much smaller elastic deformation because a work roll surface is much harder than a metal strip [3]. As such, the microscale asperities of the work roll can indent into the strip surface, resulting in a texture transfer from the work roll surface to that of the strip surface. In addition, relative sliding at the roll-strip interface occurs when the strip is drawn into the rolling bite [4,5]. The sliding from the entry of the rolling bite to its exit can bring about a continual topography evolution of the strip surface as well, which further complicates the texture transfer process. To obtain a desired surface topography of a rolled strip, it is necessary to reach an in-depth understanding of the microscale asperity deformation of a strip when it is travelling through the rolling bite in its skin pass. ⁎
In the past decades, experimental studies and theoretical investigations have been conducted to try to reveal the mechanisms of the texture transfer process. Experimental measurement of the surface topography of rolled strips can provide certain information about the resultant surface topography of a strip corresponding to a given surface texture of a work roll. Yet, such experiment is the easiest to perform. Thus, extensive experimental studies have been carried to investigate the effects of rolling conditions on the strip surface finish [1–3,6–10]. However, this approach cannot bring about the understanding of the texture transfer mechanisms in the rolling bite. On the other hand, numerical simulations, e.g., those using the finite element (FE) method, have also been carried out to try to uncover the microscale deformation process of asperities which cannot be realised by experimental observations. To do this, however, a very fine FE mesh is required to describe the microscale surface texture of a work roll, which often results in significant computational problems. To overcome such difficulties, the FE simulations on the texture transfer analysis to date have been using the plane-strain deformation assumption [11–16]. To further reduce the computational cost, roll surface asperities are simplified by circle segments, although circle segments cannot describe correctly the random shapes of surface asperities. The microscale surface features of a work roll are three-dimensional
Corresponding author. E-mail address: [email protected] (L. Zhang).
https://doi.org/10.1016/j.wear.2018.12.020 Received 3 September 2018; Received in revised form 8 December 2018; Accepted 11 December 2018 0043-1648/ © 2018 Elsevier B.V. All rights reserved.
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[10,13,17,18], generated by a surfacing process, e.g., by the pulsed laser dispersing method [18]. Some FE investigations have attempted to simulate the three-dimensional texture transfer in skin pass rolling [19]. Again, the geometry of a surface asperity was assumed to be simply-shaped, evenly-distributed grains [19] because of the limitation on the computational capability. Although statistical methods have been used to investigate the rough surface contact in cold rolling [9,20–22], the application of the statistical approach to investigating the texture transfer in skin pass rolling is a challenge. This is because the surface topography parameters of both the roll and strip surfaces are the input quantities for the statistical characterisation; but these quantities/parameters are changing from the inlet to the outlet of the rolling bite due to the surface asperity deformation. This paper develops a new method for predicting the three-dimensional surface texture transfer in the skin pass rolling of metal strips. The FE analysis will be used to obtain the overall distributions of the contact pressure and strip surface speed in the rolling bite. For an arbitrary contact point at the roll-strip interface, microscale randomlyrough surfaces of the strip and work roll will be used to model the contact. Incorporating with the instant contact pressure and relative sliding at a contact point, the plastic deformation at the random asperity tips will be treated by the material redistribution method. A penalty method will be used to predict the contact and contact pressure. An iterative method will be used in conjunction with the penalty approach to reflect the instant surface topography change in the rolling bite induced by the instant asperity deformation. To model the overall texture transfer process throughout the rolling bite, a time increment approach will be adopted to draw the control surfaces of the strip and the work roll to the rolling bite.
on these asperities, the surface is discretised by uniform cubic bars as shown in Fig. 1(b). For an arbitrary cubic bar, the contact pressure pi on its surface can be calculated by using a penalty method, i.e.,
pi =
0 K (h i , s H
hi , r
(h i , s h i , r d ) 0 d) 0 < (hi, s hi, r d ) (h i , s h i , r d ) > h
h (1)
where i represents the ith surface height, K is the penalty stiffness; hr and hs are the surface heights measured from their reference planes, respectively; H is the strip surface hardness; and ∆h is the critical interference defined by
h=
H K
(2)
The surface hardness H can be related to the yielding strength S of the strip material by [21] (3)
H = 2.8S
In practice, the yielding strength of metal sheet is dependent on the strain where S is a function of strain ε [23,24]. The calculations considering the variations of flow stress with strain are presented in Appendix A. Since the main purpose of this manuscript is to demonstrate the capability of the established approach in predicting the surface texture transfer in skin pass rolling, an elastic-perfectly plastic material of metal sheet, i.e., a constant yielding strength S, was used to simplify the analysis. The real contact area A can be calculated by [22]
A= A
f (h
d ) dh
(4)
where ∆A is the uniform surface area associated with those cubic bars. Function f(h-d) is given by
2. Modelling The rough surface contact at the roll-strip interface in a skin pass rolling is demonstrated by Fig. 1(a). Since the hardness of the roll material is much greater than that of the strip, it is reasonable to assume that the roll surface is rigid. As shown in Fig. 1(b), the rough surfaces will contact intimately when the distance d between the reference planes of the roll and strip surfaces decreases. Therefore, some asperities in the roll surface will indent into the strip surface, causing the deformation of the strip asperities. In rolling, relative sliding also occurs at the roll-strip interface, as shown in Figs. 2(a) to 2(b). These require that both the indentation and sliding should be coupled in the analysis to reflect the instant variation of the strip surface topography, and in turn the instant change of the contact stresses throughout the rolling bite. Because of the surface asperities, the rolling force is carried by the asperities in contact in the rolling bite. To calculate the contact pressure
f h
d =
0 h 1 h
d 0 d>0
(5)
where h = hs – hr. Therefore, the nominal contact pressure P at the rough surface contact can be determined by
P=
A
p (h
d ) f (h A
d ) dh
(6)
where A′ is the nominal contact area associated with the control surface. As the rigid rough surface of the work roll indents into the strip surface, the material on the contact zone flows to the surrounding noncontacting zone [22]. This inevitably results in a material redistribution on the surface heights and brings about a surface topography variation.
Fig. 1. (a) Rough surface contact, (b) asperity deformation. 1247
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Fig. 2. Sliding-induced asperity deformation (a) 2D (b) 3D.
To take this into the account, it assumes that the non-contact zone has a uniform rise [22], and that the plastic flow initiates when the contact pressure at the surface height reaches hardness H. Hence, the uniform rise ∆u at the non-contact zone can be determined by
u=
A A
f1 (h
(1
and Vr, respectively. To complete the analysis, the nominal contact pressure Pt and strip surface speed Vs at point m were determined by using the distributions of pressure and speed obtained above, respectively. A time t, the surface height on the roll surface is hr,t. Thus its location in the rolling bite xr,t can be determined as
d) dh
f (h
d)) dh
d
h h
(7)
f1 h
d =
0 h
h
d h d> h
(9)
x r , t = x r ,0 + Vr t
where
The distribution of the roll surface heights in the rolling bite is governed by Eq. (9). With hr,t, hs,t-∆t and Pt, an iterative calculation of the material redistribution at the strip surface heights could be carried. For a given reference distance d, the updated height hsk, t at the kth iteration can be given by
(8)
It should be noted that some surface asperities in the initially noncontact zone will come into contact due to the deformation-caused uniform rise and bring about new contact areas. Therefore, an iterative process is required to complete the material redistribution. Relative sliding occurs at the roll-strip interface, which changes the strip surface topography. Thus, to reliably predict the texture transfer process, both the asperity indentation and relative sliding need to be integrated. This will be outlined in the following sections.
hsk, t
1
hsk, t = hsk, t
1
t
hk
+ u
t
hsk, t 1 t
(hsk, t 1 t
d
h)
1
d
0
0 < hk
1
d
hk 1
d> h
h (10)
This process was repeated until the new uniform rise ∆u became zero, ensuring that the surface height hsk, t 1 t + u was no longer in contact with the rigid roll surface. With the converged surface height hsk, t , the nominal contact pressure P was determined by Eq. (6). Then, the reference distance d was adjusted according to the difference between P and Pt, i.e.,
3. Implementation As discussed above, the overall distributions of the nominal contact pressure and strip surface speed are important to the texture transfer analysis. A commercially available finite element code, ABAQUS EXPLICIT, was used to obtain the distributions along the rolling bite. In this part of the analysis, however, the effect of surface roughness is negligible. For a point m at the entry of the rolling bite, as shown in Fig. 3, the control surface of the strip was defined at the point when it was drawn into the rolling bite. The rough surface heights were then superimposed to the rigid roll. The roll surface speed, Vr, along the rolling direction was constant because the roll was considered to be rigid. To draw the point m and roll surface to the rolling bite, a time increment method was used. Within each time increment, the control surface at point m and the rigid roll surface move to the rolling bite with surface speeds Vs
dn = d n
1
+
(P n
1
(11)
Pt )
where ω is a relaxation factor. With the updated reference dn, the material redistribution process of surface height hs, t was conducted again by using Eq. (10). This process was repeated until Pt converges to P, i.e.,
=
Pn
Pt Pt
1.0 × 10
4
(12)
Then, the updated surface height hs, t at the time t was used as the input parameter for the texture transfer analysis at t + ∆t. Such incremental process was repeated until point m had completely gone through the rolling bite. With the obtained heights hs, t on the control surface, its distribution and surface roughness were then predicted. The overall calculation process is outlined in Fig. 4. 4. Results and discussion The above model has been well verified by a systematic finite element analysis as detailed in Appendix A at the end of the paper. Base on the discussion and comparison in Appendix A, it can be seen that the developed approach can successfully predict the surface texture transfer in skin pass rolling. However, it is worthwhile mentioning that the FEA is very time consuming because a very fine mesh is usually required to
Fig. 3. Control surface in the analysis model. 1248
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where λx and λy are the wave length along the x (rolling direction) and y directions, respectively. The dimension of the control surface on the metal strip is 24λx × 5λy. To investigate the effect of wave length on the surface topography in the rolled strip, four wave lengths were used in the analysis. To facilitate the comparison, the strip surface before rolling was smooth. Fig. 5 shows the rolled strip surface predicted by the established approach with λx = λy = 20 µm. Fig. 5(a) demonstrates the constructed sinusoidal surface on the rigid work roll with the highlighted wave length. Fig. 5(b) illustrates the overall strip surface topography after rolling. The total number of peaks along x and y directions are 24 and 5, respectively, bounded by the dimension of control surface selected above. Fig. 5(c) gives some details on the distribution of surface heights within a smaller dimension. To compare the rolled surface topography, 2D surface profiles along the cross-section n1, as shown in Fig. 5(a), n2 and n3, as shown in Fig. 5(c), are plotted in Fig. 5(d). By comparing the 2D profiles on the metal strip and that on the rigid roll, it can be seen that only part of the roll surface texture has been transferred to the metal strip. The height of the non-contact region rises due to the material redistribution aforementioned. As the wave length is reduced to 10 µm and 5 µm, the non-contact area decreases (Figs. 6 and 7). The surface texture transfer becomes more complete. When the wave length is further reduced to 2 µm, the non-contact area vanishes, as shown in Fig. 8 that the rolled strip surface mimic the roll surface texture very well. Because of the relative sliding at the roll-strip interface, however, the surface height rise along the cross-section plane n3 is no longer uniform. The above analysis shows that the wave length of the roll surface texture has an obvious effect on the texture transfer process to the strip surface. Under the same rolling force and speed, a smaller wave length results in a more complete texture transfer. 4.2. Rough surface contact Fig. 4. Flow chart.
4.2.1. Generation of rough surface The surfaces of any roll and strip in manufacturing practice are randomly rough. Thus, a rolling contact in production is always a random contact. To investigate such a process, three-dimensional rough surfaces were generated, whose asperity heights followed a Gaussian distribution, i.e.,
reflect the microscale surface pattern. To obtain the results given in Fig. 7(a), for example, the CPU time was about 9 h on a desktop PC with Intel(R) Core(TM) i5–6500 CPU@ 3.20 GHz. Clearly, the CPU time problem will become a more significant barrier for FEA if a three-dimensional of texture transfer is to be analysed. By contrast, the CPU time for the current model was only 20 min using the same desktop PC. Thus using the new method established in this study, a three-dimensional surface topography evolution in skin pass rolling will no longer be a problem. This will be detailed in the following sections.
f (h ) =
In the calculation of Pt and Vs, the half strip thickness at the entry of the strip was taken as 0.5 mm while the radius of the roll was 225 mm. The strip material was elastic-perfectly plastic with the yielding strength of 0.5 GPa. The rolling speed was fixed at 5 m/s and the reduction ratio was 0.6%. In addition, a constant friction coefficient of 0.1 was applied to the roll-strip interface. By using the Pt and Vs obtained above, a three-dimensional texture transfer was conducted with a regular surface pattern on the rigid roll surface. Here, the pattern height ha (mm) is governed by (13)
fy =
=
1 x
(14)
(16)
Rq, r Rq, s1
(17)
where Rq,r is the roll surface roughness and Rq,s1 is the surface roughness on the strip before rolling. The calculation of surface roughness and the ratio of surface texture transfer are detailed in Appendix A. Fig. 11 demonstrates the effect of ζ on the surface topography of the
1 y
1 (h / R q ) 2 2
4.2.2. Roughness ratio effects In skin pass rolling, the roll surface roughness is usually higher than that of the strip. To understand the effect of the roughness difference on the texture transfer process, let us introduce a roughness ratio ζ defined by
where
fx =
exp
To do this, the distribution of the control points in the rough surface was first generated randomly using Eq. (16), as shown in Fig. 9(a) in which the average spacing lengths between the control points along the x and y directions are denoted by ∆x and ∆y, respectively. Those control points were then interpolated by cubic splines, as shown in Fig. 9(b), such that the height of thus generated rough surface also follows the Gaussian distribution. The rough work roll surface was also generated in the same way, as shown in Fig. 10. These surfaces were used for the texture transfer analysis detailed in the sections below.
4.1. Effect of wave length on texture transfer in rolling
ha = 0.003(sin(2 fx x ) + sin(2 f y y ))
1 Rq 2
(15) 1249
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Fig. 5. Three-dimensional surface topography with λx= λy= 20 µm.
Fig. 6. Three-dimensional surface topography with λx= λy= 10 µm.
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Fig. 7. Three-dimensional surface topography with λx= λy= 5 µm.
Fig. 8. Three-dimensional surface topography with λx= λy= 2 µm. 1251
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Fig. 9. (a) Surface heights of the control points (b) generation of rough surface by using cubic spline interpolation.
Fig. 10. (a) rough surface of work roll (b) surface height distribution.
rolled strip. The average spacing length ∆x and ∆y between control points are 20 µm. The reduction ratio and rolling speed are 2.74% and 5 m/s, respectively. The strip surface roughness before rolling, Rq,s1, is 0.78 µm. It can be seen that the strip surface has experienced a dramatic change due to the texture transfer in rolling, although its surface height distribution after rolling also approximates a Gaussian distribution.
With a larger roughness ratio ζ, the surface roughness of the rolled strip is also greater. Fig. 12 presents the strip surface topography after rolling at smaller average spacing length of ∆x = ∆y = 5 µm. Comparing the results in Figs. 11 and 12, it can be seen that a roll of smaller spacing lengths ∆x and ∆y produces smoother strips. 1252
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Fig. 11. Surface prediction with ∆x= ∆y= 20 µm (a) Rq,s= 0.815 µm, ζ = 1, (b) Rq,s= 1.64 µm, ζ = 2 (c) Rq,s= 2.4 µm, ζ = 3.
4.2.3. Reduction ratio effects Fig. 13 shows the effect of reduction ratio on the surface topography of rolled strips, in which Rq,s1 and ζ are fixed at 0.78 µm and 3, respectively. The average spacing lengths ∆x and ∆y are 5 µm and the rolling speed is 5 m/s. It is interesting to note that at a smaller reduction ratio the distribution of the surface asperity heights after rolling is not Gaussian, as shown in Figs. 13(a) to 13(b). However, as the reduction ratio increases, it turns to be Gaussian again approximately, As shown in Fig. 13(c) and Fig. 12(c). Fig. 13(d) shows that when the reduction ratio ψ is below 1%, increasing the reduction ratio will bring about a
rougher rolled strip surface. However, when the reduction ratio further increases, the ψ variation stabilizes because of the bounding by the surface topography of the rigid roll. These results demonstrate that reduction ratio can be used as an effective rolling parameter to produce desirable surface roughness in the skin pass rolling of strips. 4.2.4. Effect of rolling speed Fig. 14 shows the effect of rolling speed on the surface texture transfer, in which ∆x = ∆y = 5 µm, and Rq,s1 and ζ are 0.78 µm and 3, respectively. The range of rolling speeds used in the simulation are from 1253
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Fig. 12. Surface prediction with ∆x= ∆y= 5 µm (a) Rq,s= 0.718 µm, ζ = 1, (b) Rq,s= 0.1.42 µm, ζ = 2 (c) Rq,s= 2.12 µm, ζ = 3.
2.5 to 20 m/s. It can be seen that the rough surface profiles after rolling generally follow Gaussian distributions. In addition, the increase in the rolling speed only leads to small variations on the ratio of surface texture transfer, as show in Fig. 15. This is because the overall distribution of contact pressure is not significantly affected by the rolling speed, as evidenced by Fig. 16. When the relative sliding occurs in rolling, the rigid roll asperity indents into the strip surface and brings about the sliding process in the contact zone. Therefore, the overall sliding distance l, as shown in Fig. 16, can be used to assess the effect of rolling speed. It should be noted that l is dependent on the reduction
ratio rather the rolling speed. Thus, the variation of rolling speed has a negligible effect on the texture transfer process. 5. Conclusions This paper has developed a new method for predicting the threedimensional surface texture transfer in the skin pass rolling of metal strips. In this method, the FE analysis was used to obtain the distributions of the contact pressure and strip surface speed in the rolling bite. In addition, the plastic deformation of the random asperities was 1254
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Fig. 13. (a) Reduction ratio 0.049%, (b) reduction ratio 0.584%, (c) reduction ratio 1.14% (d) Rq,s vs reduction ratio.
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Fig. 14. (a) rolling speed 2 m/s, (b) rolling speed 5 m/s, (c) rolling speed 10 m/s, (d) rolling speed 20 m/s.
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Fig. 15. Effect of rolling speed on the surface roughness of rolled metal strips.
Fig. 16. Effect of rolling speed on the surface texture.
treated by the material redistribution at asperity tips. A time increment scheme was used to draw the rough surfaces of work roll and strip into the rolling bite. Within each time increment, an iterative method was used in conjunction with the penalty approach to reflect the instant surface topography change in the rolling bite induced by the instant asperity deformation. The established approach has overcome the difficulties due to the complex contact of randomly distributed surface asperities in both the roll and strip surfaces. A systematic comparison with the corresponding finite element analysis confirmed that the method developed is much cost-effective and can reliably predict the surface texture transfer processes in the skin pass of metal strip production. The obtained results revealed that the surface topography of a metal strip can be significantly affected by the texture transfer from the work roll and that the work roll surface topography is critical to the formation of desired strip surfaces. A higher ratio of the roll surface roughness
to that of the strip before rolling brings about a rougher strip surface after rolling. In addition, a larger reduction ratio results in a higher texture transfer ratio. As the reduction ratio increases, the texture transfer ratio will increase gradually before reaching a stabilized state bounded by the roll surface topography. Compared with the reduction ratio, the effect of rolling speed on the texture transfer is negligible. These results demonstrate that reduction ratio can be used as an effective rolling parameter to produce desirable surface roughness (and thus texture) in the skin pass rolling of metal strips. Acknowledgement This research forms part of the Baosteel Australia Research and Development Centre (BAJC) portfolio of projects and has received financial support from the Centre through Project BA15001.
Appendix A Because of the microscale surface roughness, it is challenging to experimentally investigate how the dynamic process of texture transfer occurs in the rolling bite. To investigate such processes [11,12,15], FEA has been widely used. Therefore, the feasibility of established model in predicting the texture transfer process will be verified by the relevant FEA in this study.
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A.1 Indentation simulation: FEM vs current model To verify the established model, a systematic finite element analysis (FEA) was conducted to simulate an indentation process between a rigid indenter with a regular surface pattern and a material with a smooth surface, as shown in Fig. A1 (a). The element type used for the former was R2D2, and that for the latter was CPE4R. The dimension of the deformable block was 2.4 mm× 0.5 mm. For simplicity, the FEA was under the planestrain condition. A sinusoidal function was used to construct the regular surface pattern of the rigid indenter. Let the height of the surface pattern be ha (mm), i.e.,
ha = 3 × 10
3
(A-1)
sin(2 fx )
Fig. A1. (a) Dimension of indentation simulation and boundary conditions in FEA, (b) Von-Mises stress at loading and unloading stage with an indentation depth of 4.5 µm, (c) Comparison of contact pressure.
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where
f=
1
(A-2)
where λ is the wave length, which is 0.2 mm in this verification case study. In the FEA, a y-symmetrical condition was applied to the edge BC while an x-symmetrical condition was applied to edges AB and CD. The indentation depth was ∆d. The block under indentation was an elastic-perfectly plastic material with a yielding strength of 0.5 GPa. A constant friction coefficient of 0.1 was applied to the indenter-block interface. The indentation simulation consisted of two steps, i.e., loading and unloading stages. The surface profile of the deformed block after unloading was used to compare with the prediction given by the established method. Fig. A1(b) shows the distributions of von-Mises stress at the end of the loading stage as well as after complete unloading. The maximum indentation depth used was ∆d = 4.5 µm. It can be seen that the regular surface pattern of the rigid indenter brings about a regular pattern of vonMises stress within the deformed material associated with a regular surface pattern. The total indentation force F at the maximum loading was used as the input parameter to our texture transfer model described by using Eqs. (6), (7) and (10)–(12)). For comparison, an identical surface pattern of the indenter was constructed by Eq. (A-1). Then, the total force Fn on the indenter was determined by
Fn = A
p (h
d) f (h
d) dh
(A-3)
The reference distance d was adjusted until Fn converged to F. During the calculation, the penalty stiffness K was set to be 1.0 × 1018 Pa/m. Fig. A1(c) shows the comparison of the distributions of contact pressure given by the FEA and Eq. (1). A dimensionless contact pressures, P/S, was plotted in Fig. A1(c). It can be noted that the prediction given by Eq. (1) agrees well with those given by the FEA although a lower maximum contact pressure was predicted by the current model. This is because, unlike the FEA, the contact pressure in Eq. (1) is limited by the surface hardness H, as shown in Eqs. (2)–(3), and that the elastic deformation is not taken into the consideration in Eq. (1). Fig. A2(a) demonstrates that the surface profiles predicted by Eq. (10) agree with those given by the FEA with different indentation depths. However, an obvious difference between these results can be noted for a lower indentation depth. Again, this is because the FEA considered both the elastic and plastic deformation. As a result, the elastic deformation also contributes to the texture transfer process in the indentation simulation, as evidenced by Fig. A2(b). Under a smaller indentation depth, a non-uniform pile-up can be observed on the non-contact zone due to such elastic deformation, as confirmed by Fig. A2(a). As the indentation depth increases gradually, the contribution of the elastic deformation becomes negligible due to the large plastic deformation, as demonstrated in Fig. A1(b). A better agreement between the surface profiles given by Eq. (10) and that given by the FEA is therefore reached, as shown in Fig. A2(a). With the obtained surface profiles above, the ratio of texture transfer from the rigid indenter to the material can be calculated. Since those profiles are represented by digital form, the surface roughness is calculated by
Rq =
1 N
N
(hi
have )2
(A-4)
i=1
where hi is the surface height on the digital point, have is the average surface height, and N is the total number of digital points. Thus, the texture transfer ratio can be defined by
=
Rq, s (A-5)
Rq, r
where Rq,s is the surface roughness of block after unloading, Rq,r is the surface roughness of rigid indenter. Fig. A2(c) plots texture transfer ratio ψ against the indentation depth. As expected, the ψ increases when the indentation depth increases. Overall, the ψ given by the established approach agrees well with those from the FEA. A.2 Rolling simulation-FEM vs current model To verify the surface topography predicted by Eq. (10) in the skin pass of rolling, an FEA was further conducted by using a rigid roll with a constructed sinusoidal surface profile which was governed by Eq. (A-1). The half strip thickness at the entry of the strip was 0.5 mm while the radius of the roll was 225 mm. The strip material was elastic-perfectly plastic with the yielding strength of 0.5 GPa. The rolling speed was fixed at 5 m/s and a constant friction coefficient of 0.1 was applied to the roll-strip interface. To reduce the computation cost, a y-symmetrical boundary was applied to edge AB (see Fig. A3(a)). Fig. A3(a) shows the prediction by the FEA. It can be seen that some texture on the roll surface has been transferred to the strip surface after rolling, demonstrated by the regular pattern on the strip surface. Because of the roll surface pattern, the distributions of contact pressure P and shear stress T in the rolling bite are no longer smooth. The corresponding distribution of von-Mises stress demonstrates that a regular pattern of residual stresses is caused by the coining of the roll pattern. To calculate the reference reduction ∆Y, a reference plane on the rolled strip surface can be obtained by averaging the surface heights over the selected region, as shown in Fig. A3(a). With the same ∆Y, i.e., 3 µm, a skin pass rolling with a smooth work roll was also carried out, as shown in Fig. A3(b). The overall distributions of contact pressure P and strip surface speed Vs along the rolling bite shown in Fig. A4(a) were the input parameters for the current model, and the analysis of texture transfer followed the procedure in Fig. 4. Fig. A4(b) demonstrates that the strip surface profile given by Eq. (10), as shown in blue line, agrees with that given by the FEA. A3 Effects of strain-dependent yielding strength In practice, the yielding strength of metal sheet is dependent on the strain ε [23,24]. Thus, Eq. (3) can be extended to describe the variations of flow stress with strain. To do this, the yielding strength S is considered to be dependent on strain, i.e., [21] (A-6)
S=f( ) 1259
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Fig. A2. (a) Comparison of surface profile between current model and FEA, (b) Elastic deformation vs plastic deformation for a lower ∆d, (c) Comparison of texture transfer ratio.
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Fig. A3. (a) FEA of skin pass rolling with constructed sinusoidal surface, (b) FEA of skin pass rolling with a smooth work roll.
As long as strain is known, the surface hardness H will be updated accordingly by Eqs. (A-6) and (3). To investigate the effects of strain-dependent yielding strength on the texture transfer, a strain-stress curve experimentally measured by Yoshida et al. [23] as show in Fig. A5, was used as an input parameter in the FEA. The initial yielding instability caused by strain localisation, which is known as the Lüders' bands, were well reflected by such strain-stress curve [23]. At the end of the FEA, the strain distribution in the rolling bite was obtained, and was used to calculate the surface hardness H of point m by Eqs. (3) and (A6). This process was repeated until the control surface completely passed the rolling bite. Fig. A6 demonstrates the predicted surface topography with or without considering the variations of flow stress with strain. For the elasticperfectly plastic material shown in Fig. A6(a), the yielding strength is obtained from the sharp yielding point shown in Fig. A5. Rolling speed and reduction ratio were fixed at 5 m/s and 2.7%, respectively. To simplify the analysis, the strip surface before rolling was regarded to be smooth. The roughness of the work roll, whose height followed a Gaussian distribution, was 0.81 µm. A lower surface roughness was predicted when considering the variation of flow stress with strain. Fig. A7 (a) further demonstrates that the strain-dependent yielding strength has a strong influence on the distribution of surface height after skin pass rolling because the surface hardness H is dependent on the yielding strength. In addition, the initial yielding instability caused by the strain localisation leads to a lower contact pressure at the entry, as shown in Fig. A7(b). However, a higher contact pressure at the exit can be noted due to the work-hardening of sheet metal. Overall, the proposed approach can successfully deal with the variations of yielding strength with strain in analysing the surface texture transfer in skin pass rolling.
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Fig. A4. (a) Distribution of contact pressure and strip surface speed in the rolling bite (b) Comparison of surface profile predicted by the current method and that given by FEA.
Fig. A5. Strain-stress curve [23].
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Fig. A6. Predicted surface topography (a) Elastic-perfectly plastic, 0.76 µm (b) Strain-dependent yielding strength, 0.69 µm.
Fig. A7. (a) Surface height distribution, (b) Contact pressure comparison. 1263
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