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Journal of Materials Processing Technology 213 (2013) 1764– 1771

Contents lists available at SciVerse ScienceDirect

Journal of Materials Processing Technology

jou rn al h om epage : www.elsev ier .com/ locate / jmatprotec

nfluence of roll radius on contact condition and material deformation inkin-pass rolling of steel strip

ideo Kijima ∗

olling and Processing Research Department, JFE Steel Corporation, 1, Kokan-cho, Fukuyama, Hiroshima, Japan

a r t i c l e i n f o

rticle history:eceived 1 October 2012eceived in revised form 25 April 2013ccepted 27 April 2013vailable online 6 May 2013

a b s t r a c t

Skin-pass rolling (or temper rolling) is the final forming step in the production of cold rolled steel sheets.Although a large roll radius compared to the contact length is one of the characteristics of skin-passrolling conditions, numerous studies have been conducted thus far using laboratory mills with smallradius rolls. In this paper, the influence of roll radius on the contact condition and material deformation

eywords:kin-pass rollingemper rollinginite element analysis

in skin-pass rolling is examined and clarified by numerical analysis by an elastic–plastic FEM analysisas well as experimental rolling tests, which were performed to verify the result of the analysis. Somecharacteristics of skin-pass rolling related to pressure distribution, contact condition and material defor-mation are not properly simulated using small radius rolls. Considering characteristic skin-pass rollingconditions, two cases using simplified models, i.e., vertical compression and rolling with a circular, rigidroll, were analyzed.

. Introduction

Skin-pass rolling (or temper rolling), usually following thennealing process, is the final operational step in the productionf cold rolled steel sheets, and has a great influence on mechani-al properties including Lüderband prevention, surface topography,trip flatness and so on. The parameter settings in skin-pass rollingre quite different from those in conventional plate rolling due tohe small reduction (app. 1%), large contact length compared to theheet thickness, large roll radius compared to the contact lengthnd high friction. Considering those conditions, it is expected thataterial deformation will not be uniform in the through-thickness

irection, and the influence of the elastic deformation of the rollsn material deformation will be crucial.

Most of the early literatures on theoretical/numerical modelingf skin-pass rolling simplified either the inhomogeneous materialeformation or the elastic deformation of the rolls. In the formerpproach, the slab method combining precise, non-circular elasticnalysis of work roll deformation (Jortner et al., 1960) was used toalculate the rolling force for certain conditions. Fleck et al. (1992)eveloped a realistic model to describe an aluminum foil rollingrocess which includes a long flat region where the strip thickness

oes not change. Their model had been used to develop modelsf skin-pass rolling. Krimpelstätter et al. (2004) utilized a regular-zed Coulomb friction law to express a sliding region and a sticking

∗ Tel.: +81 84 945 4162; fax: +81 84 945 3840.E-mail address: h-kijima@jfe-steel.co.jp

924-0136/$ – see front matter © 2013 Elsevier B.V. All rights reserved.ttp://dx.doi.org/10.1016/j.jmatprotec.2013.04.011

© 2013 Elsevier B.V. All rights reserved.

region (a no-slip-zone). The thickness distribution inside the rollgap is similar to the results from FE simulation mentioned later,qualitatively. Domanti et al. (1994) analyzed a wet skin-pass rollingcondition, in which the friction coefficient was modeled as around0.05, with the foil rolling model. They showed the same thick-ness distribution pattern in wet skin-pass rolling as in foil rolling.Matsumoto and Shiraishi (2008) separately calculated a skin-passrolling condition with the long flat region, and proposed a model tostabilize the convergence calculation which allows elastic deforma-tion of the strip in the flat region. All those studies focused on andsucceeded in practical rolling force calculations, respectively, byconsidering the non-circular deformation of the work roll, whereasthe mechanism of the material deformation was not clarified.

In the latter approach, the rolls were modeled as circular andrigid with a flattened radius, and inhomogeneous, elastic–plasticdeformations of the material were analyzed by FE model. Yaritaand Itoh (2008) compared the calculated rolling load with the slabmethod and concluded that the both results showed good agree-ment under a small friction and small roll condition. Kijima andBay (2006, 2007) showed that the skin-pass rolling process can bemodeled as plane strain upsetting of a sheet strip with long narrowtools and clarified the basic mechanism of inhomogeneous materialdeformation and the contact condition for the case of high frictionand smooth tool surfaces.

Although a large roll radius compared to the contact length is

one of the characteristics of skin-pass rolling, numerous studieshave been conducted thus far using laboratory mills with smallradius rolls, mainly in order to investigate roughness transfer fromthe rolls to the material. Kimura et al. (2009) showed that the large

H. Kijima / Journal of Materials Processing Technology 213 (2013) 1764– 1771 1765

Large roll

Small roll

R250x 15 0

R50 x 100

0.69t x 80w x 300L

0.69t x 50w x 300 L

rNhd

dprsatctttbi

cesipsucaeped

2

2

s2tmtwit

R

h0/2 z

R/2

Rigid

Elastic

Rotation

Roll centerWork roll

Workpiece

Deformed

Elastic-plastic

for normal penetration as well as tangential sliding (Kijima and

Fig. 1. Laboratory rolling mills.

oll could enlarge the roughness transfer for the same elongation.evertheless, the influence of roll radius on roughness transferas not been discussed in relation to the mechanism of materialeformation in the literature.

Recently, some papers have shown that skin-pass rolling con-itions can be successfully analyzed by commercial FE analysisrograms, even combining elastic–plastic deformation of the mate-ial and elastic deformation of the roll. Sun et al. (2009) numericallyhowed the effects of elongation, friction coefficient, yield stressnd entry/delivery tension on the pressure and shear stress dis-ributions and elastic roll deformation patterns. As a result, theyoncluded that any factor that increases the rolling load may leado elongation of the central flat region, but they did not discusshe mechanism of material deformation. Akashi et al. (2008) inves-igated the jumping phenomenon in wet skin-pass rolling with aright work roll (Imai et al., 1980) and proposed a mechanism for

ts occurrence in relation to the friction coefficient.In the present paper, the influence of roll radius on the contact

ondition and material deformation is investigated numerically andxperimentally in rolling of relatively soft, medium-to-heavy gaugeteel strip with relatively smooth rolls, as a basis for clarifying thenfluence of roll radius on roughness transfer, for which the authorlans to report an experimental investigation in future. Here, theame material, namely, an annealed carbon steel strip, is rolledsing two laboratory mills with different work roll radii. A numeri-al analysis, combining elastic–plastic deformation of the materialnd elastic deformation of the roll, is conducted to simulate thexperimental conditions using commercial FE software. The appro-riateness of two simplified models, i.e., simple compression withlastic roll and skin-pass rolling with a rigid circular roll, is alsoiscussed.

. Experimental apparatus and FEM analysis

.1. Experimental conditions

Experiments were carried out with two laboratory mills. Ashown in Fig. 1, one was a 2Hi mill with a work roll radius of50 mm as an example of the operational size (hereinafter referredo as “large roll”) and a 150 mm barrel width. The other was a 4Hi

ill with a work roll radius of 50 mm as an example of the labora-ory size (hereinafter referred to as “small roll”) and 100 mm barrel

idth. The roll material is high chromium steel, SUJ2 as provided

n JIS G 4805 (similar to AISI E52100), which was hardened andempered to HRC 65. The roll surface was ground to 0.2 �m Ra.

x

Fig. 2. Schematic outline of skin-pass rolling model.

The workpiece is an annealed low carbon steel strip of which dis-continuous yield behavior is practically negligible in order to easeFEM calculations to obtain convergence. Its mechanical propertieswere modeled as described in the next section. The dimensions ofthe workpiece strips were thickness, h0, 0.69 mm, length, 300 mmand width, 80 mm and 50 mm, respectively for the large and smallrolls.

Before rolling, the roll and workpiece surfaces were both care-fully degreased with petroleum benzin to achieve dry frictionconditions.

In order to measure elongation, the workpiece surface wasmarked with two scratched lines in the cross-width direction, witha spacing of 150 mm in the longitudinal direction. The distancein the longitudinal direction was measured before and after theexperiment with a microscope equipped with a micrometer device.

The rolling velocity was 5 m per minute.

2.2. Conditions in FEM analysis

The FEM analysis simulating the experiments described abovewas carried out by the two-dimensional, plane strain, static implicitmethod in Abaqus standard ver.6 to predict the contact conditionand the deformation pattern. Fig. 2 shows a schematic outline of themodel. The upper half of the roll and the workpiece were modeledconsidering the symmetry around the horizontal center line in theworkpiece thickness.

The central part of the roll corresponding to the half radius wasmodeled as rigid to stabilize the analysis and to shorten the simu-lation time. The workpiece length in the FEM model was decidedto be more than 10 times the expected contact length which wasdetermined by preliminary analyses with a shorter model length.Loading was modeled by applying a certain vertical downward dis-placement of the roll on the front tip of the workpiece in the firststep. Thereafter, the roll was rotated around its center, which wasconsidered to be fixed at the position of displacement.

The roll was modeled as an elastic body with Young’s mod-ulus E = 205.8 GPa and Poisson’s ratio � = 0.3. The workpiece wasassumed to be elastic–plastic with Young’s modulus E = 205.8 GPa,Poisson’s ratio � = 0.3 and initial yield stress �0 = 165.8 MPa. Workhardening was determined by connecting the dotted points on thetensile test of the workpiece used in the experiment as shown inFig. 3. The Von Mises criterion was used. Adopting Coulomb’s law, afriction coefficient of 0.3 was used to simulate the dry friction con-dition (Kijima and Bay, 2007). The contact problem between theroll and the workpiece was solved adopting the penalty method

Bay, 2007).The mesh for the workpiece was square and 1/16th the size of

the workpiece thickness. The mesh for the contact surface region of

1766 H. Kijima / Journal of Materials Processing Technology 213 (2013) 1764– 1771

0

100

200

300

400

500

0 0.02 0.04 0.06 0.08 0.1 0.12

Yiel

d st

ress

[MPa

]

tRtqwe

tttw(twbcatt

3

3

sf

Rd

h0/2

x

z

Rigid

Rotation

Roll center

Work roll

WorkpieceElastic-plastic

Equ ivalent plastic strain [ -]

Fig. 3. Work hardening model of material.

he roll was also square and twice the size of the workpiece mesh.ougher rectangular meshes were applied at distances further fromhe contact surface. The element type of the roll was an 8-nodeuadratic plane strain element (CPE8 in Abaqus) and that of theorkpiece was a 4-node bilinear, reduced integration plane strain

lement with hourglass control (CPE4R in Abaqus).Two additional cases were also modeled and analyzed to discuss

he characteristics of the contact condition and material deforma-ion in skin-pass rolling. One was simple vertical compression byhe roll with the same vertical load as in skin-pass rolling, whichas modeled as vertical downward displacement of an elastic roll

Pawelski et al., 1993), as shown in Fig. 4. The second used the condi-ions of a rigid circular roll with a certain flattened radius Rd, whichas decided to approximate the thickness distribution in the roll

ite, as shown in Fig. 5. This case was added to evaluate the resultsalculated with the rigid roll model in the previous papers. All thenalytical conditions except the rotational movement of the roll forhe first case and the rigidity of the roll for the second case werehe same as those mentioned above.

. Results and discussion

.1. Comparison between experiment and analysis

Fig. 6, presented previously in Kijima, 2012b, shows the mea-ured elongation from the experiment and the calculated resultsrom the FEM analysis of the first model set up in Fig. 2,

R

h0/2

x

z

R/2

RigidElastic

Roll centerWork roll

WorkpieceElastic-plastic

Verticaldownwarddisplacement

Fig. 4. Schematic outline of vertical compression model.

Fig. 5. Schematic outline of skin-pass rolling model of rigid circular roll with certainflattened radius.

determined by the nominal strain in the longitudinal direction asthe average value in the through-thickness direction. The calcu-lated values show good agreement with the experimental resultsfor both the large roll and the small roll. This result verifies thefact that the value of the friction coefficient, 0.3, is appropriate forsimulating the dry friction condition, and is the same as in planestrain upsetting with small reduction (Kijima and Bay, 2007). Theappropriate coefficient might be changed in relation to the surfaceroughness of the work roll, lubrication and rolling speed, and isexpected to affect the analytical results in the following sections.

Under these conditions, the relationships between rolling forceand elongation are rectilinear for both the large roll and the smallroll, but quantitatively, the results for the two rolls are quite differ-ent.

3.2. Influence of roll radius on contact condition and materialdeformation

Fig. 7 shows the calculated thickness distribution of the mate-rial and the corresponding elastically deformed shape of the rollcircumference. The zero point on the abscissa corresponds to theposition of the roll center. Approximated circular arcs, which will be

applied to describe the flattened, rigid roll in the additional anal-ysis in the following section, were also shown. Those arcs weredetermined so that the entry and exit points and the minimumthickness of the roll bite are the same as those under the elastic roll

0

1

2

3

4

5

0 1 2 3 4

Rolling force [kN/mm]

Elo

ngat

ion

[%]

R250 R50FEMExperiment

Fig. 6. Relationship between rolling force and elongation (Kijima, 2012b).

H. Kijima / Journal of Materials Processing Technology 213 (2013) 1764– 1771 1767

0.97 5

0.98 0

0.98 5

0.99 0

0.99 5

1.00 0

1.005

-4 -3 -2 -1 0 1 2 3 4

In � Roll ing direction x/h0 [-] � Out

Pla

te th

ickn

ess

h/h 0

[- ]

Elongation

0.44 %1.08 %2.11 %

RollCircular arc

Material

(a) Large roll

0.97 5

0.98 0

0.98 5

0.99 0

0.99 5

1.00 0

1.00 5

-1.5 -1 -0.5 0 0.5

In � Roll ing direction x/h0 [-] � Out

Pla

te th

ickn

ess

h/h 0

[-]

Elon gation

0.32 %1.12 %1.75 %

RollCirc ular arc

Material

(b) S mall rollFe

co

usac

sdifoc

TR

-2-1012345678

-4 -3 -2 -1 0 1 2 3 4

In Roll ing direction x/h0 [-] Out

Pres

sure

p/σ

0S

hear

stre

ss τ

/ σ0

[-]

Elon gation2.11 % 1.08%0.44 % Pressure

Shear stress

(a) Large roll

-1

-0.5

0

0.5

1

1.5

2

2.5

-1.5 -1 -0.5 0 0.5

Pres

sure

p/ σ

0Sh

ear s

tress

τ/ σ

0[-]

In Roll ing direction x/h0 [-] Out

1.75 % 1.12 %0.32 %

Elon gationPress ure

Shea r stress

(b) Small roll

ig. 7. Calculated thickness distribution, corresponding deformed roll circumfer-nce and approximated circular arc, (a) large roll, (b) small roll.

ondition. The radii of the arcs are listed in Table 1 as ratios to theriginal radius.

No intrusion of the roll surface into the material can be seennder any conditions. Characteristic deformation patterns can beeen for the large roll (Sun et al., 2009), as a certain concavity occursround the center of the roll bite. On the other hand, the deformedircumference of the small roll remains circular.

Fig. 8 shows the distribution of the normal pressure and thehear stress on the contact surface, and Fig. 9 shows its ratio asetermined by the shear stress over the normal pressure. This ratio

mplies a nominal friction coefficient. The region for the absolute

riction coefficient value of 0.3, which is same as the input valuef Coulomb friction, is the sliding region, and the area for frictionoefficients <0.3 is the sticking region (Peric and Owen, 1992).

able 1adius of approximated circular arcs.

Roll radius R (mm) 1 2 3

250 (Large roll) Elongation (%) 0.44 1.08 2.11Flattened radius Rd/R (−) 3.26 3.51 3.95

50 (Small roll) Elongation (%) 0.32 1.12 1.75Flattened radius Rd/R (−) 1.53 1.33 1.33

Fig. 8. Normal pressure and shear stress distribution at interface between materialand roll (�0: initial yield stress, 165.8 MPa), (a) large roll, (b) small roll.

The pressure distributions for the large roll are a type of frictionhill, even for the small elongation of 0.44%. The point of peak pres-sure corresponds to the concavity of the elastic deformation of theroll. Sliding regions exist at the entry and exit, and a large stickingregion exists in the center of the roll bite. Inside the sticking region,the shear stress distribution gradually transfers to the oppositedirection. These tendencies coincide with plane strain upsettingwith small reduction as reported by Kijima and Bay (2006, 2007).

On the other hand, the pressure distributions for the small rollare quite different from the above-mentioned friction hill. The peakpressure exists at the entry region. The sliding region at the entryside is quite short, and most of the entry side is in the stickingregion. The absolute values of the pressure and the contact lengthare both much smaller than those of the large roll.

To clarify the difference in the material deformation patterns,the distributions of plastic strain in the rolling direction and equiv-alent plastic strain on the material surface and on the symmetrycenter in the thickness direction are shown in Figs. 10 and 11.

In all cases, surface plastic strain in the rolling directionincreases earlier than plastic strain at the center. The two lines forthe surface and the center cross inside the sticking region. Theirvalues at the exit are almost the same for each case. On the otherhand, the distribution of equivalent plastic strain is quite different,depending on the roll radius. In the case of the large roll, the equiv-alent plastic strain on the surface increases steeply after zero onthe abscissa and becomes much larger at the exit than the equiva-

lent plastic strain at the center and the plastic strain in the rollingdirection. This implies that some additional shear strain occurs,especially in the surface region. In the case of the small roll, the

1768 H. Kijima / Journal of Materials Processing Technology 213 (2013) 1764– 1771

-0.4-0.3-0.2-0.10.00.10.20.30.4

-4 -3 -2 -1 0 1 2 3 4

In Rolling direction x/ h0 [-] Out

[erusserp/raehsfo

oita]

Elongation2.11% 1.08%0.44%

(a) Large roll

-0.4-0.3-0.2-0.1

00.10.20.30.4

-1.5 -1 -0.5 0 0.5

Elongation1.75% 1.12%0.32%

In Rolling direct ion x/ h0 [-] Out

(b) Small roll

Fig. 9. Ratio between shear stress and normal pressure as nominal friction coeffi-cient, (a) large roll, (b) small roll.

0

0.005

0.01

0.015

0.02

0.025

-4 -3 -2 -1 0 1 2 3 4

Pla

stic

stra

inin

rollin

g di

rect

ion

[-]

In Rolling direct ion x/ h0 [-] Out

Elongation

2.11%

1.08%

0.44%

SurfaceCenter

(a) Large rol l

1.75%

1.12%

0.32%

Elongation

SurfaceCenter

0

0.005

0.01

0.015

0.02

0.025

-1.5 -1 -0.5 0 0.5

In Rolling direction x/ h0 [-] Out

Pla

stic

stra

inin

rollin

g di

rect

ion

[- ]

(b) Small roll

Fig. 10. Plastic strain in rolling direction on material surface and central symmetryline, (a) large roll, (b) small roll.

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

-4 -3 -2 -1 0 1 2 3 4

In Rolling direction x/ h0 [-] Out

0.44%

1.08%

2.11%ElongationSurface

Center

(a) Large rol l

0

0.005

0.01

0.015

0.02

0.025

-1.5 -1 -0.5 0 0.5

In Rolling direction x/ h0 [-] Out

1.75%

1.12%

0.32%

Elongation

SurfaceCenter

(b) Small roll

Fig. 11. Equivalent plastic strain on material surface and central symmetry line, (a)large roll, (b) small roll.

distribution of the equivalent plastic strain resembles that of theplastic strain in rolling direction. Therefore, the value at the exit forthe small roll is almost the same on the surface and at the centerfor the each elongation, implying that the additional shear strain ismuch less than that with the large roll and occurs almost uniformlyin the through-thickness direction.

3.3. Inhomogeneity of workpiece deformation

Fig. 12 shows the calculated deformed shape of the originallyvertical cross section before the roll bite for elongation of ca. 1.1%.The lateral position at the axis of symmetry is the reference. Theshape of each line is exaggerated 200 times in the rolling direction(Kijima and Bay, 2006, 2007). In Fig. 12b, the shape in the case ofthe large roll in Fig. 12a is also shown corresponding to the rangeon the abscissa. The shape displays inhomogeneous deformation, incontrast to the hypothesis of the slab method in the classical rollingtheory. Additional shear strain can be recognized as leaning fromthe vertical, straight line.

The differences of equivalent plastic strain in the surface and thecenter region in Fig. 11a and b can be explained from the deformedshape of the vertical cross section in Fig. 12. In the case of the largeroll, the change in the leaning angle can be seen in the surfaceregion, where the equivalent strain is larger than at the center, tothe right and returning back to almost vertical. Actually, the lines ofthe cross section remain almost straight and vertical near the sym-metry center throughout the roll bite. In contrast to this, in the case

of the small roll, the line of the cross section leans almost uniformlyaround the center of the roll bite and returns to a nearly straight lineat the exit. Considering the difference of the range in the abscissa,the deformation of the small roll is smaller than that of the large

H. Kijima / Journal of Materials Processing Technology 213 (2013) 1764– 1771 1769

0

0.1

0.2

0.3

0.4

0.5V

ertic

al d

irect

ion

z/h 0

[-]

-4 -3 -2 -1 10 32 4

In Rolling direct ion x/ h0 [-] Out

Center

Surface

(a) Large roll

(b) Small roll

0

0.1

0.2

0.3

0.4

0.5

Verti

cal d

irect

ion

z/h 0

[-]

-1.5 -1 -0.5 0 0.5

In Rol ling direct ion x/h0 [-] Ou t

Center

Surface

Fvr

rei

p

012345678

-4 -3 -2 -1 0 1 2 3 4

Pres

sure

p/σ

0[-]

In Roll ing direction x/h0 [-] Out

Elon gation2.11 % 1.08 %0.44 %

Skin-passCompressHertzian

ig. 12. Shape of cross section (Kijima, 2012a,b) (h0: initial thickness, 0.69 mm, z:ertical position in thickness from the center), (a) large roll, (b) small roll (gray: largeoll in (a)).

oll. This is considered to be the reason for the smaller difference of

quivalent plastic strain after rolling in the surface and the centern Fig. 11b compared to the large roll in Fig. 11a.

Fig. 13 shows the shear stress distributions inside the work-iece for elongation of ca. 1.1%. Even with an inhomogeneity in the

-1-0.8-0.6-0.4-0.2

00.20.40.60.8

-4 -3 -2 -1 0 1 2 3 4

Shea

r stre

ss τ

x z/σ

0[-]

In Rolling direct ion x/ h0 [-] Out

Surfaceh0/8h0/4

3h0/8

(a) Large roll

h0/2

Surfaceh0/8

3h0/8h0/4

Conta

Center

(c) LineFig. 13. Shear stress distribution inside the workpiec

Fig. 14. Pressure distribution in skin-pass rolling, simple compression and Hertzianelastic contact (Kijima, 2012c).

deformation, the tendency of the distribution is the same through-out the thickness for the large roll, resulting in the pressuredistribution, as shown in Fig. 8, of the typical friction hill. On theother hand, the shear stress distributions are quite different forthe small roll. The point of the peak pressure seems to correspondto the point where the shear stress distribution inside the work-piece crosses the zero line. Although the rolling condition is quitedifferent, a similar pressure distribution was experimentally mea-sured under a condition of small contact length to average thickness(Motomura and Shimamura, 1975).

Considering these contact conditions and material deformationpattern, the experimental skin-pass rolling with small radius rolls

is not appropriate for simulating skin-pass rolling in operation withlarge radius rolls. Actually, the rolling condition with the large rollin this study belongs to the second category (Fleck et al., 1992), inwhich an increased deformation of the roll occurs near the center of

-0.5-0.4-0.3-0.2-0.1

00.10.20.30.4

-1.5 -1 -0.5 0 0.5

In Rolling direct ion x/ h0 [-] Out

Shea

r stre

ss τ

x z/σ

0[-]

Surfaceh0/8h0/4

3h0/8

(b) Small roll

x

z

ct region

s of datae, (a) large roll, (b) small roll, (c) lines of data.

1770 H. Kijima / Journal of Materials Processing Technology 213 (2013) 1764– 1771

012345678

0 1 2 3 4

Rolling / compression f orce [kN/mm]

CompressionHertzian contact

Skin-pass rolling

Peak

pre

ssur

e p/σ

0[-]

tgcidfs

3

3

reolrpaa

tTrbpid

0

0.1

0.2

0.3

0.4

0.5

0-1-2-3-4 1 2 3 4

Rg.El.

Ver

tical

dire

ctio

n z/

h 0[-]

In Rol ling direct ion x/h0 [-] Ou t

Center

Surface

Fig. 17. Shape of cross section for elongation of 1.08% (h0: initial thickness, 0.69 mm,

Fig. 15. Comparison of peak pressure.

he roll bite, while that with the small roll belongs to the first cate-ory, in which the deformed circumference of the work roll remainsircular. The basic mechanisms in workpiece deformation observedn the large roll skin-passing, as regards the contact condition andeformation pattern, especially the large plastic strain in the sur-ace region, are qualitatively similar to plane strain upsetting withmall reduction (Kijima and Bay, 2006, 2007).

.4. Simplification of analysis

.4.1. Simple compressionFig. 14 shows the calculated pressure distribution for skin-pass

olling with the large roll and simple compression with the samelastic roll. The pressure distributions of the Hertzian elastic contactf the roll on a rigid, flat surface are also drawn. The compressionoad and the Hertzian contact load are the same as the skin-passolling conditions. The position on the abscissa for simple com-ression is adjusted so that the left-hand side of the distributionlmost matches that of skin-pass rolling. The peak pressure valuesre compared in Fig. 15.

The pressure distribution in skin-pass rolling can be satisfac-orily approximated by simple compression with the same load.his is a great aid in reducing the burden of simulating skin-passolling conditions just to know the approximate pressure distri-

ution. In addition, the Hertzian theory provides the value of peakressure with certain precision. This is due to the fact that, consider-

ng the characteristics of skin-pass rolling conditions, the pressureistribution is largely dominated by the elastic deformation of the

-2-1012345678

-4 -3 -2 -1 0 1 2 3 4

In Roll ing direction x/h0 [-] Out

Pre

ssur

e p/σ

0S

hear

stre

ss τ

/σ0

[-]

Elon gation2.11 % 1.08 %0.44 % Pr essure

Shear stress

RigidElastic

Fig. 16. Effect of simplification of rigid roll on stress distribution.

z: vertical position in thickness from the center).

roll, as the thickness change in the roll bite is quite small comparedto the contact length in skin-pass rolling. This simplification will beutilized to clarify the influence of roll radius on roughness transfer,for which the author plans to report an experimental investigationin the future. For that purpose, the interactions and local deforma-tion of the asperities will be one of the key parameters which needto be added to the analysis (Dixon and Yuen, 2006).

3.4.2. Rigid roll modelFig. 16 shows the pressure and shear stress distributions for the

rigid roll condition of the large roll shown in Fig. 8, in comparisonwith the elastic roll condition. In Figs. 17 and 18, the deformationpattern of the vertical cross section and the equivalent plastic defor-mation for the elongation of 1.08% are compared respectively asFigs. 12 and 11. The calculated elongation and rolling force coin-cide with the elastic roll in each case. Considering these results,even with the admitted difference in thickness distribution, dueto the characteristic concave shape of the elastic deformation ofthe roll, it is obviously reasonable to simplify the analysis with arigid, circular roll determined to be the same contact length andthe minimum thickness. Actually, when the author calculated var-ious additional conditions, the contact length is dominant for therolling force under a certain elongation condition, and the mini-mum thickness is the important parameter for elongation. In otherwords, the thickness distribution is not crucial to the contact con-dition and material deformation. Hence, the previously calculated

results with a rigid circular roll in the literature could be acceptedqualitatively (Kijima and Bay, 2007).

0

0.00 5

0.01

0.01 5

0.02

-4 -3 -2 -1 0 1 2 3 4

[niarts

ci tsal pt ne lav iuqE

-]

In Roll ing direction x/h0 [-] Out

RigidElastic

Surface

Center

Fig. 18. Difference of equivalent plastic strain by roll model.

sing T

4

dardttucmgdrw

seudatrdamtdrs

firwbTAtradtmnd

H. Kijima / Journal of Materials Proces

. Conclusion

The influence of roll radius on the contact condition and materialeformation in skin-pass rolling was investigated experimentallynd analytically. Experimental skin-pass rolling with two labo-atory mills was conducted with the same workpiece and twoifferent work roll radii under a dry friction condition to investigatehe effect of roll size. In one case, the roll radius corresponded tohe size in operational mills, and in the other, the radius was a sizesed in laboratory mills. FEM analysis, which was performed using aommercial software program and combining elastic–plastic defor-ation of the workpiece and elastic deformation of the roll, showed

ood agreement with the experimental results. Under these con-itions, the relationships between rolling force and elongation areectilinear for both the large roll and the small roll, but the resultsith the two roll sizes are quantitatively quite different.

With the large roll, a friction hill shape of pressure distribution ishown, and a central sticking region and two sliding regions at thentry and exit exist, as estimated by an investigation of plane strainpsetting with small reduction. With the small roll, the contact con-ition is quite different, as are the absolute values of the pressurend the contact length. The pressure peak occurs at the entry andhe sliding region at the exit is quite short. The difference in mate-ial deformation was clarified by the plastic strain distribution andeformation pattern of the vertical cross section. With the large roll,dditional shear strain concentrates in the surface region, whereasore uniform plastic strain occurs in the through-thickness direc-

ion at the exit in the case of the small roll. Considering suchifferences, it can be concluded that the experimental skin-passolling with the small radius roll is not appropriate for simulatingkin-pass rolling in actual operation.

Considering the characteristics of skin-pass rolling, two simpli-cations were calculated. The pressure distribution in skin-passolling can be approximated by simple compression of an elastic rollith the same load. Moreover, the peak pressure can be estimated

y the Hertzian theory of elastic contact on a rigid, flat surface.his simplification is beneficial for reducing the burden of analysis.

rigid, circular roll model, in which the radius is decided so thathe entry and the exit point and the minimum thickness inside theoll bite are the same as those under the elastic roll condition, waslso calculated. Even with the admitted difference in the thicknessistribution, it is quite reasonable to simplify the analysis using

he rigid, circular roll. This implies that the contact length and the

inimum thickness are the important parameters and the thick-ess distribution is not crucial to the contact condition and materialeformation.

echnology 213 (2013) 1764– 1771 1771

Appendix A. Supplementary data

Supplementary data associated with this article can be found, inthe online version, at http://dx.doi.org/10.1016/j.jmatprotec.2013.04.011.

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