Analysis of Asymmetrical Rolling Process for Bimetal
Sheet with Rigid Plastic FEM
Shi Qingnan He Yanling
Wei Wei
Kunming University of Science and Technology
Kunming, P.R.China, 650093
Abstract
A set of super-thin element model on the composite surface is established with rigid plastic FEM
in the strip composite rolling process. The functional equations of super-thin element have been
formulated adopting the theories of velocity discontinuities and equivalent strain energy etc.. On the
basis of this model, the relevant software is designed using OOP(Object Oriented Programming)
method, and simulating the technological process of asymmetrical rolling for bimetal flat sheet.
Meanwhile, the variations of deformation flow and composite surface are analyzed, and simulated in
the bimetal strip metrical and asymmetrical rolling process.
Keywords: bimetal, composite rolling, asymmetrical rolling, FEM
1. Introduction
FEM is used extensively in the bimetal strip composite rolling. In comparison with
other metalworking processes, the bimetal strip composite rolling is a relatively difficult
process in which composite surface, inhomogeneous deformation and friction result in
asymmetry. Hence, a set of super-thin element model on the composite surface is
established with rigid plastic FEM in the strip composite rolling process. The functional
equations of super-thin element have been formulated adopting the theories of velocity
discontinuities and equivalent strain energy etc.. On the basis of this model, the relevant
software is designed using OOP(Object Oriented Programming) method. The variations
of deformation flow and composite surface are analyzed, and simulated in the bimetal
strip metrical and asymmetrical rolling process.
2. Super-thin element model on the composite surface
2.1 Super-thin element
The theory of rigid plastic FEM has been basically perfect to analyze the various
metalworking processes. However, some technical problems remain to be better solved in
the process of really analyzing. For example, how to handle composite surface has been a
key problem. The methods of handling composite surface will directly effect accuracy
and precision of results, otherwise which lead to divergence that exists with iterative
calculation.
The variations of composite surface in rolling will be beneficial to how to handle the
composite surface. Before rolling two contact surfaces have been polished by wire brush,
its smooth finish is minor. Under the low strain rate, just entering the rolls, the contact
surfaces are sliding acutely, where is slip zone. When the strain rate attains a certain value
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in the later period of rolling, the contact surfaces are sliding little, on which two metals
are packed tight each other to be turned into one mixed material, where is composite
zone. In the middle of the two above zones, contact surfaces are not only sliding but also
biting and being split, where is transitional zone. In the light of the above analysis, the
authors establish a set of super-thin element to handle bimetal composite surface adopting
rigid plastic FEM. The element thickness is optimal from 8 to 15. The elements in slip
zone are mainly sheared, similar to a velocity discontinuity. In composite zone the
elements are composed of two mixed metals, whose functional equation is different from
that of any other materials. The element in transitional zone is a pattern between slip zone
and composite zone.
2.2 Establishing elements and analyzing theoretically
A coat of super-thin element is shown in figure 1, which is a quadrilateral (such as
EFGH) similar to a common element, each node has only a velocity. There are indications
that shearing rate in slip zone (AB) is great by calculation, whose equivalent strain rate is
tens, hundreds or thousands' times than that of common elements. In composite zone
(CD) strain rate is almost identical. How to establish the elements of each zone can be
explained theoretically as following.
Fig.1
Fig.2
1) Slip zone
Two dimensions are mainly taken into consideration. Figure 2 shows the area D of a
super-thin element (EFGH). To be relevant fictitious velocity field its functional equation
 ( m) may be expressed as:
I
 ( m)    s  dA  
I
D
dy1
0

dx1
0
 s  dxdy
(1)
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Where  s is the effective stress.
 is the effective strain.
2

3


  x yz
xy 

1 v x v y
(

)
2  v y v x
In AB zone the velocity differences between E and G, as well as F and H, are great
in the X-direction. When dy1 tends to zero, super-thin element will be a velocity
discontinuity s. Because v y is a continuous function of Y, v x is no need to be
v x
may be a function of  , namely it may be
v y
continuous. According to J.B.Martin,
zero on the side of s, but a certain value on the other side of s. The item containing
xy will incline to boundless. As a result, in these elements shearing rate xy is great,
and a primary item, x and  y occupying little. Therefore, the functional equation may
approximately expressed as:
1
 ( m)  
I
dy1
0
2
2
 
 v x v y   
dx1
1  2
2
0  s  3 x y   v y  v x   dxdy
 
 
1

dy1
0

dx x
0
or
 1  v  2  2
dx1
x
0  s  3  v y   dxdy



1
 s  vx
3
y 0
 vx
y  dy1

(2)
1
2
2
 dx
dx1
 ( m)    x v x dx
I
(3)
0
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where v x  v x
y 0
 vx
y  dy1
,  x is friction stress. x may be evaluated from
 x   x  y
(4)
 x is the coefficient of composite surface
where
 y is the Y-direction stress of contact element
2) Composite zone
Figure 1 shows that the strain rate of super-thin element is almost identical to that of
common element in CD segment. In the light of the equivalent strain energy, the plastic
deformation power U(m) is expressed as:
U(m) =

D
  dA
(5)
From the standpoint of equivalent deformation energy, the above equation may be
written as
U(m) = U1(m) + U2(m)
(6)
where U1(m) =   11dA is the deformation power of component 1.
D1
U2(m) =   2 2 dA is the deformation power of component 2.
D2
The strain of element is discontinuous due to cavities. So force is transferred from
one component to another.
Hence
1  2  
(7)
If two materials are well distributed in elements, and each of them accounts for 50
per cent, so
U1(m) =   11dA =
1
 1dA
2 D
(8)
U2(m) =   2 2 dA =
1
 2 dA
2 D
(9)
D1
D2
By substitution Eqs. (8) and (9) into Eq.(5),

D
  dA 
1
(  1dA    2 dA)
D
2 D
(10)
Then by substitution for the stress- strain curve  = A+B in Eq.(7)
  A1  B1  1  A2  B2   2
(11)
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 can be obtained from Eqs.(10) and (11), the functional equation III may be
written as
 III     dA
D
3) Transitional zone
The functional equation can be obtained from Eqs.(1) and (5) in transitional zone (BC
segment).
 II  N 1 (Im)  N 2  (IIIm)
(12)
where N 1 is coefficient as well as N 2 and the sum of them is 1.
Fig.3
BC segment is a transitional zone in fig.3, one of whose elements is EFGH, O is
midpoint of GH. If transitional zone is linear, N 1 and N 2 can be obtained respectively
from Eqs.(13) and (14).
N1 
OC
BC
(13)
N2 
BO
BC
(14)
4) Definition of demarcation points of B and C
A key problem is how to define the demarcation points of B and C. The element
EFGH is still be analyzed as an example. Considering the velocity field, the strain rates of
E and G can be calculated, analyzed as following.
Assuming the greater absolute value  as denominator, If absolute value  of
point G(or H) is greater than that of E(or F), then
when
F
E
  , EFGH is a slip zone element; when
  , it is a composite
H
G
zone element; If
E
F
  and
  , EG is the boundary line of slip zone and
G
H
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transitional zone, B is on the EG. If
E
F
  and
  , C is on the line FH, which
G
H
is the boundary line of slip zone and transitional zone.
3. Analysis
3.1 Calculation and analysis of velocity field, equivalent strain rate field and stress
field
To calculate and analyze the velocity field, equivalent strain rate field and stress
field, experiments are performed with the materials of Steel and Aluminum. On entrance
and exit side, the thickness of steel is 0.483mm and 0.23mm, and that of Aluminum is
0.548mm and 0.25mm. In the experiment Steel is closed to the upper roll. The
experimental equipment involves a two-high mill, whose two diameters of rolls ratio is
150/170.
Figure 4 shows the distributions of velocity, equivalent strain rate and stress. The
relationships between each isogram value and its color are shown in figure 5.
Figure 4 (a) is for the velocity vectors. It is seen that the variations of each node
velocity and direction are different in two cases. At entrance of roll-gap velocity of Steel
is slightly greater than that of Aluminum, and the lower layer(Aluminum) is distinctly
sliding downward, then distinctly moving downward and back sliding of nodes on the
interface appears simultaneously after rolling. While in exit roll-gap velocity of nodes is
more or less the same on the interface, which obviously inclines to the lower layer.
Figure 4 (b) is for the isogram of equivalent strain rate. At entrance of roll-gap near
the interface there is a zone where the equivalent strain rates of the lower layer vary
severely, which also vary largely near the interface and the zone of contact rolls, and vary
slightly in the middle of each layer. In addition, as seen in the figure, the equivalent strain
rate of the lower layer varies rapidly than that of the upper layer.
Figure 4 (e) is for isogram of shearing strain rate, which shows the zone where the
shearing strain rate of the lower layer near the interface varies severely. Besides, the zone
is continuously existing in the rolling process.
Figure 4 (f), (g) and (h) are for the isogram of stress. As seen in the figures, stresses
of the upper layer are greater than those of the lower layer. Nevertheless, there exists a
zone near the interface where stresses vary largely at a specified reduction. In this zone it
possesses the minimal stress and the maximal hydrostatic pressure.
Fig.4 (a) Velocity vectors
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Fig.4 (b) Isogram of equivalent strain rate
Fig.4 (c) Isogram of equivalent strain rate in X-direction
Fig.4 (d) Isogram of equivalent strain rate in Y-directio
Fig.4 (e) Isogram of shearing strain rate
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Fig.4 (f) Isogram of stress in X-direction
Fig.4 (g) Isogram of stress in Y-direction
Fig.4 (h) Isogram of shearing stress
Fig.5 Relationships between values and colors
3.2 Analysis and investigation on the flat sheet
Traditionally flat sheets may be produced through many tests of regulating
asymmetrical ratio, thickness ratio and so on. The cost of using these traditional methods,
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however, is very high and the manufacturing process is not suitable for mass production.
With the development of computers, simulation experiments that may be carried out with
the aid of them, economize and guide the actual manufacturing. The software in this
paper possesses the function of simulating the rolling process. For the purpose of
obtaining the flat composite sheet, simulation calculation can be conducted with
computers.
Figure 6 shows an example of simulation calculation on the composite sheet. It is seen
that at the exit velocities in Y-direction tend to zero and equivalent strain rates vary
gently. Under those circumstances, the flat sheets can be conveniently produced.
(a) Velocity vectors
(b) Grey graph of equivalent strain rate
Fig.6
4. Conclusions
A rigid plastic finite element method using OOP method to design the relevant
software is established. Many composite rolling experiments on bimetal sheet have been
conducted with this software. This study can be summarized as follows:
1) The software is applied simply. Grey graph or isogram can be adopted
to indicate results according to specific condition.
2) Analysis and investigation of the specific example show:
 In bimetal sheets rolling process, the shearing strain of the soft layer varies
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severely near the interface. If the effect of composite rolling is relatively poor, the
shearing strain will vary greatly; otherwise, it will vary gently in the later period of
rolling.
 In bimetal sheet rolling process, there exists a zone near the interface where it
possesses the maximal hydrostatic pressure, and beneficial to bimetal composition. The
above results further show asymmetrical technology is feasible and superior to producing
bimetal sheets.
3）The results from study of the shearing strain show that the shearing strain rate in
the later period of rolling is great. Nevertheless in the earlier period of rolling there exists
two zones, comparison with experimental results, one is slip zone and the other is
transitional zone. This also tests the soundness of the super-thin element model.
Acknowledgements -The authors would like to extend their thanks to the Natural Science Fund
Committee of Yunnan Province Under grant No. 95E026M.
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