European Congress on Computational Methods in Applied Science and Engineering
ECCOMAS 2000
Barcelona, 11-14 September 2000
© ECCOMAS
SIMULATION OF DEFORMATION AND TEMPERATURE
IN CALIBER ROLLING
Kazutake Komori
Department of Mechanical Engineering, Daido Institute of Technology
2-21 Daido-cho, Minami-ku, Nagoya-shi, Aichi-ken, 457-8531 Japan
e-mail: [email protected], web page: http://www.daido-it.ac.jp
Key words: Material Forming, Caliber Rolling, Finite-Element Method, Finite Difference
Method, Deformation, Temperature.
Abstract. Simulation of deformation and temperature in caliber rolling (square-oval-square
rolling) is performed. The method of analyzing deformation is the conventional threedimensional rigid-plastic FEM. In the analysis of temperature, three-dimensional FEM is
utilized mainly, which requires a large amount of computation time and memory capacity,
hence two-dimensional FEM and one-dimensional FDM are also utilized. First, the effect of
the finite-element mesh in the cross section of the material is examined. As a result, the finiteelement mesh derived from the mesh of concentric circle is proved to be more appropriate than
the finite-element mesh derived from the mesh of square grid in the analysis of square-ovalsquare rolling. Second, the analysis of 6-pass caliber rolling in an actual rolling process is
performed. Due to the turn of the material in the region between two rolling mills, not the onefourth of the material but the whole material is analyzed. The shape of the material obtained
by the analysis agrees well with the shape of the product. Hence, the validity of the simulation
is demonstrated.
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1 INTRODUCTION
Pass design of caliber rolling is necessary to obtain products of high accuracy and high
quality. Accuracy means the accuracy of the shape of the final products and quality means the
quality of the microstructure of the final products. The distribution of deformation resistance,
which depends on the strain, the strain rate and the temperature, and frictional stress affects
the shape of the material after rolling. Moreover, the grain size, the recrystallization fraction
and the dislocation density depend on the strain, the strain rate and the temperature, and they
affect the microstructure of the material after rolling. Hence, simulation of deformation and
temperature in caliber rolling is indispensable. Several simulations have been performed
recently using various methods 1,2.
It is easy to analyze the temperature by finite-element method, the finite-element mesh of
which coincides with the finite-element mesh of the analysis of the deformation. However, in
hot rolling, since heat conduction occurs between the high-temperature material and the roomtemperature roll, the temperature distribution near the material-roll contact surface changes
drastically. Hence, the temperature distribution must be analyzed by finite-element method,
the finite-element mesh of which is much finer than the finite-element mesh of the analysis of
the deformation, to obtain the temperature distribution precisely.
Although it is possible to analyze the temperature in two-dimensional rolling by this
finite-element method, it is difficult to analyze the temperature in three-dimensional rolling by
this finite-element method, because large amounts of computation time and memory capacity
are required. On the other hand, the temperature in two-dimensional rolling has been analyzed
by one-dimensional finite difference method, in which the heat conduction in the rolling
direction is neglected and the heat conduction in the thickness direction is considered 3. Since
the temperature distribution obtained by the finite difference method coincides well with the
temperature distribution obtained by the two-dimensional finite-element method, the finite
difference method is effective 4.
Hence, we have proposed a method of numerical simulation for hot caliber rolling 5,6, in
which the deformation of the material is analyzed by the conventional three-dimensional
rigid-plastic finite-element method 7, while the temperature distribution in the material is
calculated using a new combined method of three-dimensional finite-element method, twodimensional finite-element method and one-dimensional finite difference method. In other
words, the temperature in the region near the material-roll contact surface is analyzed by onedimensional finite difference method, the temperature in the region other than the region near
the material-roll contact surface is analyzed by three-dimensional finite-element method and
the temperature in the region between two rolling mills is analyzed by two-dimensional finiteelement method.
In the numerical simulation of caliber rolling, the type of finite-element mesh in the cross
section of the material influences calculated results. There are some occasions that reasonable
results are not obtained in the use of some types of finite-element mesh. Hence, in this study,
first, we discuss two types of finite-element mesh in the cross section of the material; one type
of finite-element mesh is derived from the mesh of square grid and the other type of finite-
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Kazutake Komori
element mesh is derived from the mesh of concentric circle.
In our previous papers 5,6, one-fourth of the material or one-sixth of the material was
analyzed because of symmetry. However, there are some occasions that the material does not
have lines of symmetry, in which the whole material has to be analyzed. Hence, in this study,
next, we develop a computer program that can analyze the deformation and temperature of the
material that does not have lines of symmetry. We perform simulation of hot caliber rolling in
an actual rolling process and the validity of the developed program is demonstrated.
2 METHOD OF ANALYZING TEMPERATURE
The x-axis, y-axis and z-axis are set to coincide with the direction of rolling, the direction of
width and the direction of thickness, respectively. The origin is set at the center of the exit
cross section of the roll gap region. Although the whole region is analyzed, the method of
analyzing temperature in the xz-plane and along the z-axis is explained as follows. Figure 1
shows the xz-plane and the temperature distribution along the z-axis.
Figure 1: xz-plane and temperature distribution along z-axis
2.1 Region near the contact surface (one-dimensional finite difference method)
In the region near the material-roll contact surface, which is surrounded by one dotted
broken line in Figure 1, since the temperature gradient in the tangential direction of the contact
surface is much smaller than the temperature gradient in the normal direction of the contact
surface, the former temperature gradient is assumed to be negligible. Hence, the temperature
in the region is analyzed by one-dimensional finite difference method. The equation of thermal
conductivity and the boundary condition are
( tR / ô)=(ëR /c RñR ) ( 2tR / z’2 ) (z’ >0)
2
2
(1)
( tM / ô)=(ëM /c MñM ) ( tM / z’ )+(qd / c MñM ) (z’ <0)
(2)
-ëR ( tR / z’ )=-ëM ( tM / z’ )+ qf and tR= tM (z’ =0)
(3)
where z’ is the axis normal to the contact surface, t denotes temperature, ô denotes time,
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ëdenotes thermal conductivity, c denotes specific heat, ñ denotes density, R denotes roll, M
denotes material, qd is the heat generated by plastic deformation per unit time and volume, and
qf is the heat generated by friction per unit time and area. qd is calculated from the energy of
plastic deformation and qf is calculated from the energy of friction. The energies are assumed
to be completely transformed into heat. The region of z’ >0, the region of z’ =0 and the region
of z’ <0 denote the roll, the contact surface and the material, respectively.
Attention is focused on the temperature gradient in the region z’<0 at x=0 obtained by the
analysis. The distance between the contact surface and the point where the temperature
gradient becomes almost zero is defined as ä. The imaginary face, which is at distance ä from
the contact surface, is assumed to be adiabatic face.
2.2 Region other than the region near the contact surface (three-dimensional finiteelement method)
The temperature in the region other than the region near the contact surface, which is
surrounded by broken line in Figure 1, is analyzed by the three-dimensional finite-element
method 8. The equation of heat conduction is
c MñM• ( vx• ( tM / x)+ vy• ( tM / y)+ vz• ( tM / z))
=ëM• (( 2tM / x2 )+( 2tM / y2 )+( 2tM / z2 ))+ qd
(4)
where vx, vy and vz are the components of velocity in the x, y and z directions, respectively.
The finite-element mesh used in the analysis of deformation is utilized. The nodes which are
on the contact surface are moved in the negative direction along the z’-axis. The amount of
movement is ä. The boundary condition of the nodes is assumed to be adiabatic. Heat transfer
and radiation are assumed to occur on the stress free surface of the material.
2.3 Region between two rolling mills (two-dimensional finite-element method)
In the region between two rolling mills, since the temperature gradient in the cross section is
much larger than the temperature gradient in the rolling direction, the latter temperature
gradient is negligible. Hence, the temperature in this region is analyzed by the twodimensional finite-element method. The following equation of heat conduction is calculated
and the temperature distribution is obtained.
ëM• (( 2tM / y2 )+( 2tM / z2 ))=0
(5)
The finite-element mesh of the exit cross section of the roll gap region used in the analysis is
utilized. The temperature distribution at the exit cross section of the roll gap region is made to
coincide with the temperature distribution at the entrance cross section of the region between
two rolling mills. However, the temperature distribution calculated by the finite difference
method with a fine mesh cannot be matched precisely with the temperature distribution to be
represented by the finite-element method with a coarse mesh. Hence, the temperature of nodes
on the surface of the material is assumed such that the total heat capacity at the exit cross
section of the roll gap region coincides with the total heat capacity at the entrance cross
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Kazutake Komori
section of the region between the two rolling mills, as shown in Figure 1. Heat transfer and
radiation are assumed to occur on the stress free surface of the material.
3 ANALYTICAL RESULTS
3.1 Square-oval-square rolling
Simulation of square-oval-square rolling has been performed. In other words, the initial
material shape in the cross section of the material is square, the caliber shape of the first
rolling mill is oval and the caliber shape of the second rolling mill is square. The relationship
between the stress and strain of the material is assumed to obey Shida’s empirical formula 9,
which is,
ó=ót•óå•(å/10)m (MPa)
ót=2.7•exp(5000/ tM -0.01/(C+0.05)) ( tM > t*)
ót=2.7•(30.0•(C+0.90)•( tM -950•(C+0.49)/(C+0.42))2+(C+0.06)/(C+0.09))
•exp(5000/ t*-0.01/(C+0.05)) (tM < t*)
0.41-0.07C
óå=1.3•(å/0.2)
-0.3•(å/0.2)
m=(-0.019C+0.126) tM +(0.075C-0.050) ( tM > t*)
m=(0.081C-0.154) tM +(-0.019C+0.207)+0.027/(C+0.320) (tM < t*)
t*=950•(C+0.41)/(C+0.32) (K)
(6)
(7)
(8)
(9)
(10)
(11)
(12)
where C denotes the carbon content of the material (%) and is assumed to be 0.23. Further, the
condition of sticking friction is assumed between the material and the roll. Table 1 shows the
thermal analytical condition.
Specific heat of material c M (J/kgK)
Density of materialñM (g/cm3)
Thermal conductivity of materialëM (W/mK)
Temperature of material tM (K) (initial value)
Specific heat of roll c R (J/kgK)
Density of rollñR (g/cm3)
Thermal conductivity of rollëR (W/mK)
Temperature of roll tR (K) (initial value)
Emission coefficient from material to atmosphere
Heat transfer coefficient from material to atmosphere (W/m2K)
Atmospheric temperature (K)
644
7.86
27.2
1273
485
7.86
51.0
373
0.7
20
293
Table 1:Thermal analytical condition
According to the shape of caliber roll in an actual rolling process, the shape of the material
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Kazutake Komori
in the cross section becomes square in every two rolling mills. Hence, the shape of caliber roll
is designed so that square billets are produced from themselves after they are rolled twice.
Table 2 shows the rolling condition for square-oval-square rolling.
Initial thickness of material (mm)
Initial width of material (mm)
Caliber depth of first rolling mill (mm)
Caliber width of first rolling mill (mm)
Caliber depth of second rolling mill (mm)
Caliber width of second rolling mill (mm)
Roll radius (mm)
Circumferential roll velocity (m/s)
Distance between two rolling mills (m)
100
100
64
128
92
92
300
2
1
Table 2:Rolling condition for square-oval-square rolling
Two types of finite-element meshes in the cross section of the material are utilized. One
finite-element mesh is derived from the mesh of square grid and the other mesh is derived
from the mesh of concentric circle. Figure 2 shows the finite-element mesh obtained by the
simulation for the first rolling mill and the second rolling mill.
(a) Square grid (first rolling mill)
(b) Concentric circle (first rolling mill)
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Kazutake Komori
(c) Square grid (second rolling mill)
(d) Concentric circle (second rolling mill)
Figure 2:Finite-element mesh
Figure 3 shows the equivalent strain distribution at the exit cross section of the roll gap
region. The equivalent strain distribution obtained using the square grid finite-element mesh is
almost the same as the equivalent strain distribution obtained using the concentric circle finiteelement mesh. However, the magnitude of the equivalent strain near the region that is initially
the corner part of the material in the case of the square grid finite-element mesh is larger than
that in the case of the concentric circle finite-element mesh. The reason of the difference of the
magnitude is as follows. As is shown in Figure 2, the finite-element shape near the region
deteriorates in the case of the square grid finite-element mesh, while the finite-element shape
near the region does not deteriorate in the case of the concentric circle finite-element mesh.
(a) Square grid (first rolling mill)
(b) Concentric circle (first rolling mill)
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Kazutake Komori
(c) Square grid (second rolling mill)
(d) Concentric circle (second rolling mill)
Figure 3:Equivalent strain distribution
Figure 4 shows the temperature distribution at the exit cross section of the region between
two rolling mills. The temperature distribution obtained using the square grid finite-element
mesh is almost the same as the temperature distribution obtained using the concentric circle
finite-element mesh. Since heat conduction in the cross section of the material and heat
transfer from the stress free surface of the material to the atmosphere occur between two
rolling mills, the deterioration of the finite-element shape does not affect the temperature
distribution at the exit cross section of the region between two rolling mills.
(a) Square grid (first rolling mill)
(b) Concentric circle (first rolling mill)
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Kazutake Komori
(c) Square grid (second rolling mill)
(d) Concentric circle (second rolling mill)
Figure 4:Temperature distribution (K)
3.2 An actual rolling process
Simulation of an actual hot rolling process of steel bars is performed to verify the method of
analysis. Figure 5 shows a pass schedule (the shape of the caliber) in an actual rolling process.
Figure 5:Pass schedule in an actual rolling process
The initial shape in the cross section of the material is a square of 100×100 mm and the final
shape is a square of 28×28 mm. The number of rolling mills is 6. In the first, third and fifth
rolling mills, square-oval rolling is performed, in other words, the material shape is square and
the caliber shape is oval. On the other hand, in the second, fourth and sixth rolling mills, ovalsquare rolling is performed. In the region between the first and second, the third and fourth,
and the fifth and sixth rolling mills, the material is subjected to a fourth turn about the axis in
the rolling direction. On the other hand, in the region between the second and third, and the
fourth and fifth rolling mills, the material is subjected to an eighth turn about the axis in the
rolling direction. In the previous section, since the shape of the material in the cross section
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Kazutake Komori
has two lines of symmetry, one-fourth of the material is sufficient to be analyzed. In this
section, however, since the shape of the material in the cross section has point symmetry, the
whole material must be analyzed.
The relationship between the stress and strain of the material is assumed to obey Shida’s
empirical formula and the condition of sticking friction is assumed between the material and
the roll. The thermal analytical condition shown in Table 1 is utilized. In the previous section,
it is demonstrated that the shape of finite elements deteriorates in the case of the square grid
finite-element mesh, and that the shape of finite elements does not deteriorate in the case of
the concentric circle finite-element mesh. Hence, the concentric circle finite-element mesh is
utilized in this section.
(a) Fifth rolling mill
(b) Sixth rolling mill
Figure 6:Equivalent strain distribution
(a) Fifth rolling mill
(b) Sixth rolling mill
Figure 7:Temperature distribution (K)
Figure 6 shows the equivalent strain distribution in the fifth and sixth rolling mills. Figure 7
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shows the temperature distribution in the fifth and sixth rolling mills. The equivalent strain
and temperature are high at the center of the cross section of the material, while the equivalent
strain and temperature are low at the surface of the material.
(a) Equivalent strain
(b) Temperature (K)
Figure 8:Equivalent strain and temperature distributions (sixth rolling mill)
Figure 8 shows the equivalent strain and temperature distributions at the exit cross section of
the sixth rolling mill. Due to the eighth turn of the material about the axis in the rolling
direction in the region between two rolling mills, the equivalent strain and temperature
distributions in the cross section of the material do not have line symmetry but have point
symmetry. The shape of the exit cross section of the material obtained by the analysis agrees
well with the shape of the product of a square. Hence, the validity for the simulation in which
the whole material is analyzed is confirmed.
4. CONCLUSIONS
Simulation of deformation and temperature in caliber rolling (square-oval-square rolling) is
performed. The method of analyzing deformation is the conventional three-dimensional rigidplastic FEM. In the analysis of temperature, three-dimensional FEM is utilized mainly, which
requires a large amount of computation time and memory capacity, hence two-dimensional
FEM and one-dimensional FDM are also utilized. First, the effect of the finite-element mesh
in the cross section of the material is examined. As a result, the finite-element mesh derived
from the mesh of concentric circle is proved to be more appropriate than the finite-element
mesh derived from the mesh of square grid in the analysis of square-oval-square rolling.
Second, the analysis of 6-pass caliber rolling in an actual rolling process is performed. Due to
the turn of the material in the region between two rolling mills, not the one-fourth of the
material but the whole material is analyzed. The shape of the material obtained by the analysis
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Kazutake Komori
agrees well with the shape of the product. Hence, the validity of the simulation is
demonstrated.
REFERENCE
[1] K. Karhausen, R. Kopp and M.M. de Sauza, “Numerical simulation method for designing
thermomechanical treatments illustrated by bar rolling”, Scandinavian Journal of Metallurgy,
20, 351-362 (1991).
[2] P. Montmitonnet J.L. Chenot C. Bertrand-Corsini C. David T. Iung and P. Buessler,
“A coupled thermomechanical approach for hot rolling by a 3D finite element method”,
Transactions of the ASME Journal of Engineering for Industry, 114, 336-343 (1992).
[3] ISIJ (Editor), Theory and Practice of Flat Rolling, ISIJ, (1984).
[4] Y. Kojima, M. Takeyama and T. Mizuno, “Calculation of interface temperature in cold
strip rolling” Journal of the JSTP, 30, 1004-1009 (1989).
[5] K. Komori, “Simulation of deformation and temperature in multi-pass caliber rolling”,
Journal of Materials Processing Technology, 71, 329-336 (1997).
[6] K. Komori, “Simulation of deformation and temperature in multi-pass three-roll rolling”,
Journal of Materials Processing Technology, 92-93, 450-457 (1999).
[7] K. Komori, “Rigid-plastic finite-element method for analysis of three-dimensional rolling
that requires small memory capacity”, International Journal of Mechanical Science, 40, 479491 (1998).
[8] G. Yagawa, An Introduction to Finite-Element Method for Fluid and Heat, Baifukan,
(1983).
[9] S. Shida, “Empirical formula of flow-stress of carbon steels”, Journal of the Japan Society
for Technology of Plasticity, 10, 610-617(1969).
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