A unified method for characterizing multiple lubrication regimes involving plastic deformation of surface asperities

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A unified method for characterizing multiple lubrication regimes involving plastic deformation of surface asperities Chuhan Wu a, Liangchi Zhang a,n, Shanqing Li b, Zhenglian Jiang b, Peilei Qu b 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

art ic l e i nf o

a b s t r a c t

Article history: Received 19 July 2015 Received in revised form 11 November 2015 Accepted 13 November 2015

This paper presents a unified method for characterizing the hydrodynamic, mixed and boundary lubrication regimes in cold strip rolling. The analysis involves random surface asperities, multi-scale plastic deformation of both the asperities and workpiece, and a statistical modeling of an equivalent interface between the lubricated surfaces in contact sliding. This new method enables us to understand the effect of asperity plastic deformation on the interface lubrication and on the surface topography evolution of a strip surface during rolling. It is expected that the method is also applicable to other contact sliding processes involving complex lubrication and plastic asperity deformation. & 2015 Elsevier Ltd. All rights reserved.

Keywords: Multiple lubrication regimes Asperity plastic deformation Surface topography evolution

1. Introduction Roll–strip interface lubrication plays a crucial role in predicting the rolling force and surface finish in cold strip rolling. Thus a satisfactory characterization of the roll–strip interface, such as friction and lubrication, is of great importance to the improvement of rolling performance and product surface quality. Engineering surfaces are rough microscopically. The real solid–solid contact at the roll–strip interface is between asperity tips. In metal rolling, the surface asperities in contact usually carry a very large portion of the interface load, which inevitably makes the asperities deform plastically. Such large plastic deformation of the surface asperities significantly affects the friction/lubrication in the roll–strip interface, leads to strip topography variation, and in turn governs the rolling performance and surface finish of a strip rolled. Although extensive efforts have been placed to investigate the interface friction in metal rolling, a comprehensive understanding of the effect of asperity plastic deformation on roll–lubricant–strip interaction is not available. Because of the random asperity contacts, a statistical characterization of roll–workpiece interface is required to estimate the asperity contact pressure and real contact area. With the involvement of lubricant, hydrodynamic, mixed and boundary lubrication will take place in different parts of the rolling bite and will be governed by the coupled effect of solid–solid (asperity–asperity) and solid–liquid (asperity–lubricant) contacts. Therefore, the micro-scale asperity contact and asperity/lubricant n

Corresponding author. Tel.: þ 61 2 9385 6078. E-mail address: [email protected] (L. Zhang).

interaction at the roll–strip interface and the macroscopic bulk plastic deformation of the strip should be taken into account. The investigations on the flattening of one-dimensional asperities of a ridge-like shape at the roll–strip interface in metal forming have been carried out [1–2]. Many studies [3–4] have also been conducted to develop mixed lubrication models of rolling, considering both one-dimensional asperity flattening and effect of lubricant. Since theoretical investigations usually impose many limitations, numerical analyses, such as those with the aid of the finite element (FE) method, have been widely used to explore the effect of roll–strip interface friction [5–6]. Recently, the authors have developed a multi-scale method to predict the pressure and friction distributions in cold rolling, and introduced an equivalent interfacial layer (EIL) to characterize the roll–strip interface [7]. A statistical analysis using the Greenwood–Williamson model [8] was carried out to address the elastic asperity deformation and asperity/lubricant interaction within the EIL. The FE method was performed to obtain the plastic deformation of a strip. In so doing, they have overcome the difficulty in the coupling of microscopic elastic deformation of surface asperities and macroscopic plastic deformation of the bulk strip. However, the plastic deformation of surface asperities has not been addressed. As such, the real contact in rolling cannot be accurately characterized, and the surface topography evolution of the strip during rolling cannot be analyzed. The aim of this paper is to take into account the plastic deformation of asperities. Based on this, the problems described above can be resolved; the hydrodynamic, mixed and boundary lubrication regimes can be analyzed in a unified way; and the strip surface topography evolution during rolling can be explored.

http://dx.doi.org/10.1016/j.triboint.2015.11.020 0301-679X/& 2015 Elsevier Ltd. All rights reserved.

Please cite this article as: Wu C, et al. A unified method for characterizing multiple lubrication regimes involving plastic deformation of surface asperities. Tribology International (2015), http://dx.doi.org/10.1016/j.triboint.2015.11.020i

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Lind Lcon

Nomenclature A A0 Ac As asmax

α β C d ΔHi d'p E´ E Fa Fc Fs Fsmax f(h) f´(hu) f(h0)

ϕx

h Hbulk hT h'T ho hu k Ki

solid–solid contact area unit nominal area critical area at the initial yielding point of an individual contact asperity contact area at a single asperity maximum contact radius of a single asperity at loading pressure coefficient of viscosity reduction ratio of strip rolling constant for ωc mean separation or EIL thickness interference between a slave surface node and master surface gap distance between a slave surface node and master surface reduced Young's modulus elastic modulus total contact force on a nominal unit area of asperity contact critical force at the initial yielding point of a contact asperity contact force on a single asperity maximum contact force on a single asperity at loading asperity height distribution asperity height distribution on a deformed rough surface original asperity height distribution average flow factor asperity height deformable bulk thickness average film thickness at loading average lubricant film at unloading original asperity height asperity height after unloading material shear strength under plane-strain compression penalty stiffness

2. Modeling 2.1. Overall methodology The multi-scale plastic deformation is based on an EIL [7] at the roll–strip interface to integrate hydrodynamic pressure with the plastic deformation of surface asperities. Since lubricant can only flow outside the solid–solid (asperity–asperity) contact area in the EIL, the normal contact pressure Pn on the unit nominal area (A'¼1) can be given by the force equilibrium equation [7] 0

A P n ¼ AP a þ ð1  AÞP f

ð1Þ

where A denotes the ratio of real contact area by direct asperity– asperity contacts to the unit nominal area A0 ; Pa is the average asperity contact pressure and Pf is the hydrodynamic pressure due to the pressurized lubricant given by [4] ! 3 ∂ hT ∂P f U r þ U s ∂hT ð2Þ φx ¼ ∂x 12η ∂x 2 ∂x where η is the lubricant viscosity; Ur and Us are the roll surface speed and strip surface speed, respectively. For a given surface, the lubricant flow can be significantly affected by the surface roughness at the contact interface. In order to consider such an effect in

indenter length estimated rolling bite length Coulomb friction coefficient total friction coefficient at the roll–strip interface asperity density interference of an asperity in contact critical interference for purely elastic contact maximum interference of a single asperity at loading interference of a single asperity at unloading average asperity contact pressure hydrodynamic pressure normal contact pressure over the nominal area given by the penalty method normal contact pressure average interface pressure given by slab method total shear stress average asperity radius roll radius asperity radius of a plastically deformed asperity at unloading yielding strength standard deviation of asperity heights root mean square surface roughness frictional stress due to direct asperity contact lubricant shear traction interface friction stress given by the slab method roll surface speed strip surface speed rolling speed horizontal coordinate in the finite element (FE) model vertical coordinate in the FE model local strip thickness in the slab method local strip thickness initial strip thickness lubricant viscosity lubricant viscosity at atmosphere pressure Poisson’s ratio ratio of rolling bite length to half strip thickness

μa μr

N

ω ωc ωsmax ωsres

Pa Pf Pi

Pn Ps Pt R Rroll Rsres Sy

σ σs τa τf τs

Ur Us Vroll x y ys Y Y0

η η0 ν ζ

calculating the hydrodynamic pressure, the average flow been introduced in Eq. (2), which can be defined by [4] ( pffiffiffi pffiffiffi 2 3σ =hT hT o 3σ pffiffiffi φx ¼ 1 þ 3ðσ =hT Þ2 hT Z 3σ

ϕx has ð3Þ

where σ is the surface roughness and hT is the average film thickness determined by [7]: Z hT ¼

d 1

ðd  hÞf ðhÞdh

ð4Þ

where h is the asperity height; f(h) is the function of asperity height distribution and d is the EIL thickness (or the mean separation ) measured from the reference plane of the rough strip surface with respect to that of the smooth roll surface. Similarly, the total shear stress Pt within the EIL is given by [7]: A0 P t ¼ Aτa þ ð1  AÞτf

ð5Þ

where τa is the frictional stress due to the sliding of asperity contact and τf is the lubricant shear traction generated by the lubricant layer, given by [7]:

τ a ¼ μa P a

ð6Þ

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Fig. 1. The illustration of rough surface contact. (a) The contact between a smooth rigid surface and a rough surface (b) The statistical characterization of rough surface contact.

τf ¼ η

U r U s hT

ð7Þ

where μa is the Coulomb friction coefficient for the direct solid– solid (asperity–asperity) contact. The asperity contact stress Pa, τa and the contact area ratio A due to the asperity plastic deformation can be determined statistically, as to be detailed in Section 2.2. Under an isothermal condition, lubricant viscosity can be expressed by the Barus law:

η ¼ η0 eαPf

ð8Þ

where η0 is the lubricant viscosity at atmosphere pressure and α is the pressure coefficient of viscosity. It should be mentioned that the interface temperature between the roll and strip surfaces can increase because of the large plastic deformation of the strip and the interface friction. This temperature rise can lead to a decrease in lubricant viscosity. A complete analysis of the thermal effect will involve complex calculations such as heat generation due to strip plastic deformation and interface sliding. These are beyond the scope of the present paper but will be addressed in further research. Moreover, it has been realized that the Barus law is applicable to a limited range of low pressures [9]. Thus, it is

assumed in this study that the lubricant viscosity will not further increase when Pf is beyond 0.1 GPa. 2.2. Statistical modeling of elasto-plastic deformation of asperity contact The contact between two rough surfaces can be equivalently treated as the contact between a smooth surface and a rough surface [10]. Therefore, it is assumed that all asperities are on the strip surface while the roll surface is smooth. Since the roll hardness is much higher than that of a strip material, it is reasonable to assume that the roll is rigid when the strip is not too thin. The contact between a rigid smooth surface and a deformable rough surface is schematically illustrated in Fig. 1(a). The asperity height h is usually measured with respect to the reference plane of asperity heights. It is assumed that all asperities have an identical radius R. The standard deviation of asperity heights σ can be related to the standard deviation of surface height or the root mean square surface roughness σs by [11]:

σs ¼ σ þ

3:717  10  4 N 2 R2

ð9Þ

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where N is the asperity density on the unit nominal area. Generally, σ can be considered to be identical to σs. In practice, the asperities in contact are rarely found when the mean separation d is larger than 2σ. Thus, the asperity height distribution is in the range of (  2σ, 2σ). With the decrease in mean separation d, some asperities will be subjected to an indentation with an interference ω. As a result, the asperities on a strip surface can be categorized into the following groups according to their differences in the indentation interference ω [11]: (1) When ω r0, there is no contact. (2) When 0 o ω r 1.9ωc, an asperity deforms elastically. (3) When ωc o ω, an asperity undergoes elasto-plastic deformation. where ωc is the critical interference for the purely elastic contact, at which the asperity starts yielding. According to the Jackson– Green (JG) model [11], ωc is given by   π CSy R ð10Þ ωc ¼ 0 2E where E0 is the reduced Young's modulus defined by E0 ¼

E 1  υ2

ð11Þ

where E and ν are the elastic modulus and Poisson's ratio of the asperity material, respectively; Sy is the yielding strength of the asperity material and constant C is given by [11]: C ¼ 1:295expð0:736νÞ



 CSy R 2 2E0

ð14Þ

According to the JG model, the deformation of a single asperity in contact could be described by the Hertzian solution when 0 o ω∕ωc r1.9. At an indentation interference of ω∕ωc 41.9, the contact force Fs and contact area As of a single asperity are [11] (" !#    1 ω 5=12 ω 3=2 F s ðωÞ ¼ F c exp  4 ωc ωc "  5=9 !# ) 4H G 1 ω ω þ 1  exp  ð15Þ 25 ωc ωc CSy As ðωÞ ¼ π Rω



ω 1:9ωc

B ð16Þ

where B ¼ 0:14expð23ey Þ ey ¼

ð17Þ

Sy E0

ð18Þ 2

0

HG ¼ 2:8441  exp@  0:82 Sy

ð21Þ Z A¼

d þ 1:9ωc d

A0 N π Rωf ðhÞdh þ

Z

þ1 d þ 1:9ωc

A0 NAs ðωÞf ðhÞdh

ð22Þ

where ω ¼ h d. Since the asperity height distribution is in the range of ( 2σ, 2σ), 2σ can be used to replace the infinity in Eqs. (21) and (22). Thus, the average asperity contact pressure Pa is determined by Pa ¼

Fa A

ð23Þ

Under dry friction, the normal contact pressure Pn on the unit nominal area A0 ¼1 in Eq. (1) reduces to A0 P n ¼ AP a ¼ F a

ð24Þ

Without lubricant, the total asperity contact force Fa and hence the normal contact pressure Pn are the result of direct asperity contacts. When lubricant is applied, a fraction of the normal pressure Pn will be shared by the pressurized lubricant. 2.3. Statistical characterization of unloading

ð12Þ

The critical force Fc and critical area Ac at the onset of yielding of an asperity are [11]:    3 4 R 2 C Fc ¼ π S ð13Þ y 3 E0 2 Ac ¼ π 3

Because of the random asperity distribution, a statistical analysis is required to get the total asperity contact force Fa and real contact area A over the unit nominal area A' ¼1. As schematically illustrated in Fig. 1(b), Z d þ 1:9ωc Z þ1 4 0 0 1=2 3=2 A NE R ω f ðhÞdh þ Fa ¼ A0 NF s ðωÞf ðhÞdh 3 d þ 1:9ωc d

13  !  0:7 ω ω B=2 A5 R 1:9ωc

As an asperity undergoes plastic deformation, the asperity cannot fully recovery to its original undeformed geometry after full unloading (see Fig. 2(a)). Thus, asperity plastic deformation will lead to the topography variation of a strip surface after rolling, which eventually determines the surface finish of a strip rolled. It should be noted that the rough surface contact at unloading is different from that at loading. Before investigating the unloading of a plastically deformed rough surface, it is necessary to understand the unloading of individual asperities. An analytical solution [12] to the elastic recovery of a plastically deformed asperity is particularly helpful. For a single asperity with the interference of ωsmax at loading, the asperity contact force Fsmax and contact radius asmax can be obtained by the JG model [11]. As indicated in Fig. 2(a), Fsmax and asmax decrease to 0 with a new interference ωsres measured from the loaded profile to the fully unloaded profile. Since unloading is perfectly elastic, it is reasonable to consider that the subsequent loading from the completely unloaded profile to the original loaded profile obeys the Hertzian solution of a perfect sphere with radius Rsres (Fig. 2(b)). The final contact force and contact radius on the completion of subsequent loading to the originally loaded profile with a new interference ωsres should be exactly the same as the Fsmax and asmax given by the previous loading step [12]. With the further increase in the interference from the originally loaded profile in the subsequent loading, the analysis repeats. The varied radius Rsres and the new interference ωsres for a single asperity in the unloading process can be given by [12] Rsres ¼

rffiffiffiffi

ð19Þ

To calculate the asperity contact force and real contact area, it is assumed that the asperity height follows the Gaussian distribution, i.e.,   1 1 f ðhÞ ¼ pffiffiffiffiffiffiexp  ðh=σ Þ2 ð20Þ 2 σ 2π

4E0 a3s max F s max

ð25Þ

a2s max Rsres

ð26Þ

ωsrec ¼

In the unloading stage, radius Rsres for the plastically deformed asperity becomes different from the original asperity radius R; and plastically deformed asperities have different radiuses Rsres at unloading due to their different interferences in the loading process.

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Fig. 2. Multiple loading/unloading of a single asperity. (a) Loading and unloading of a single asperity (b) Subsequent loading after the unloading.

Fig. 3. The variation in the asperity height distribution. (a) Original distribution of asperity height (b) Asperity distribution after unloading.

The complexity of unloading is due to the variation of asperity height distribution. To get the total asperity contact force and contact area ratio at unloading, it is necessary to investigate how a plastically deformed asperity affects the asperity height distribution. It is noted that the integral of the probability function over the whole range of the asperity height distribution on a rolled strip surface should be equal to that given by the initial strip surface. As indicated in Fig. 3, the varied distribution of asperity height f´(hu) due to asperity plastic deformation can be related to the original distribution of the asperity height f(h0) by Z hu2 Z ho2 0 f ðho Þdho ¼ f ðhu Þdhu ð27Þ

2.4. Surface topography evolution

ð28Þ

In cold rolling, lubricant is usually applied to reduce friction. Hence, hydrodynamic pressure can build up in the rolling bite due to the pressurized lubricant entrapped in surface valleys. The coupling of the asperity/lubricant interaction at the roll–strip interface can cause complex loading–unloading cycles on an asperity, because both the asperity contact pressure and hydrodynamic pressure vary in the rolling bite. This eventually determines the rolled surface finish, as illustrated in Fig. 4. To take into account such asperity/lubricant interaction and multiple loading– unloading processes simultaneously, let us consider an arbitrary position xi at the roll–strip interface within the rolling bite, as demonstrated in Fig. 5(a). If the total asperity contact force Fa at xi is given by the loading process, which gives the current EIL thickness dmin (see Fig. 5(d)), two possible situations can occur when xi moves to xi þ 1:

Hence, a statistical analysis can predict the total asperity contact force and contact area ratio during unloading, provided that the hu distribution is known. Now, the total asperity contact force and contact area ratio in loading and unloading can be obtained by using Eqs. (10)–(28).

(1) Further increase in the asperity contact force from Fa to F'a, which results in a decrease in EIL thickness from dmin to d'min, Fig. 5(b). As indicated in Fig. 5(d), the average asperity contact pressure Pa, contact area ratio A and average thickness of lubricant film hT, etc. can be obtained by the loading process.

ho1

hu1

where ho is the original asperity height and hu is the corresponding asperity height after complete unloading. For a plastically deformed asperity, hu can be related to ho by (Fig. 2) hu ¼ ho  ðωs max  ωsres Þ

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(2) Unloading starts from Fa to F''a, which leads to an increase in EIL thickness from dmin to d'u, Fig. 5(c and d). This can occur when a high hydrodynamic pressure builds up in the rolling

bite or when the strip approaches the outlet of the rolling bite. As a result, the asperity contact force F''a and its corresponding contact area ratio can be determined by the unloading

Fig. 4. The evolution of a strip surface in the rolling bite.

Fig. 5. The multiple loading/unloading in the rolling bite.

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formulas derived above. Because the asperity plastic deformation has changed the asperity height distribution, the average film thickness at unloading no longer follows Eq. (4). The average lubricant film h'T at unloading is given by 0

hT ¼

Z

d0u 1

0

0

ðdu  hu Þf ðhu Þdhu

ð29Þ

As aforementioned, strip surface finish is determined by the multiple loading/unloading processes and the solid–fluid interaction in the rolling bite. To determine whether position xi þ 1 in the rolling bite is under loading or unloading, the asperity contact force Fa at position xi þ 1 should be compared with the maximum asperity contact force recorded from x0 (inlet point) to xi (see Fig. 5(a)). The strip surface topography such as surface roughness and asperity density on a unit nominal area are influenced by the bulk plastic deformation of a strip. The strip elongation in the rolling bite will permanently change the surface topography due to the plastic flow of the strip. As detailed in the Appendix of this paper, the surface roughness variation σi at xi in the rolling bite is related to the initial strip surface roughness σ by

σi ¼

Yi σ Y0

ð30Þ

where Yi is the strip thickness at xi and Y0 is the initial strip thickness. Similarly, the asperity density variation Ni on a unit nominal area at xi is given by Ni ¼

Yi N Y0

ð31Þ

where N is the initial asperity density. During the calculations, the strip surface roughness and asperity density will be updated instantly when the strip thickness varies in the rolling bite.

7

3. Implementation The overall computation was done by using ABAQUS EXPLICIT with a user-subroutine to take into account the multi-scale contact deformation at the roll–strip interface. The penalty method in ABAQUS EXPLICIT was applied to deal with the surface contact to get the total asperity contact force Fa. From Fig. 6(a), it can be seen that the contact pressure Pi at node i on the slave surface is given by [13]: P i ¼ K i ΔH i

ð32Þ

where Ki is the penalty stiffness and ΔHi is the interference between the master and slave surfaces (corresponding to the roll and strip surface), respectively. The selection of the penalty stiffness is based on the following considerations: (1) The interference ΔHi should be as small as possible, which requires that the penalty stiffness should be as large as possible. (2) The finite element computation should be stable, which means that penalty stiffness cannot be too large since a higher penalty stiffness tends to increase the computation instability. It can be seen that the above two considerations impose contradictory constraints on the selection of the penalty stiffness. Normally, ABAQUS EXPLICIT will choose an appropriate penalty stiffness for contact analysis based on the above criteria. The distance measured from the slave surface nodes with respect to the master surface is the output of ABAQUS EXPLICIT, which will be passed onto the user-subroutine. If the interference between node i and master surface is detected, contact pressure Pi is computed by the penalty method. The integral of Pi on the unit nominal area A0 ¼1 is considered to be equal to the total asperity contact force Fa

Fig. 6. Penalty method and FE setup. (a) Penalty method in ABAQUS EXPLICIT (b) EIL thickness d for non-contact nodes (c) The FE mesh for the multi-scale analysis.

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Table 1 Parameters used in simulation.

Specify the rolling conditions: such as Rroll, Vroll, y0 & material properties Finite element analysis Multi-scale treatment and passing the relevant parameters, i.e. surface speeds, to the User-subroutine Set initial value for R, N, η0, α, μa, E, Sy, ν at the first entry Initializing the discretization of the loading/unloading process for different strip thickness (or different surface structure) at the first entry

Name

Symbol

Value

Young's modulus Poisson' ratio Yielding strength Roll radius Strip thickness Frictional coefficient Asperity density Roughness Asperity radius Lubricant viscosity Pressure coefficient of viscosity Reduction ratio Rolling speed

E υ Sy Rroll Y0 μa N σ R μ0 α β Vroll

2.1  1011 Pa 0.3 1  109 Pa 0.225 m 0.002 m 0.1 4.8  1010/m2 1.0  10  6 m 2.5  10  5 m 0.0005–0.05 Pa s 1.0  10  8 Pa  1 4%,14% and 29% 5.0 m/s

Using Eq. (32-33) to obtain Fa for the contact nodes and using Eqs. (35-36) to obtain the d and then A & hT Using Eq. (34) to obtain hT for non-contact nodes Solve Pf from Eq. (2) Solve Pn & Pt from Eqs. (1) & (5) Update Pn & Pt on the strip surface nodes No

Completed? Yes Output Pn, Pt A &hT

Fig. 7. Flow chart of the overall computation. Fig. 8. The discretization of the loading/unloading process. (a) The comparisons in the interface pressure (b)The effect of strain rate distribution on normal pressure.

given by the statistical analysis. This gives rise to A0 P i ¼ F a ¼ AP a

ð33Þ

For a given total asperity contact force Fa on a unit nominal area, its corresponding EIL thickness d can be calculated by the above loading–unloading analysis. As soon as d is known, the contact area ratio A and average lubricant thickness hT can be found by Eqs. (22), (4) or (29). From Fig. 6(b), it can be seen that the EIL thickness di for a non-contact node can be given by 0

di ¼ 2σ þdp

ð34Þ

where d'p is the gap distance between the slave surface node and master surface, which is also the output of ABAQUS EXPLICIT. Thus EIL thickness d will be equal to 2σ when the asperity contact between the roll and strip surfaces starts. The overall computation is illustrated in the flowchart, Fig. 7. Eqs. (10)–(24) were programmed to determine asperity contact force Fa, average asperity contact pressure Pa and contact area ratio A in the loading process. Meanwhile, Eqs. (25)–(28) were used to calculate the quantities at unloading. Eqs. (4) and (29) were used to compute the lubricant film hT at loading and unloading, respectively. The instant roll surface speed Ur and strip surface speed Us were the output of ABAQUS. They were in turn passed onto the subroutine at the end of each time increment. Then, Eqs. (2), (3) and (8) were used to compute the hydrodynamic pressure Pf. The finite difference method was used to solve Eq. (2). The details of the finite difference mesh in EIL for solving the average Reynolds equation can be found in [7]. In the plastic deformation zone, the range for the film pressure Pf, t þ Δt is 0 rPf, t þ Δt o Pn, t þ Δt

from time increment t to tþΔt. During the iteration, Pf, t þ Δt would be set to an appropriate limit if it was outside the above range [4]. This condition can be easily satisfied by using 0 rPf, t þ Δt o0.7Pn, t where Pn, t is the normal pressure obtained in the previous time step. The iterative computation continued until the average deviation reached the prescribed convergence criterion (1.0  10  5 in the current study). Eqs. (1) and (5) were used to get the contact stresses at the roll–strip interface and to update the stresses on the corresponding finite element nodes on the strip surface. The strip bulk deformation using the FE analysis was then determined by these surface stresses. The FE meshing is given in Fig. 6(c), noting that only half of the strip thickness was taken into account due to the deformation symmetry. The strip material was assumed to be elastic–perfectly plastic, and its deformation was considered to be plane-strain. All the parameters used in the simulation are listed in Table 1. Special numerical methods were used in the simulation to speed up the overall analysis. For a given surface topography, the loading was discretized with respect to EIL thickness d and the unloading process from the loading point was discretized over d as well. As indicated in Fig. 8, the loading process is highlighted by the blue solid line and its corresponding discretization is denoted by blue solid dots. The unloading is highlighted by the red solid line and its corresponding discretization is denoted by red solid dots. For a total asperity contact force F'a(1) in loading, which is denoted by hollow blue circle, its corresponding EIL thickness d'1 can be obtained by a linear interpolation from the neighboring

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data, i.e., 0

d1 ¼ dði;1Þ þ ðdði þ 1;1Þ  dði;1Þ Þ

F 0að1Þ  F aði;1Þ F aði þ 1;1Þ  F aði;1Þ

ð35Þ

When unloading starts from F'a(1), its corresponding discretization, which is denoted by the hollow black circle, can be obtained by F 0aðjÞ ¼ F aði;jÞ þðF aði þ 1;jÞ  F aði;jÞ Þ

F 0að1Þ  F aði;1Þ F aði þ 1;1Þ F aði;1Þ

ð36Þ

Similarly, the corresponding EIL thickness d0 j and d0 j þ 1 in unloading can be obtained by using Eq. (35). Therefore, EIL thickness d'' for the asperity contact force F''a in unloading from F'a(1) can be obtained as 00

0

0

0

d ¼ dj þðdj þ 1  dj Þ

F 00a  F 0aðjÞ 0 F aðj þ 1Þ  F 0aðjÞ

ð37Þ

The contact ratio A and average lubricant film thickness hT can be computed in a similar way. Because the surface structure varies in the rolling bite due to the strip plastic deformation, a series of different strip surface topographies was computed based on the strip thickness variations. The discretization of the loading/unloading process were repeated for each strip surface topography and an additional interpolation was required to compute EIL thickness d, contact ratio A and average lubricant thickness hT based on the varied strip thickness. The computation time was about 1 h for a complete multi-scale analysis on a desktop PC with a processor of Intel (R) Core i5-3470 CPU @3.2 GHz.

4. Results and discussions 4.1. Model verification

Fig. 9. Multi-peak pressure distribution.

In order to verify the statistical model established above, the calculated results were compared with those given by the slab method [4] which has been widely used in the theoretical modeling of cold rolling. In the slab method, it is assumed that the strip deformation is homogenous across the strip thickness and the strip material is rigid-perfectly plastic. Thus, the average interface pressure Ps in the plastic deformation zone in the rolling bite is governed by the equilibrium equation: dy dP s þ 2τ s ¼ 0 2k s  ys dx dx where ys is strip thickness and determined by

τs ¼ μa P s

ð38Þ

τs is the interface friction stress ð39Þ

The x–y coordinate system used in the slab method was identical to that used in the statistical FE analysis shown in Fig. 5(a). Variable k in Eq. (38) is the material shear strength under planestrain compression [14], i.e., pffiffiffi k ¼ Sy = 3 ð40Þ where Sy is the uniaxial yielding stress associated with the uniaxial equivalent strain. The average interface pressure Ps should be equal to the plane-strain yielding stress 2k in the inlet and outlet points, which gives rise to P s;entry ¼ 2k

ð41Þ

P s;exit ¼ 2k

ð42Þ

Thus Eq. (38) can be solved numerically by starting from boundaries given by Eqs. (41) and (42) (double shooting method

[4]). Fig. 9 presents the comparison between the results of the present model and those by the slab method, under a dry friction condition. Fig. 9(a) shows that the interface pressure distribution agrees well overall. However, the slab method predicts a lower pressure near the outlet. This is due to the difference in friction transition. The slab method has a neutral point and hence friction changes its direction suddenly there. The current model however predicts a transition zone (or a neutral zone) through which the change of the friction direction is gradual. Extensive experimental measurements in cold rolling [15,16] have confirmed that such friction transition zone does exist. Moreover, the current model predicts a longer rolling bite. This is because the present model includes elastic deformation but the slab method deals with a rigid-perfectly plastic material. The elastic recovery of the deformed strip near the outlet of the rolling bite makes the contact length greater. The total rolling forces given by the slab method and the current model are 17.757 KN and 20.375 KN, respectively. However, the present model shows that the pressure distribution contains multiple peaks. Experimental measurement [15,16] on cold rolling demonstrated that multiple pressure peaks do exist, but the mechanism is unclear. The present simulation shows that the formation of the multiple peaks is due to the inhomogeneous strip deformation in the rolling bite, influenced by many rolling conditions such as strip thickness, reduction ratio and roll radius. Fig. 9(b) shows the strain rate distribution corresponding to such inhomogeneity. It is interesting to see that the strain rate distribution is not uniform and has a regular pattern across the strip thickness, indicating that the strip material does not flow uniformly within the rolling bite. A higher strain rate at the roll–strip interface corresponds to a local pressure valley, and

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Fig. 10. The strain rate pattern effect on normal pressure. (a) The strain rate pattern in the bulk when Lind /Hbulk ¼2 (b)The strain rate pattern in the bulk when Lind /Hbulk ¼ 4 (c)The effect of the reduction ratio on the multi-peak pressure.

vice versa (see Fig. 9(b)). Hence, it is the variation of the strain rate that brings about the multiple pressure peaks. A consequent question is therefore: How does the regular pattern of strain rate in the bulk strip occur in cold rolling? If we connect the strain rate pattern across the bulk strip by a broken line shown in Fig. 9(b), it is interesting to see that the broken line forms approximately an angle of 45° with respect to the x-axis. This can be due to the fact that the plastic flow of the strip tends to flow more easily in this direction, generating a high strain rate in that direction. This phenomenon is similarly evidenced by indentation simulations under plane-strain conditions as shown in Fig. 10 (a) and (b), in which the indenter was considered to be rigid and a symmetrical boundary condition was applied to the bottom surface of the deformable bulk. These figures demonstrate the effect of different ratio of indenter length Lind to bulk thickness Hbulk on the formation of the strain rate pattern. It is clear that a similar strain rate

pattern forms in the deformable bulk and that its distribution is dependent on the ratio of Lind to Hbulk. This indicates that the multiple peaks of the normal pressure in cold rolling may have a relationship with the ratio of the roll–strip contact length to the strip thickness, ζ. If the roll is rigid, the roll–strip contact length Lcon can be approximately estimated by qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð43Þ Lcon ¼ Rroll Y 0 β where β is the reduction ratio and Y0 is the initial strip thickness. Thus, ζ can be given by

ζ¼

Lcon Y 0 =2

ð44Þ

Fig. 10 (c) shows the influence of ζ, by changing the reduction ratio β on the pressure peak location. It can be seen that the peak

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Fig. 11. Plastic asperity model vs elastic asperity model. (a) The normal pressure and shear stress (b) Contact ratio A (c) Average lubricant thickness hT (d) Before rolling, roughness Rq 1.593 μm (e) After rolling, roughness Rq 0.926 μm.

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Fig. 12. The lubricant viscosity effect on the interface pressure.

number increases when ζ rises. At a small ζ, the normal pressure predicted by the slab method differs more from the present model. The two predictions converge when ζ increases. However, the slab method cannot reveal the multiple pressure peaks and the deformation inhomogeneity. 4.2. Effect of plastic deformation of asperities The plastic deformation of asperities in cold strip rolling was ignored in previous works [7], and hence its effect on cold rolling performance cannot be addressed. Fig. 11 compares the results under dry friction with and without asperity plasticity. It can be seen that without including asperity plasticity significantly underestimates the contact ratio, Fig. 11(b), but overestimates lubricant film thickness hT, Fig. 11(c), and asperity contact pressure, leading to an inaccurate ratio of hydrodynamic pressure to normal pressure. As experimental measurements have shown, Fig. 11(d) and (e), the topography of a strip surface varies in cold rolling. Therefore, the present model that includes asperity plasticity is necessary in application. 4.3. Lubricant viscosity effect Fig. 12 shows the distributions of the non-dimensional normal pressure Pn and hydrodynamic pressure Pf in the rolling bite. The lubricant viscosity was varied from 5.0  10  4 Pa s to 5.0  10  2 Pa s with other rolling conditions unchanged. In the inlet zone, no asperity is in contact; thus rolling is in a full film lubrication regime where the normal pressure is carried solely by the lubricant. As the strip is drawn into the rolling bite, normal pressure Pn at the roll–strip interface rises significantly. At this stage, both asperity contacts and hydrodynamic pressure contribute to the normal pressure, giving rise to a transition from full film lubrication to mixed lubrication. The normal pressure reaches its maxima in the neutral zone and decreases to zero when the strip approaches to the outlet of the rolling bite. It can be seen that the hydrodynamic pressure decreases to zero before the normal pressure does. In this stage, the entire normal pressure is carried by the contact asperities, and the mixed lubrication eventually approaches boundary lubrication. The transition from the full film lubrication, mixed lubrication, to boundary lubrication from the inlet to the outlet of the rolling bite is all governed by the asperity/ lubricant interaction. Increasing the lubricant viscosity results in a significant increase in hydrodynamic pressure Pf and leads to the decrease in normal pressure. A large drop in the maximum normal pressure is favorable in production, because a sharp pressure peak is detrimental to rolling performance. With a low lubricant viscosity, the

Fig. 13. The shear stress and friction coefficient.(a)The total friction stress (b) The resulted friction coefficient.

hydrodynamic pressure in the rolling bite is very small; its influence on the normal pressure is negligible. With increasing the lubricant viscosity, the hydrodynamic pressure increases significantly. In this case, a large fraction of the normal pressure Pn is shared by the hydrodynamic pressure. For example, when the viscosity is 0.02 Pa s, the hydrodynamic pressure in the rolling bite flattens and the normal pressure at the roll–strip interface decreases noticeably. When the viscosity increases to 0.05 Pa s, it is of great interest to note that the hydrodynamic pressure also exhibits multiple peaks but with its maximum value decreasing. The reason is that hydrodynamic pressure is limited by the total interface pressure controlled by the plastic formation of the bulk strip and by the friction in the rolling bite. Fig. 13(a) shows the friction stress Pt at the roll–strip interface. It can be seen that Pt near the inlet of the rolling bite gradually changes its direction in the neutral zone. When the lubricant viscosity is low, the friction stress is largely due to the direct asperity contacts as the hydrodynamic pressure effect is negligible. With increasing the viscosity, friction stress decreases significantly

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Table 2 The roughness on the rolled strip surface (original roughness 1.0 μm). Lubricant viscosity (Pa s)

The final surface roughness (μm)

5.0  10  4 5.0  10  3 8.0  10  3 2.0  10  2 5.0  10  2

0.4531 0.4641 0.4754 0.5029 0.5083

the surface roughness predictions. It can be seen that when the surface roughness of the strip before rolling is 1 μm, it is reduced significantly after rolling. This is confirmed by the experimental measurements shown in Fig.11(d) and (e). The lubricant viscosity effect on the surface roughness variation is also demonstrated in Table 2. It is clear that with a very low lubricant viscosity, the surface roughness variation is mainly the result of the asperity plastic deformation. As the lubricant viscosity increases, the surface roughness reduction is less.

5. Conclusions This paper has developed a unified method for characterizing the multiple lubrication regimes in cold rolling which involves multiscale plastic deformation. The microscopic asperity plastic deformation and asperity/lubricant interaction have been successfully integrated with the macroscopic bulk plastic deformation of a strip. The investigation has brought about the following findings:

Fig.14. The distribution of the contact ratio and dimensionless hT. (a) The contact ratio A (b) Dimensionless average lubricant thickness.

and the neutral zone shifts to the outlet of the rolling bite. The total friction coefficient μr at the roll–strip interface can be defined by

μr ¼

Pt Pn

ð45Þ

The distribution of μr is given in Fig. 13(b). It can be seen that μr increases significantly at the rolling bite inlet where the strip starts yielding. A lower friction coefficient can be obtained by increasing the lubricant viscosity, because it gives rise to a higher hydrodynamic pressure. Fig. 14 shows the contact ratio A and lubricant film thickness hT in the rolling bite. Due to the inhomogeneous bulk deformation of the strip, the multiple peaks also appear in the contact ratio variation throughout the rolling bite. Increasing the lubricant viscosity leads to a smaller contact ratio, because in this case less asperities will be in contact. At the same time, the average lubricant film thickness shows a significant decrease. A larger lubricant film thickness can be obtained as the lubricant viscosity increases. The surface topography is changed mainly by the plastic deformation of asperities and the bulk elongation of the strip. Using the present model, the distribution of the asperity height on a rolled strip surface can be easily determined; therefore, the roughness on a rolled strip surface can be predicted. Table 2 lists

(1) The transition from full film lubrication, mixed lubrication to boundary lubrication from the inlet to the outlet of a rolling bite is governed by the asperity/lubricant interactions. The present new method can characterize all these lubrication regimes in a single step. (2) The multiple pressure peaks is caused by the inhomogeneous bulk plastic deformation of the strip, reflected by the regular strain rate pattern across the strip deformation zone. The ratio of the contact length in the rolling bite to the strip thickness plays a crucial role in the formation of the strain rate pattern. (3) If asperity plasticity is ignored (i.e., elastic asperities), the contact area ratio will be significantly underestimated, but the average asperity pressure be overestimated. This will bring about inaccurate predictions of hydrodynamic and normal pressure. (4) A higher lubricant viscosity leads to a lower contact ratio, low friction coefficient and low normal pressure with a reduced peak value. (5) The plastically deformed asperity, asperity/lubricant interaction and bulk plastic deformation of the strip all contribute to the surface roughness variation of a strip in rolling. The method developed in this study can reveal such variation. A higher lubricant viscosity produces rougher surfaces. It is expected that the new method can be applied to other contact sliding processes involving lubrication and plastic asperity deformation.

Acknowledgments This research was supported by a BAJC project, BA12003. C. H. Wu is financially supported by China CSC and Australia UNSW scholarships.

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point h'i on the deformed rough surface (see Fig. A1(b) ) will be given by 0

hi ¼

hi a

ðA3Þ

Thus, the reference plane of the deformed rough surface can be given by H0 ¼

N N N 1X 1X hi 1 1 X H 0 ¼ h ¼ h ¼ Ni¼1 i Ni¼1 a aNi¼1 i a

ðA4Þ

The corresponding roughness on the deformed rough surface σ' can be given by vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u N u N  u N 2 u1 X 0 u u1 X X 1 h H 1 σ i 0 2 0 t t  σ ¼ ðh  H Þ ¼ ðhi  HÞ2 ¼ ¼ t Ni¼1 i Ni¼1 a a a Ni¼1 a ðA5Þ Consider a control volume with the base area L  W and H, as indicated in Fig. A2. Assume the asperity density on the original rough surface is N. The asperity density N' on the deformed rough surface after the control volume is elongated in the length direction can be given by Fig. A1. The surface topography variation (a) The original surface profile (b)The deformed surface profile..

N0 ¼

NLW N ¼ a L0 W

ðA6Þ

References

Fig. A2. The asperity density variation.

Appendix As indicated in Fig. A1(a), the reference plane of a rough surface H measured with respect to an arbitrary plane within the bulk material can be given by H¼

N 1X h Ni¼1 i

ðA1Þ

where hi is the height of the measured point on a rough surface and N is the total number of the measured points. Therefore, the surface roughness σ can be given by vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u N u1 X σ¼t ðh  HÞ2 ðA2Þ Ni¼1 i

[1] Sutcliffe MPF. Surface asperity deformation in metal forming processes. Int J Mech Sci 1988;30(11):847–68. [2] Wilson WRD, Sheu S. Real area of contact and boundary friction in metal forming. Int J Mech Sci 1988;30(7):475–89. [3] Sheu S, Wilson WRD. Mixed lubrication of strip rolling. STLE Tribol Trans 1994;37(3):483–93. [4] Chang DF, Marsault N, Wilson WRD. Lubrication of strip rolling in the lowspeed mixed regime. Tribol Trans 1996;39(2):407–15. [5] Liu C, Hartley P, Sturgess CEN, Rowe GW. Simulation of the cold rolling of strip using an elastic-plastic finite element technique. Int J Mech Sci 1985;27(11–12):829–39. [6] Hwu YJ, Lenard JG. A finite-element study of flat rolling. J Eng Mater Technol 1988;110(1):22–7. [7] Wu CH, Zhang LC, Li SQ, Jiang ZL, Qu PL. A novel multi-scale statistical characterisation of interface pressure and friction in metal strip rolling. Int J Mech Sci 2014;89:391–402. [8] Greenwood JA, Williamson JB. Contact of nominally flat surfaces. Proc R Soc Lond Ser A – Math Phys Sci 1966;295(1442):300–19. [9] Kioupis LI, Maginn EJ. Impact of molecular architecture on the high-pressure rheology of hydrocarbon fluids. J Phys Chem B 2000;104:7774–83. [10] Jin XL, Zhang LC. A statistical model for material removal prediction in polishing. Wear 2012;274–275:203–11. [11] Jackson RL, Green I. A finite element study of elasto-plastic hemispherical contact against a rigid flat. J Tribol 2005;127(2):343–54. [12] Etsion I, Kligerman Y, Kadin Y. Unloading of an elastic–plastic loaded spherical contact. Int J Solids Struct 2005;42(13):3716–29. [13] ABAQUS, Abaqus 6.10 analysis user’s manual 2010, Dassault Sys. Simulia Corp: USA. [14] Alexander JM. On the theory of rolling. Proc R Soc Lond 1972;326:535–63. [15] Al-Salehi FAR, Firbank TC, Lancaster PR. An experimental determination of the rollpressure distribution in cold rolling. Int J Mech Sci 1973;15(9):693–710. [16] Lenard JG. Measurement of friction in cold flat rolling. J Mater Shap Technol 1991;9(3):171–80.

If the bulk material is elongated in one direction and its length is increased by a factor of a from L to L', the height of the measured

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