1. a Model for Evaluating Thermo-mechanical Stresses

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J Eng Math (2012) 72:73–85

DOI 10.1007/s10665-011-9462-8

A model for evaluating thermo-mechanical stresses

within work-rolls in hot-strip rolling

A. Sonboli · S. Serajzadeh

Received: 14 February 2010 / Accepted: 15 February 2011 / Published online: 2 March 2011

© Springer Science+Business Media B.V. 2011

Abstract A mathematical model is proposed for the determination of the thermo-mechanical stresses in work-

rolls during hot-strip rolling. The model describes the evolution of the temperature fields in the work-roll and in the

work-piece, and in the latter the plastic heat generation is taken into account. The frictional heat generated on the

contact surface is also included. The problem is treated in two steps. First, a numerical method is developed for

the analysis of the coupled thermal problem of the temperature distributions within the work-roll and metal being

rolled, while an admissible velocity field is employed to estimate the heat of deformation. In the second step, the

finite-element method is employed to determine the resulting thermo-mechanical stresses within the work-rolls. A

slab method is used to obtain the mechanical boundary conditions in the contact region between the work-piece

and the work-roll. The model takes into account the effects of different process parameters such as the initial tem-

perature of the strip, reduction, and the rolling speed on the thermo-mechanical stresses and their distributions. The

numerical predictions are compared with experimental results.

Keywords Hot rolling · Mathematical modelling · Thermo-mechanical stress · Transient temperature distribution

1 Introduction

Hot rolling of metals and alloys is an important industrial process for the manufacturing of plates and sheets. In

this process, the behavior of metal being rolled and the working rolls are always important to mill designers for

designing an appropriate rolling layout. In this regard, mathematical modeling of hot-rolling operations has been

widely employed as a valuable tool in predicting metal/work-roll behavior and consequently many publications

have been devoted to the modeling and simulation of various aspects of hot rolling. For instance, Tseng et al. [1]have determined the thermal behavior of metal and work-rolls in hot-strip rolling of aluminum alloys. Atack and

Robinson [2] have studied temperature variations in work-rolls for the control of thermal cambering in a single

reversing stand. Panjkovic [3] has proposed a model to evaluate the temperature distribution in a finishing mill

for strip-rolling processes. Tudball and Brown [4] have employed a three-dimensional transient model to predict

temperature variations in hot rolling of steels. Colas [5] has developed a thermal-microstructural model for hot

rolling of carbon steels using a two-dimensional finite-difference technique. Huang et al. [ 6] have predicted the

A. Sonboli · S. Serajzadeh (B)

Department of Materials Science and Engineering, Sharif University of Technology, Azadi Ave., Tehran, Iran

e-mail: [email protected]

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74 A. Sonboli, S. Serajzadeh

surface temperature of work-rolls in a steel-rolling mill. Guo [7] has presented a two-dimensional transient model

for calculating the temperature distribution in work-rolls. Hsu et al. [8] have developed a model to estimate surface

thermal behavior of work-rolls utilizing a finite-difference method and an inverse analysis. Yuen [9] has developed

an analytical model to determine the temperature distribution in a fast-rotating cylinder that may be used in hot-

rolling mills for the determination of the temperature field in the working rolls. Troeder et al. [10] have predicted

temperature variations and resulting thermal stresses in work-rolls using an analytical method. Lai et al. [11] have

presented a two-dimensional thermo-elastic model to determine transient thermal stresses within work-rolls. Sunet al. [12] have presented a thermo-mechanical model to consider the thermo-mechanical response of work-rolls in

a hot-strip rolling process. They employed a two-dimensional finite-element method for the determination of the

temperature and strain fields within work-rolls and rolling metal. Chang [13] have presented a thermo-mechanical

model to predict the work-roll temperature profile and the thermal stresses using a finite-difference method. Lee

et al. [14] have employed a three-dimensional finite-element method under unsteady-state rolling conditions to

predict the thermal history within work-rolls during deformation as well as during the idling cycles. Cavaliere [15]

developed an Eulerian rigid-viscoplastic model for metal being rolled as well as a Lagrangian elastic model of the

work-roll deformation. Li et al. [16] have developed a thermal-mechanical microstructural model to incorporate

various phenomena during hot-strip rolling of steels. Arif et al. [17] have developed a thermo-mechanical model to

evaluate stresses and strains within working rolls in the process of cold strip rolling. Corral et al. [18] have predicted

the thermo-elastic response of the working rolls in hot-strip rolling of steels. Sonboli and Serajzadeh [19] developeda mathematical model to evaluate thermal stresses during single- and multi-pass strip hot rolling considering idling

roll revolutions.

Regarding publicationsdealing with themodeling of strip-rolling processes, although severalof these concentrate

on the prediction of work-roll deformation and stresses, particularly thermal stresses, this issue needs further study

for a better understanding of the thermo-mechanical behavior of work-rolls. In the present paper, the temperature

variations and deformation of work-rolls are determined by employing a finite-element analysis. The thermal rela-

tionship between the work-rolls and the rolling metal are considered in the model through simultaneous calculation

of temperature fields in the work-roll and rolling metal. In order to implement the effect of the heat generated by

plastic deformations, an admissible velocity field is used in solving the heat-conduction problem of the strip; also,

to determine mechanical boundary conditions on the work-roll, the slab method is employed. The proposed model

considers the effects of different process parameters such as reduction, rolling speed and initial strip temperature.Further, the model may be utilized in a single and a multi-pass rolling schedule as well as different roll-cooling

layouts.

2 Mathematical model

The thermo-mechanical modeling of a work-roll can be divided into two parts including thermal and mechanical

models. In the thermal model, the temperature distribution within the work-roll and the rolling metal are calculated

at the same time and then the results of the thermal model are implemented in the mechanical part to predict the

thermo-mechanical stresses developed within the works-rolls.

2.1 Heat-conduction model

Heat flows from the hot metal being rolled to the working rolls owing to a high interfacial heat-transfer coeffi-

cient, as well as the high temperature gradients that exist between the metal and the work-rolls. As a result, the

temperature of the surface layer of the work-roll increases when the work-rolls and the metal come in contact.

However, after exiting the deformation zone, the rolls are cooled by water sprays that decrease the temperature of

the outer layer. These temperature variations may take place in successive revolutions leading to the formation of

thermal stresses. On the other hand, the roll pressure causes deformation and mechanical stresses within the work

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A model for evaluating thermo-mechanical stresses within work-rolls in hot-strip rolling 75

rolls. Therefore, thermo-mechanical stresses are developed in working rolls during hot-rolling operations. In order

to assess the thermo-mechanical stresses, mechanical boundary conditions acting on the work-roll/metal interface

as well as transient temperature variations should be predicted. In this regard, it is vital to solve the governing

heat-conduction equations for both the strip and the work-roll simultaneously. The basic heat-transfer equation in

cylindrical coordinates for a work-roll can be written as Eq. 1 below [20, Chap. 1], ignoring heat conduction along

the roll axis, i.e., the z-direction; also the thermal conduction at the top and bottom sides of the strip has been taken

identical;11

r

∂

∂r

k r r

∂ T

∂r

+

1

r 2∂

∂θ

k r

∂ T

∂θ

= ρr cr

∂ T

∂ t . (1)

Here ρr , cr , and k r are the density, the specific heat, and the thermal conductivity of the work-roll, respectively. The

initial temperature of the work-rolls is taken as 25◦C and the following boundary conditions are employed on the

work-roll, assuming that half of the heat generated by friction flows into the work-roll.

−k r

∂ T

∂r |r = R = hc(T − T sur) −

1

2ητ fricv (deformation zone),

−k r

∂ T

∂r |r = R = hc(T − T sur) (other zones), (2)

−k r

∂ T

∂r |r =r ∗ = 0;here T sur is the surrounding temperature while in the deformation zone it is the strip-surface temperature. The

term “hc” represents the convection heat-transfer coefficient that takes different values in successive thermal zones

such as the deformation zone, the air cooling zone and the water-spray zone. Note that in (2) τ fric is the friction

shear stress, v is the relative velocity, and η is the conversion factor. The term r ∗ is the radius of the work-roll

in which temperature variations are insignificant. Accordingly, it can be assumed that only the outer layer of the

work-rolls with the thickness of R − r ∗ experiences considerable temperature changes. The thickness of this layer

can be estimated by using the Peclet number calculated as Pe = R2ωρr cr /k r , where ω is the angular velocity of

the work-roll and “ R” is work-roll radius [1]. The above heat-conduction problem can be solved by a variational

method and the finite-element method; details of the numerical procedures can be found in [21, Chap. 7]. It should

be noted that there is a strong thermal relationship between the work-roll and the strip and, therefore, it is necessary

to calculate the temperature distribution of the strip at the same time. In this regard, the process of hot rolling maybe divided into consecutive sections consisting of the interstand section and the deformation zone. The governing

heat conduction equation may be employed as follows [22].

∂

∂ x

k s

∂ T

∂ x

+

∂

∂ y

k s

∂ T

∂ y

+ q̇ = ρs cs

∂ T

∂ t ; (3)

here “ x ”and“ y” denote longitude and thickness directions, respectively. Further ρs, cs, and k s represent the density,

the specific heat, and the thermal conductivity of the metal being rolled, respectively, and q̇ is the rate of heat of

deformation and is equal to zero in the interstand and descaling zones and for the deformation zone this term can

be expressed as follows:

q̇ = 0.95 σ̄ ˙̄ε; (4)

here σ̄ and ˙̄ε are the effective stress and effective strain rate, respectively. The following boundary conditions have

been employed for the strip. It should be noted that radiation boundary conditions are assumed in the interstand

zone where the slab is being cooled in the air. This boundary condition is particularly important for hot rolling of

steels where radiation heat transfer is important at high temperatures, i.e., above 700◦C while this may be ignored

for hot rolling of aluminum alloys taking place at relatively low temperatures, i.e., temperatures less than 500◦C.

−k s∂ T

∂n= hc(T − T r ) −

1

2ητ fricv (deformation zone), (5)

−k s∂ T

∂n= hc(T − T sur) + λσ B

T 4 − T 4sur

(interstand zone) . (6)

1 See Appendix list of symbols.

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Here “n” represents the direction normal to the strip surface, λ is the emissivity and σ B is Stephan-Boltzmann

constant; T r is the work-roll surface temperature. An iterative scheme has been used to simultaneously determine

temperature fields in the work-roll and in the strip at the same time.

2.1.1 Derivation of the velocity field

It should be noted that in (4), the rate of heat of deformation has been calculated using an admissible velocity

field proposed by Tselikov [23, pp. 120–127] where for the derivation of the velocity field, the principle of volume

constancy for an incompressible material is employed. Accordingly, it implies that the following equation should

be satisfied:

+h/2 −h/2

v x d y = vnx hn = . (7)

Here h and hn are the thicknesses at an arbitrary point in the deformation zone and at the neutral point, respectively,

represents the volume rate passing through each position of the deformation zone, and vnx expresses the velocity

component along the rolling direction at the neutral point assuming a uniform velocity distribution at this position.In the next stage, in order to estimate the velocity field, a parabolic velocity distribution along y-direction was taken

based on the assumption made by Tselikov [23, p. 124] while the velocity boundary conditions were defined under

fully sticking friction conditions as follows:

v x | y=0 = v xc, (8)

v x | y=h x = v xs = vr cos ϕ; (9)

here vr is the velocity of the work-roll. v xc and v xs are velocities at the center and the surface, respectively. Note

that the other velocity component, namely the velocity along the y-direction, can be determined by employing the

rule of volume constancy. Details of the derivation of the velocity field may be found in a previous work by the

authors [19].

2.2 Deformation model for the work-roll

The work-roll deformation is predicted by employing the predicted temperature variations with imposing the

mechanical boundary conditions in the deformation zone. In order to analyze the thermal stresses, it has been

assumed that strains in the axial direction may be ignored because the roll length is very long compared to the

depth of the surface layer where severe temperature variations take place. Thus, the work-roll deformation may be

considered as a two-dimensional thermo-mechanical problem. Regarding the minimum potential-energy principle,

the displacement field can be determined by the following minimization:

δπ =

δε T σ dV −

δuT F dS = 0; (10)

here π represents the energy functional, F denotes surface traction, σ and ε are stress and strain vectors, respectively,

which are expressed as follows:

σ = {σ rr σ θ θ σ r θ }T , ε = {εrr εθ θ εr θ }

T . (11)

Note that in (9) the effect of body force, i.e., mass effect of the work-roll, has been ignored and Hooke’s constitutive

equations were employed in this equation as follows:

σ = D(ε − ε0) + σ 0, (12)

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R

σρ

θ

h0 /2

h1 /2h/2

φ

ph+d(ph) ph

qRdφ

τRdφ

Work-roll

Roll bite angle

y

x

τRdφ

qRdφ

Fig. 1 Equilibrium diagram acting on a thin slice in the deformation zone

where σ 0 is the residual-stress vector which, for the case of the present problem, describes the residual stresses

produced in the previous revolutions and D is the matrix of elastic constants. ε0 is the vector of thermal strains. By

using the constitutive equations, it is possible to rewrite (9) in a finite-element formulation as follows [24, Chap. 5].

K eae = f eth + f est − f ere; (13)

here “ae” is the nodal-displacement vector, K e represents the stiffness matrix, f eth, f ere and f est are nodal force

vectors relating to thermal stress, surface traction and residual stress respectively. The first nodal force vector is

determined based on the predicted temperature distribution within the work-roll. To determine the surface traction

nodal vector it is necessary to know the surface-traction distribution. In this regard, the slab method is employed to

assess the surface-traction vectors in the deformation zone acting on the contact region between work-roll/metal.

Considering the equilibrium on a thin slice in the deformation zone as displayed in Fig. 1, one may derive thefollowing differential equation, assuming sticking friction conditions [25, Sect. 7.9].

d

dϕ(hp) = 2 R(q sin ϕ ± τ cos ϕ); (14)

here “ R” is the effective work-roll radius, “h” the thickness, “ϕ” the contact angle as shownin Fig. 2, “q” the normal

pressure and τ the shear-flow stress. In addition, the following equation can be derived from the yield condition

describing the relationship between the normal and the horizontal pressures:

q − p =π

2τ. (15)

Substituting Eq. 15 in Eq. 14, one can estimate the variation of the normal pressure by the following Eq. [25, Sect.

7.9]:

dq

2τ =

π

4

dτ

τ +

dh

h

±

R cos ϕdϕ

h1 + 2 R(1 − cos ϕ). (16)

A numerical method is employed to solve the above problem while the boundary conditions at both entry and exit

positions can be expressed as:

p = 0, q =π τ

2. (17)

Note that in the above problem the upper sign holds for the exit region and the lower sign for the entry zone.

Thus, two separate differential equations must be solved, namely one for the entry region and the other one for

the exit region by use of the boundary condition given in Eq. 17. The results are then used in the thermo-elastic

finite-element analysis as the surface traction to determine the thermo-mechanical stresses within the work-roll.

123




Fig. 2 The mesh system

used in the model

2.3 Solution procedure

In this work, as discussed above, two separate models are coupled to calculate the thermo-mechanical stresses in

the work-rolls. Accordingly, the computational steps can be summarized as follows:

1. the temperature distribution in the strip is calculated;

2. the roll pressure distribution in the deformation zone is estimated using the slab method;

3. the thermo-elastic model in the work-roll is conducted to achieve temperature and stress fields;4. step 1 is repeated using the new thermal boundary conditions obtained from step 3;

5. performing steps 2 and 3 and then calculating the error norm for two successive solutions both in the work-roll

and in the strip using the following condition:ui +1 − uiui

< Error, where u =

uT · u1/2

,

where u denotes the Euclidean norm of vector “u” i.e., the temperature or displacement vector, and the

superscript “i” represents the iteration number;

6. if the error is small enough, i.e., of the order of 10−2 for the thermal analysis and 10−6 for the stress model,

then the above calculations are followed for the next time step. If not, the calculation is restarted from step 1

employing the work-roll’s last solution as the updated boundary conditions for the strip.

It should be noted that four-node isoparametric elements are employed for both thermal and mechanical calcu-

lations; 270 nodes and 232 elements are used for the strip and 1050 nodes and 966 elements are used in analyzing

the thermo-mechanical behavior of the work-roll. Figure 2 shows the utilized mesh system. Non-uniform meshing

is used in the work-roll while finer elements are generated in the surface region and coarser elements are produced

inside the work-roll where temperature variations are smooth.

3 Results and discussion

First, hot-rolling experiments were performed to evaluate the predicted temperature fields. A commercial pure

aluminum slab with an initial thickness of 20 mm was used in the experiments. The as-received material was firstannealed at 450◦C for 2 h and then air-cooled and after that the hot-rolling samples were machined out of the raw

material. In thenext stage, hot-rolling experiments under tworolling conditions were performedand the temperature

variations were recorded by embedding thermocouples within the metal being deformed.

The employed experimental rolling conditions are listed in Table 1. The work-roll thermal conductivity and

thermal expansion have been taken as 35 (W/m K) and 1.6 × 10−5, respectively and a thermal conductivity of 220

(W/m K) and specific heat of 950 (J/kg K) have been used as the thermo-physical properties of the aluminum. In

addition, the data in [26, pp. 35–37] were used for determining flow stress behavior of the alloy. Figure 3 displays

the predicted and recorded temperature variations within the rolling sample under different working conditions. As

can be observed there generally exists a reasonable agreement between the two sets of data. However, in Fig. 3b

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Table 1 Rolling conditions used in the experiments

Sample Initial thickness (mm) Final thickness (mm) Initial temp. (◦C)

1 8 6.8 450

2 8 6.8 350

Reduction = 15%, Rolling speed = 43 rpm, Roll diameter = 150mm

0 .0 0 .4 0. 8 1. 2 1. 6 2 .0 2 .4 2 .8 3 .2 3. 6

360

380

400

420

440

460

480

PredictedMeasured

T e m p e r a t u r e ( ° C )

Time (s)

(a)

0.0 0.5 1.0 1.5 2.0 2.5 3.0

260

280

300

320

340

360

380

400

PredictedMeasured


Time (s)

(b)

Fig.3 Comparison between predictedand measured temperature variations underdifferentworking conditions, a Sample1, b Sample2

Fig. 4 Comparison

between the predicted and

the measured temperature

variations at different points

of the work-roll under the

hot-rolling conditions

mentioned in [27]

0 1 2 3 4 5

0

100

200

300

400

500

600

Initial slab temp.= 1230°C

Reduction=22.8%

Surface (Predicted)

Depth of 0.14 in (Predicted)

Surface (Measured)

Depth of 0.14 in (Measured)


Time (s)

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0.0 0.5 1.0 1.5 2.0 2.5 3.0

0

50

100

150

200

250

Depth of 4 mm

Depth of 2 mm

Surface


Time (s)

(a)

0.0 0.5 1.0 1.5 2.0 2.5 3.0

0

20

40

60

80

100

120

140

160

180

200

Depth of 4 mm

Depth of 2 mm

Surface

T e m p e r a t u r e ( ° C

)

Time (s)

(b)

Fig. 5 Work-roll temperature variations during two rotations a sample 1, b sample 2

0.0 0.5 1.0 1.5 2.0 2.5 3.0

-250

-200

-150

-100

-50

0

50

Depth of 4 mm

Depth of 2 mm

Surface

T h e r m o

- m e c h a n i c a l s t r e s s ( M P a )

Time (s)

(a)

0.0 0.5 1.0 1.5 2.0 2.5 3.0

-200

-150

-100

-50

0Depth of 4 mm

Depth of 2 mm

Surface

T h e r m o -

m e c h a n i c a l s t r e s s ( M P a )

Time (s)

(b)

Fig. 6 Thermo-mechanical stresses at different points developed during two rotations a sample 1, b sample 2

there is a relatively large difference between the two sets of results particularly in the deformation zone where

temperature changes occur abruptly. This may be attributed to the error in recording temperature variations, as well

as assuming constant thermo-physical properties of the aluminum. In addition, the predicted temperature variationswithin the work-rolls were compared with the published data. Figure 4 displays the predicted temperature cycles

of the work-roll and the measured results under the working conditions mentioned in [27]. It can be observed that

there is a reasonable agreement between the predictions and experiment.

Figure 5 shows temperature variations within the work-rolls during tworotations under working conditions listed

in Table 1. Figure 6 illustrates the variations of normal thermo-mechanical stresses along the θ -direction under the

same working conditions. As expected, the maximum temperature in the work-rolls increases for higher initial

temperatures. In addition, thermo-mechanical stress has a similar trend to that of temperature variations. Note that

a higher temperature gradient causes more severe temperature fluctuations at the outer layer of the work-roll which,

in turn, causes higher thermal stresses in this region. However, although at the higher temperatures the roll pressure

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0.0 0.5 1.0 1.5 2.0 2.5 3.0

0

100

200

300

400

500

600

700

Depth of 4 mm

Depth of 2 mm

Surface


Time (s)

(a)

0.0 0.5 1.0 1.5 2.0 2.5 3.0

-700

-600

-500

-400

-300

-200

-100

0Depth of 4 mm

Depth of 2 mm

Surface

T h e r m o - m e c h a n i c a l s t

r e s s , σ θ θ

( M p a )

Time (s)

(b)

0.0 0.5 1.0 1.5 2.0 2.5 3.0

-600

-500

-400

-300

-200

-100

0 Depth of 4 mm

Depth of 2 mm

Surface

T h e r m o - m e c h a n i c a l s t r e s s , σ

r r ( M P a )

Time (s)

(c)

-0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

-2

0

2

4

6

8

10

Depth of 4 mm

Depth of 2 mm

Surface

T h e r m o - m e c h a n i c a l s h e a r s t r e s s ( M P a )

Time (s)

(d)

Fig. 7 Hot rolling of steel strip, initial temperature 1050◦C, reduction 20%, rolling speed 1.3 m/s, a temperature variations, b thermo-

mechanical stress along the θ -direction, c, thermo-mechanical stress along r -direction c thermo-mechanical shear stress

decreases, it is seen that the thermo-mechanical stresses are larger for the rolling with initial temperature 450 ◦C.

This demonstrates that the thermal part plays an important role in the formation of thermo-mechanical stresses

compared to the mechanical deformation of the work-roll, especially at higher temperatures where the work-roll

pressure is relatively low.

Themodelcanbe employed in hot rolling of differentmetalsandalloys. Figure7 shows the temperature variationsand thermo-mechanical stresses within the work-roll during two successive revolutions in hot rolling of low-carbon

steel with an initial temperature of 1050◦C, a reduction of 20% and a rolling speed of 1.3 m/s where the roll diam-

eter is 660 mm where two water sprays are employed to cool the work-rolls with a water-spray angle of 15◦. It is

observed that larger temperature variations are found in the work-roll and the magnitude of the thermo-mechanical

stresses increases as well. However, shear stresses are small compared to the normal stresses as shown in Fig. 7b–d.

The effect of the initial temperature on the thermo-mechanical stresses is shown in Fig. 8 for hot rolling of

low-carbon steel with a rolling speed of 1.2 m/s and a reduction of 25% while a single water spray was taken in the

model. As the initial temperature increases from 980 to 1100◦C, the thermo-mechanical stresses rise about 80 MPa.

Similar to the results of hot rolling of aluminum, this shows that thermal stresses are important in the forming

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0.0 0.5 1.0 1.5 2.0 2.5 3.0

-700

-600

-500

-400

-300

-200

-100

0

100

Initial Temperature = 980°C

Depth of 4 mm

Depth of 2 mm

Surface

T h e r m o - m e c h a n i c a l s

t r e s s , σ

θ θ

( M P a )

Time (s)

(a)

0.0 0.5 1.0 1.5 2.0 2.5 3.0

-700

-600

-500

-400

-300

-200

-100

0

100

Initial Temperature = 1100°C

Depth of 4 mm

Depth of 2 mm

Surface


θ θ

( M P a )

Time (s)

(b)

Fig. 8 Thermo-mechanical stress along the θ -direction during hot rolling of steel, reduction 25%, rolling speed 1.2 m/s, a initial

temperature 980◦C, b initial temperature 1100◦C

Fig. 9 Thermo-mechanical

stress variations in the

θ -direction during five

rotations in hot rolling of

steel with initial temperature

1200◦C, reduction 20%,

rolling speed 1.3 m/s

0.0 0.8 1.6 2.4 3.2 4.0 4.8 5.6 6.4 7.2 8.0

-700

-600

-500

-400

-300

-200

-100

0

100

Reduction = 20%

Surface

Depth of 4 mm

Depth of 2 mm


θ θ

( M P a )

Time (s)

of thermo-mechanical stresses. It is seen that larger thermo-mechanical stresses compared to the hot rolling of aluminum are produced owing to higher thermal gradients existing between the work-roll and the steel. Figure 9

shows the thermo-mechanical stresses after five work-roll revolutions where the third one is an idling rotation. It is

observed that the produced stress at the surface region slightly changes in different rotations while the stresses at the

inner regions, i.e., depth of 2 mm, are increasing as shown in this figure. This may be attributed to the accumulative

effect of previous thermo-mechanical stresses produced in the earlier revolutions.

Finally, if the results of the present model are compared with the thermal stresses predicted in the previous work

by the authors [19], it may be observed that the mechanical part has shown limited influence on the thermo-mechan-

ical stresses developed within the work-rolls as well as mainly affect the stress distribution at the contact region of

roll/metal. In other words, under the hot-rolling conditions used in this work, the thermal loading mainly governs

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the distribution of thermo-mechanical stresses owing to low flow stress of the metal being rolled. However, it may

be expected that, for the case of warm-rolling operations, where the roll pressure increases and/or in four-roll stand

mills where back-up rolls are employed the elastic deformation of the work-rolls contributes considerably to the

thermo-mechanical stresses and their distributions within the work-roll.

4 Conclusions

A mathematical model has been proposed to evaluate thermo-mechanical stresses in work-rolls during hot-strip

rolling. Effects of different factors, such as the thermal relationship between the work-roll/metal, heat of deforma-

tion, and the cooling program on the work-rolls have been considered in the model. A reasonable agreement was

found between the predicted and the experimental data, both for the temperature variation in the strip as well as in

the work-roll.

Predictions show that shear stresses are small in comparison with the normal stress while the nature of the

normal stresses at the boundary layer of the work-rolls is compressive. In addition, the effect of the temperature

gradient between roll/metal is higher in comparison with the roll pressure; in other words, thermal stresses have a

significant effect on the thermo-mechanical stresses compared to mechanical ones for the rolling conditions used in

this work. The results may be employed for estimating roll life as well as designing a work-roll diameter regardingthe produced thermo-mechanical stresses.

It should be mentioned that a two-dimensional Lagrangian model has been employed in this work and it seems

that the development of a three-dimensional analyses, i.e., Eulerian and/or Lagrangian, as well as considering the

effect of back-up rolls, maybe of help for a better and more accurate understanding of thermo-mechanical responses

of works-rolls, particularly at the edge regions of the strip being rolled.

Appendix

List of symbols

ae Nodal displacement vector

cr Work-roll specific heat

cs Strip specific heat

D Matrix of elastic constants

E Elastic modulus

F Surface-traction vector

f ere Nodal residual-stress vector

f est Nodal surface-traction vector

f eth Nodal thermal-stress vector

hc Convection heat-transfer coefficient

h Thickness at an arbitrary position in deformation zone

h0 Initial thickness of the strip

h1 Finial thickness of the strip

hn Thickness at the neutral point

K e Stiffness matrix for the eth element

k r Work-roll thermal conductivity

k s Strip thermal conductivity

n Normal direction to the boundary

p Mean horizontal pressure

Pe Peclet number

q̇ Rate of heat of deformation

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q Normal pressure

qfric Rate of heat generation by friction

R Work-roll radius

r Radius direction

r * Maximum radius in which temperature variations are insignificant

S Surface boundary

T TemperatureT r Work-roll surface temperature

T sur Strip surrounding temperature

t Time

u Displacement vector

V Volume

vnx Longitudinal velocity at neutral point

vr Work-roll velocity

v x Strip velocity along rolling direction

v xc Longitudinal velocity component at the center

v xs Longitudinal velocity component at the surface

x Rolling direction y Thickness direction

v Relative velocity between the strip and the work-roll

δu Variation in displacement vector

ε Strain vector

ε0 Thermal-strain vector˙̄ε Effective strain rate

ϕ Contact angle

Rate of volume passing through an arbitrary section

η Conversion efficiency

λ Emissivity

π Energy functionalθ Peripheral direction

ρr Work-roll density

ρs Strip density

σ Stress vector

σ̄ Effective stress

σ 0 Residual stress vector

σ R Stephan’s constant

σ rr Stress component along radius direction

σ θθ Stress component along peripheral direction

τ Shear-flow stress

τ fric Friction stressτ rθ Shear stress on the surface normal to r -direction

ω Angular velocity of the work-roll

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1. a Model for Evaluating Thermo-mechanical Stresses

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