[38]predictions of roll force under heavy-reduction hot

Predictions of roll force under heavy-reduction hotrolling using a large-deformation constitutive model

S M Byon1*, S I Kim

2 and Y Lee1

1Rolling Technology and Process Control Research Group, POSCO Technical Research Laboratories, Gyeongbuk,Korea

2Automotive Steel Products Research Group, POSCO Technical Research Laboratories, Kwangyang, Korea

Abstract: A large-deformation constitutive model applicable to the calculation of roll force and torquein heavy-reduction rolling has been presented. The concept of the volume fraction of dynamicallyrecrystallized grains, which depicts the flow stress softening correctly with the level of strain, strainrate and temperature has been newly introduced in the proposed model. The material constantsrequired in the proposed model have been obtained by a series of hot-torsion tests.

A laboratory-scale hot-plate rolling experiment, together with three-dimensional finite elementanalysis coupled with the proposed model, has been performed to investigate the accuracy of theproposed constitutive model. The soundness of the proposed model has been demonstrated througha series of finite element simulations with temperature and reduction changed.

The finite element predictions of roll force based on the proposed model and the experimental resultswas shown to be in fair agreement whereas those based on the Misaka–Yoshimoto model, in whichdynamic recrystallization was not considered, failed to predict the roll force precisely at heavyreduction. The results also revealed that, for a typical reduction, the flow stress softening effect wasnot observed during deformation, whereas the effect was considerable when the material underwentheavy reduction.

Keywords: constitutive model, dynamic recrystallization, finite element analysis, heavy reduction, hotrolling, roll force

NOTATION

fi body forcehi surface traction vectorn outward unit normal vector at roll–workpiece

interfacep hydrostatic pressureT temperatureui, u velocity vector�uui prescribed velocity vectoruDi tool velocity vectorun normal component of uiuDn normal component of uDi�i j Kronecker delta�"" ð"Þ effective strain (one-dimensional strain)_�""�"" ð _""Þ effective strain rate (one-dimensional strain rate)_""i j strain rate tensor

_""0i j deviatoric strain rate tensor� coefficient of Coulomb friction�1, �2 penalty constant�i j stress tensor�0i j deviatoric stress tensor

�n normal component of surface traction�t tangential component of surface traction�� flow stress of the material

1 INTRODUCTION

In hot rolling, there are many cases in which heavyreduction is required to improve productivity and toproduce grain refinement. As a result, local strains inthe deformed body reach much higher than, say, 1.0and the material experiences dynamic recrystallization(DRX) during rolling. Hence, a constitutive modeltaking into account such large strain and DRX hasbeen in great demand by process designers to computeroll force and torque at each stand, which is one of thecrucial factors in designing the roll pass schedule.

1

B22503 # IMechE 2004 Proc. Instn Mech. Engrs Vol. 218 Part B: J. Engineering Manufacture

The MS was received on 17 November 2003 and was accepted afterrevision for publication on 18 February 2004.*Corresponding author: Rolling Technologies and Process ControlResearch Group, POSCO, Technical Research Laboratories, 1 Goedong-Dong, Nam-Gu, Pohang, Gyeongbuk 790-785, Korea.

Paper B22503

Several research groups have attempted to developconstitutive models of materials and suggested their ownformulations. Misaka and Yoshimoto [1] proposed aconstitutive model as a function of temperature, naturalstrain (reduction), strain rate and carbon content tocalculate the mean resistance of deformation, which isan average value of the flow stress. The ranges of applica-tion of this formula are as follows: carbon content, up to1.2 per cent; temperature, 750–1200 8C; strain, up to 0.5;strain rate, 30–200 s�1. The equation might be useful forpractical purposes but the range of strains is quite limitedand DRX was not considered. Johnson and Cook [2]developed a constitutive model which assumes that thedependence of the stress on the strain, strain rate andtemperature can be multiplicatively decomposed intothree separate functions that include five constants to bedetermined by experimental data obtained for a specificmaterial. Hence, the ranges of temperature, strain andstrain rate are not limited but the effect of DRX on flowstress was not included. Meyers et al. [3] modified theJohnson–Cook equation to incorporate DRX at highertemperatures by measuring the temperature-initiatedDRX. However, description of the critical strain whichtriggers DRX was missing.

Laasraoui and Jonas [4] suggested a constitutivemodel to describe DRX expressed by the Avrami equa-tion [5] and used dislocation theory to describe thestrain-hardening behaviour. However, the rate constantof DRX was not expressed systematically in terms ofstrain rate and temperature. Furthermore, the effect ofDRX on metal flow was rarely taken into account,which could affect the accuracy of solution substantially,especially at high strain levels.

In this study, a constitutive model for AISI 316 ispresented which experiences large strain at elevatedtemperatures and subsequently DRX during deforma-tion, on the basis of the Voce work-hardening (WH)model [6]. Three-dimensional finite element analysiscoupled with the proposed constitutive models hasbeen carried out to calculate workpiece deformationand the rolling force. The accuracy of the proposedconstitutive model of the steel has been demonstratedby comparing experimentally measured roll forceswith predicted values when the steel undergoes a largestrain at elevated temperatures. A series of numericalsimulations has then been performed to understandbetter the flow stress behaviour and to investigate theeffect of the distribution of softening on the roll-gapprofile when temperature and reduction in a given passare changed.

2 CONSTITUTIVE MODEL

2.1 Flow stress behaviour

Hot-torsion tests were conducted to investigate theeffects of temperature and strain rate on the flow stressof AISI 316 steel in the temperature range 1000–1200 8C and strain rate range 5� 10�2–5� 100 s�1. Thechemical composition of the steel is presented inTable 1. Torsion test specimens were machined with agauge section of 20mm length and 5mm radius. Thespecimen was heated to the desired temperature usingradiation from halogen lamps inside the furnace andheld for 3min to produce a uniform distribution oftemperature across the specimen. The grain size of speci-mens annealed at 1200 8C for 5min before deformationwas approximately 90 mm.

Among the various results of torsion tests in theexperimental range above, Fig. 1 typically shows themeasured stress–strain curves at different strain rates atconstant temperatures of 1000 and 1100 8C. TheFields–Backofen equation [7] was used to calculate thesurface shear stress from the torque. The general charac-teristics of the flow stress curve are similar to those oftypical characteristics observed in many steels, whichundergo DRX [8, 9]. The flow stress curve exhibits abroad peak that is different from the plateau and reachesa steady state. The change in flow stress is attributed tothe evolution of microstructure through DRX. Thus,the evolution of DRX can be analysed from the slopeof flow curve which manifests the rate of WH. Theonset and finish of DRX and the strain for maximumrate of softening can be determined from the inflectionpoints of WH rate–strain curves or WH rate–stresscurves [10].

2.2 Formulation

2.2.1 The Misaka–Yoshimoto model

The Misaka–Yoshimoto model [1] has generally beenused in the analysis of the hot-rolling process in thesteel-making industry because of conciseness in formula-tion and ease of determining the material-dependentconstants. It is expressed as

�� ¼ c1"n _""m exp

�c2T

�ð1Þ

where c1, c2, n and m are material constants fitted from

Table 1 The chemical composition of AISI 316 stainless steel

Element C Mn Si Cr Ni Mo P S Fe

Amount (%) 0.08 2.00 1.00 17.2 11.8 2.3 0.045 0.030 Balance

2 S M BYON, S I KIM AND Y LEE

Proc. Instn Mech. Engrs Vol. 218 Part B: J. Engineering Manufacture B22503 # IMechE 2004

experimental data. The values of these constants werefound to be 51.7382, 1530.87, 0.2508 and 0.0525 respec-tively through extensive tests. The Misaka–Yoshimotomodel, however, does not take into account DRX andtherefore cannot depict the flow stress softening whenthe material undergoes a large strain at high tempera-tures.

2.2.2 Proposed model

In this study, a constitutive model for predicting soften-ing of AISI 316 is proposed by introducing a functiondescribing the volume fraction of dynamically recrystal-lized grains into the Voce WH model [6].

The important metallurgical phenomena occuringduring hot deformation of steel are WH, dynamicrecovery (DRV) and DRX. Thus, the flow stress curvecan be divided into the regions of WHþDRV and

DRX, as shown schematically in Figs 2a and b. Duringdeformation, the flow stress decreases from the peakstress �p to the steady state stress �ss with increasingstrain. "c represents the critical strain for initiatingDRX. The peak strain "p plays the role of judgingwhether DRX is active or not. In this study, it is pro-posed that the flow stress behaviour �� can be describedby the following two functions depending on strain:

�� ¼�ðWHþDRVÞ

�ðWHþDRVÞ � �DRX

at " < "p

at " > "p

(ð2Þ

In what follows, the procedure for setting up the twofunctions is described in detail.

For the WHþDRV region, the generalized Voceequation has been proposed as

� ¼ �0 þ ð�p � �0Þ½1� expð�C"Þ�m ð3Þ

Fig. 1 Representative flow stress curves obtained from hot-torsion test of AISI 316 stainless steel under variousstrain rates at (a) 1000 8C and (b) 1100 8C

Fig. 2 (a) Illustration of the stress–strain curve for steel whichdisplays DRX and (b) the measured data and stress–strain curve predicted on the basis of the proposed

model

PREDICTIONS OF ROLL FORCE UNDER HEAVY-REDUCTION HOT ROLLING 3


Figure 1 shows that the initial stress �0 is very smallrelative to peak stress �p. Thus, equation (3) wasexpressed as

�ðWHþDRVÞ ¼ �p½1� expð�C"Þ�m ð4Þ

The coefficient C and WH exponent m are dependent onstrain rate and temperature. Thus, C and m are consid-ered to be a function of the dimensionless parameterZ=A represented by the equation [8, 9]

Z

A¼ _""

Aexp

�Q

RT

�ð5Þ

where R, A and Q denote the gas constant, material-dependent constant and activation energy for defor-mation respectively. The peak stress �p is dependenton the non-dimensional parameter Z/A with theform

�p ¼ �þ � ln

�Z

A

�ð6Þ

where � and � are material constants.

2.2.3 Flow stress softening due to dynamicrecrystallization

For the region of DRX, the drop in flow stress can beexpressed by the equation

�DRX ¼ ð�p � �ssÞ�XDRX � X"p

1� X"p

�ð7Þ

where X"p represents the volume fraction of DRX atpeak strain, "p (the corresponding stress is �p), whileXDRX, which is the volume fraction of dynamically

recrystallized grains, can be expressed as

XDRX ¼ 1� exp

��"� "c"�

�m0�ð8Þ

Equation (8), which is a modified form based on theAvrami’s equation [5], means that XDRX depends onthe strain for maximum softening rate, "�, as well asthe coefficient m0, the critical strain "c and appliedstrain ". These parameters "�, "c and m0 are also depen-dent on the deformation conditions expressed by thedimensionless parameter Z/A [see equation (5)]. Sincethe DRX is a continuous process of deformation,nucleation of grains and subsequent migration of grainboundaries, XDRX increases with increasing strain. Asthe strain increases, XDRX reaches a constant value, 100per cent. At XDRX ¼ 100 per cent, the flow stressreaches a steady state, denoted by �ss. The steady statestress �ss is dependent on the non-dimensional parameterZ/A:

�ss ¼ � þ � ln

�Z

A

�ð9Þ

where � and � are material constants.Based on the stress–strain curve in Fig. 1, the various

material constants in equations (4)–(9) were obtained.The results are summarized in Table 2.

3 FINITE ELEMENT SIMULATION

A three-dimensional Eulerian finite element model wasadopted for the analysis of the steady state rigid visco-plastic deformation occurring in the roll gap. A shortoutline of the numerical analysis is summarized in

Table 2 The material parameters of AISI 316 stainless steel obtained from torsion tests

Items Equations

Zener–Hollomon parameter Z=A ¼ ½ _"" expðQ=RTÞ�=ð8:14� 1014ÞQ ¼ 414 000 J=moly, R ¼ 8:314

Volume fraction of DRX XDRX ¼ 1� exp½�fð"� "cÞ="�gm0�

"c ¼ 0:24ðZ=AÞ0:057"p ¼ 0:52ðZ=AÞ0:092"� ¼ 0:73ðZ=AÞ0:124m0 ¼ 1:258ðZ=AÞ�0:04

Flow stress �ðWHþDRVÞ ¼ �p½1� expð�C"Þ�m

�DRX ¼� ð�p � �ssÞ½ðXDRX � X"p Þ=ð1� X"p Þ�,0,

" > "p

" < "p

�� ¼ �ðWHþDRVÞ � �DRX

C ¼ 4:86ðZ=AÞ�0:028

m ¼ 0:177ðZ=AÞ0:0736�p ¼ 105þ 17:95 lnðZ=AÞ�ss ¼ 82:5þ 12:53 lnðZ=AÞ

y According to McQueen and Ryan [11], it was 460 kJ/mol.



Table 3 and more details may be found in reference [13].Among these equations, the constitutive equation ischanged with the material to be used in the analysisand generally represented by a function dependent onstrain, strain rate and temperature. In this investigation,the specific form was suggested by equations (2)–(9).

A steady state approach inherently requires a schemefor the prediction of the boundary of the analysisdomain, or free surface. In this paper, an iterativescheme in which the free surface is corrected on thebasis of the newly traced streamlines is adopted.

3.1 Streamline tracing

A neighbouring position �iþ1 on a streamline which issufficiently close to �i on the upstream part of thesame streamline can be given approximately by

�iþ1 ¼ �i þ vi �ti ð10Þ

�ti ¼ ��s

visð11Þ

where vi denotes the local velocity vector at �i defined innormalized elemental coordinates & � � �, vis denotesits magnitude and ��s denotes a marching step size.Using equations (10) and (11), starting from a givenpoint at the inlet cross-section, an entire streamlinemay be traced by sequentially predicting the final posi-tions in all the elements through which the streamlinespass.

3.2 Free-surface correction

The free surface generated by the newly traced stream-lines often violates the fundamental requirements thata portion of the free surface should be in contact withthe roll and that the free surface should not intersectthe roll. In order to meet these requirements, a projectionscheme is employed in which the primitive new stream-lines are iteratively moved to the direction minimizingthe distance between the roll surface position on theold streamline and the corresponding new contactposition to be corrected. As a result, the new correctedsurface streamline, or free surface, is produced with a

Table 3 Boundary value problem and variational equation for rigid viscoplastic deformation

Boundary value problem Find the velocity field ui satisfying the following:

Equilibrium equation �i j, j þ fi ¼ 0

Flow rule �i j ¼ �p�i j þ �0i j

�0i j ¼2��

3 _�""�""_""0i j

Constitutive equation �� ¼ f ð�"", _�""�"",TÞ y

Incompressibility ui;i ¼ 0

Boundary conditions �i jnj ¼ hi on hiui ¼ �uui on ui�n ¼ ��2ðun � uDn Þ on c

�t ¼ ��ngð�uÞ on c

where g is the special function to be selected so as to deal with bothsticking and sliding frictions [12].

Variational equation Among all the velocity fields ui satisfying prescribed boundary conditions,find ui satisfying the following variational equation for arbitrary functions!i (that vanishes on ui

):ð��0i j!

0i j d� �

ð��1 _""kk!ii d� �

ð�fi!i d� �

Xi

ð hi

hi!i d

�ð c

��2ðun � uDn Þ!n d �ð c

��2ðun � uDn Þgð�uÞ!t d ¼ 0

where !i j ¼ ð!i, j þ !j, iÞ=2 and !0i j ¼ !i j � ð!kk=3Þ�i j

Finite element approximation ui and !i are approximated by

ui ¼ NiLVL

!i ¼ NiLWL

where NiL are the finite element basis functions, and VL andWL denote uiand !i respectively evaluated at nodal point L. Substituting theseequations into the variation equation above results in a set of non-linearalgebraic equations that may be solved for VL either by the direct iterationmethod or by the Newton–Rhaphson method.

y T denotes the deforming temperature reflecting deformation heating.



portion in contact with the roll. Details regarding theseschemes may be found in reference [14].

4 EXPERIMENT

4.1 Rolling equipment

As shown in Fig. 3, a single-stand two-high laboratorymill was employed, driven by a 1200 kW constant-torque d.c. motor with a maximum rolling speed of150m/min. Ductile casting iron rolls were used, with a720mm maximum diameter and a 500mm face width.A box-type furnace with a maximum working tempera-ture of 1450 8C was employed to heat the specimens tothe desired rolling temperature.

4.2 Specimen preparation

The materials were obtained in the form of as-cast billet160mm square. The specimens to be rolled were cut andmachined into rectangular bars 30mm thick, 150mmwide and 300mm long. To study the effect of the ratioof the specimen size to the work roll diameter, specimens20mm thick, 150mm wide and 300mm long were alsomachined.

4.3 Experimental procedure

In order to measure the rolling temperature of theworkpiece, a thermocouple (type K) of 1.6mm diameterwas embedded in a hole 75mm deep drilled in themiddle side of the specimen (see Fig. 3). The specimenswere soaked at 1160 8C for 40min to ensure a homo-geneous temperature distribution. When they weretaken out of the furnace and the centre temperature ofspecimens reached 1100 8C (target temperature), thetests started.

5 RESULTS AND DISCUSSION

5.1 Comparison between the proposed and the Misaka–

Yoshimoto constitutive model

Figure 4 shows that the stress–strain curves predictedusing the proposed constitutive model are in agreementwith the experimentally measured curves. Thus, it can bededuced that the approach to obtain a constitutivemodel applicable to large strain ranges was successfuland this proposed equation might have the potential tobe used for the analysis of the hot-rolling process wherea more precise calculation of stress and subsequentlyroll force decrement due to DRX is important. However,it is noted that there are problems in application of themodel to the rolling process due to the difference betweenthe deformation conditions for rolling and torsion. Never-theless, many research groups have successfully adoptedthe torsion-based DRX model to analyse the DRX ofmaterials subject to rolling [11, 15, 16].

Figure 5 shows the measured and predicted flow stresscurves when the two different constitutive models areused for prediction. The flow stress curves calculatedusing the proposed model agree with the measuredcurves with some error, while the curves obtained fromthe Misaka–Yoshimoto model are available in thesmall-strain range only.

5.2 Roll forces

To investigate the effect of the constitutive model on theprediction of the roll force, the proposed model andthe Misaka–Yoshimoto model were incorporated intothe finite element program. The finite element simula-tions were conducted under the same condition asthose under which the rolling experiment was carriedout. The experimental conditions are described in theTable 4.

Fig. 3 Schematic diagram of a single-stand two-high laboratory mill



Figure 6 illustrates the predicted and experimentallymeasured roll forces when the entrance thicknesses are30 and 20mm. While the differences between the pro-posed model-based roll forces and the measured dataare insignificant in the total range of reduction, theresults from the Misaka–Yoshimoto model have beenshown to be up to 21 per cent higher. In heavy reduction,the drop in the accuracy of the Misaka–Yoshimotomodel compared with the proposed model is attributedto the absence of the model of flow stress softening dueto DRX.

5.3 Effect of rolling temperature on roll force, torque and

deformation

The effect of the rolling temperature on roll force, torqueand deformation is investigated for four reduction ratiosof 40, 50, 60 and 70 per cent. The process conditions aresummarized in Table 5.

Figure 7 shows that the roll force and torque increasewith an increase in reduction ratios and a decrease inrolling temperature for both the Misaka–Yoshimotoand proposed constitutive model. While the differencesdue to the two constitutive models are negligible for 40per cent reduction, the results from the Misaka–Yoshi-moto model have been shown to be up to 32.2 per cent(1100 8C) higher in the roll force and up to 19.6 percent (1100 8C) higher in the torque compared with theproposed model when the reduction is 70 per cent. It isalso noted that these deviations between two modelsdecrease as the rolling temperature decreases, up to21.7 per cent (900 8C) for the roll force and up to 12.1per cent (900 8C) for the roll torque. This indicates thatthe effect of recrystallization on the flow stress is moreconsiderable at higher rolling temperatures.

Figures 8, 9 and 10 illustrate the distributions of effec-tive strain �"", peak strain "p and flow stress softening�DRX for the two reduction ratios of 40 and 70 per centas well as the two rolling temperatures of 1100 and900 8C. These figures indicate that flow stress softeningoccurs in the region where the magnitude of effectivestrain is larger than that of peak strain, as seen in

Fig. 4 The measured data and predicted stress–strain curveson the basis of the proposed model under various

strain rates at (a) 1000 8C and (b) 1100 8C

Fig. 5 Comparison between two constitutive models together

with measured data at 1000 8C and 0.5 s�1 and at1100 8C and 0.5 s�1

Table 4 The process conditions under which the experimentalrolling was carried out

EntranceEntrance rolling Roll Rollthickness Reduction temperature diameter speed(mm) (%) (8C) (mm) (m/min)

30 30 1040 720 12040 104050 108865 1049

20 30 1008 720 12040 103450 1044



equation (2). They also explain the reason why a substan-tial improvement could be made regarding the predictionof roll force as well as roll torque when the proposed con-stitutive model is used instead of the Misaka–Yoshimotomodel.

The effective strain distributions are shown in Fig. 8 atdifferent rolling temperatures and reduction ratios. Itdemonstrates that the effective strain increases withincrease in the reduction ratio and with decrease inthe rolling temperature. At a typical reduction ratio,the calculated maximum strain is less than one over the

entire domain of the analysis, indicating that recrystalli-zation is unlikely whereas, at a heavy-reduction ratio,the maximum effective strain is about 4–10 at the outerpart of the delivery cross-section and about up to 2 atthe mid-plane, indicating that DRX would be completeat the roll gap.

To judge quantitatively whether recrystallization mayoccur, however, the peak strain "p must be considered, asshown in Fig. 9. Regardless of reduction and tempera-ture, it is seen that peak strain always occurs in the rollbite region and around the neutral cross-sectional

Fig. 6 Comparison of measured and predicted roll force for AISI 316 stainless steel when the entry thick-nesses are (a) 30mm and (b) 20mm



Table 5 Process conditions used in simulations (roll speed, 120m/min; roll diameter, 720mm)

Simulationcase number Constitutive model

Entrancethickness (mm)

Rollingtemperature (8C)

Reductionratio (%)

Frictioncoefficient

1 (2) Proposed model (Misaka–Yoshimoto model) 30 1100 40 0.3104�

3 (4) 505 (6) 607 (8) 70

9 (10) Proposed model (Misaka–Yoshimoto model) 30 900 40 0.3904�

11 (12) 5013 (14) 6015 (16) 70

� The equivalent friction coefficient dependent on the rolling temperature and velocity. The formula for the equivalent friction coefficientis expressed as [17]

� ¼ cð1:05� 0:0005T � 0:056vÞwhere c is the ratio of the coefficient of slipping to gripping friction, which is taken to be equal to 0.8 to fit the present case of rolling.

Fig. 7 Variations in (a) roll force and (b) roll torque with reduction and rolling temperature



Fig. 8 Effective strain distributions at rolling temperatures of (a) 1100 8C and (b) 900 8C

Fig. 9 Peak strain distributions at rolling temperatures of (a) 1100 8C (b) 900 8C



plane. Comparison between the effective strain distribu-tion and the peak strain distribution can be made toinvestigate the region where the effective strain exceedsthe peak strain. At the commonly used reduction (40per cent), no such regions are found while, at thehigher reduction ratios, regions satisfying this conditionoccur over most of the roll gap and delivery side, indicat-ing that flow stress softening occurs.

Figure 10 shows the flow stress softening �DRX distri-butions resulting from the two cases of reduction androlling temperature. At the commonly used reduction,as expected, there is no observable flow stress softeningregion in roll gap, which affects the solution accuracyof roll force and torque whereas, at heavy reductions,it is shown that �DRX in the roll gap is considerable,which demonstrates the importance of incorporatingthe DRX effect in predictions of the roll force andtorque in heavy-reduction hot rolling.

From these distributions, it can be expected that thecross-section of the plate might have a crown (orconvex shape) because the maximum stress exists in themiddle of width. If the flow stress softening effect is notaccounted for in the prediction of contact stress at rollgap, it overestimates the crown shape of the platecross-section, leading to a poor rolling set-up. Therefore,

it is suggested that the constitutive model with the flowstress softening effect is essential for an appropriate set-up of the bender force and roll gap to gain the desiredcrown as well as thickness of plate.

6 CONCLUDING REMARKS

In this paper, a large-strain constitutive model of AISI316 steel for the prediction of roll force and torque inheavy-reduction hot rolling is presented. In the model,the strong correlations at heavy reduction betweenDRX, strain, strain rate and temperature have beendealt with in a more rigorous manner. To verify themodel, the finite element predictions incorporated inthe constitutive model have been compared with theexperimentally measured data. The results showedgood accuracy in roll force, regardless of reduction.The model has then been applied to study the effect ofrolling temperature as well as reduction.

It has been demonstrated that the flow stress softeningof the workpiece at the roll gap is the most criticalparameter that influences the solution accuracy for theroll force at heavy reduction. The proposed model inthe present investigation might serve as a stepping

Fig. 10 Flow stress softening distributions due to recrystallization at rolling temperatures at (a) 1100 8C and

(b) 900 8C



stone towards developing an integrated analysis for rollprofile design and roll-gap set-up in heavy-reductionplate rolling.

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[38]predictions of roll force under heavy-reduction hot

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