Jtam-A4.dvi


JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

51, 3, pp. 627-637, Warsaw 2013

UPPER BOUND ANALYSIS OF BIMETALLIC ROD EXTRUSION PROCESS
THROUGH ROTATING CONICAL DIES

Heshmatollah Haghighat, Mohammad M. Mahdavi

Mechanical Engineering Department, Razi University, Kermanshah, Iran

e-mail: hhaghighat@razi.ac.ir

In this paper, an upper bound approach is used to analyze the bimetallic rod extrusion
process through rotating conical dies. By extending the kinematically admissible velocity
field formono-metal rod extrusion to thebimetallic rod extrusionprocess, the internal power
and the power dissipated on frictional and velocity discontinuity surfaces are evaluated.
Then, by equating the total power with the external power produced by axial movement of
the punch and the power induced by rotation of the die, the relative extrusion pressure is
determined. The extrusion process of bimetallic rods composed of a copper sleeve layer and
an aluminium core layer through a conical rotating die is also simulated by using the finite
element code ABAQUS. The analytical results are compared with the results given by the
finite element method. These comparisons show a good agreement.

Key words: bimetallic rod, extrusion, rotating die, upper boundmethod, FEM

1. Introduction

Bimetals are componentsmadeup of two separatemetallic units, each occupying a distinct posi-
tion in the component. Bimetal componentsmake it possibile to combine properties of dissimilar
metals to achieve a structurewith low density, good corrosion properties and high strength. The
compressive state of stress in extrusion and the possibility of producingmetallurgical bonds be-
tween the two metals make this process a suitable choice for producing bimetal rods (Berski et
al., 2004). In this process, alike othermetal forming processes, the estimation andminimization
of the extrusion pressure is important.

A number of researches have used the upper bound method and FEM to analyze the bi-
-metal extrusion process. Avitzur (1983) summarized the factors that affect simultaneous flow
of layers in extrusion of a bimetal rod through conical dies. Tokuno and Ikeda (1991) verified the
deformation in extrusion of composite bars by experimental and upper bound methods. Sliwa
(1997) described the plastic zones in forward extrusion of metal composites by experimental
and upper bound methods. Kang et al. (2002) designed the die for hot forward and backward
extrusion process of Al-Cu clad composite by experimental investigation and FEM simulation.
Hwang and Hwang (2002) studied the plastic deformation behavior within a conical die during
composite bar extrusion by experimental and upper boundmethods.

The making use of rotating dies in metal forming processes was firstly introduced by Gre-
enwood andThompson (1931). Brovrnan (1987) obtained an analytical solution based on stress
analysis for material flow through a rotating conical die excluding the circumferential slipping
effect. The so called KOBO type forming proposed by Bochniak and Korbel (1999, 2000, 2003)
applied to extrusion of tubes and wires has demonstrated essential advantages with respect to
monotonic formingprocesses.KimandPark (2003) studied the backward extrusion processwith
low die rotation to improve the problems of conventional backward extrusion process: the re-
quirement of large forming machine, the difficulty in selecting the die material caused by high
surface pressure, high cost of forming machine caused by improvement of noise and vibration,



628 H. Haghighat, M.M.Mahdavi

etc. They used the upper bound technique and FEM simulation. The results showed that the
backward extrusion with die rotation is a very useful process, because this process yields homo-
geneous deformations and lower forming load.Ma et al. (2004a,b, 2005) analysed the process of
mono-metal rod extrusion through steadily rotating conical dies, theoretically and experimental-
ly. They provided required torque for rotating the die froman external source and also supposed
that the angular velocity of thematerial inside the die changes with power relation to the radius
of each position in proportion to apex of virtual conic of the die. They inspected the effect of
slippage factor and semi die angle in extrusion pressure and finally determined the optimumdie
angle. Maciejewski and Mroz (2008) analyzed the mono-metal rod extrusion process through a
flat die assisted by cyclic torsion, which was induced by a cyclically rotating die. The evolution
of the extrusion force and torsional moment was studied with process parameters such as the
ratio of extrusion and rotation rates as well as the amplitude of die rotation.
In this study, the die rotation is proposed to reduce extrusion pressure in the bimetallic

rod extrusion process through conical dies. A velocity field for flow of a mono-metal rod during
extrusion through a rotating conical die, developed byMa et al. (2004b) is used for the bimetallic
rod extrusion process through a conical die in the upper bound model. The FEM simulation
on the extrusion of the bimetallic rod composed of a copper sleeve layer and an aluminum core
layer is also conducted.The derived upper bound solution allows the correlation of the extrusion
pressure with parameters of the process.

2. Upper bound analysis

Based on the upper bound theory, for a rigid-plastic Von-Misses material, the external power
required for material deformation is expressed as

J∗ =
2σo√
3

∫

V

√

1

2
ε̇ijε̇ij dV +

∫

Sv

k|∆V | dS+m
∫

Sf

k|∆V | dS−
∫

St

Tivi dS (2.1)

where σo is themeanflow stress of thematerial, k –material yield strength in shear, ε̇ij – strain
rate tensor, m – constant friction factor, V – volume of plastic deformation zone, Sv and Sf –
area of velocity discontinuity and frictional surfaces, respectively, St – area where the tractions
may occur, ∆V – amount of velocity discontinuity on the frictional and discontinuity surfaces
and vi and Ti are the velocity and tractions applied on St, respectively.
The rotational bimetallic rod extrusion process consists of an axial movement of the punch

and the rotational movement of the die. Figure 1 shows a schematic diagram of the bimetallic
rod extrusion through a rotating conical die shape. An initially bimetallic rod,made up of a rod
and an annular tube of two different ductile materials with the mean flow stresses σc and σs,
respectively, is considered.The initial outer and inner radius of the combined rod is R1i and R2i,
respectively. The outer radius of the extruded bimetallic rod is R1f and the interface radius of
the final extruded rod is R2f.
To analyze the process, thematerial under deformation is divided into eight zones, as shown

inFig. 1.Aspherical coordinate system (r,θ,ϕ)with theorigin O is used todescribe theposition
of the four surfaces of velocity discontinuity S1-S4 as well as the velocity in deformation zones
I and II. In zones VII and VIII, the incoming materials are assumed to flow horizontally as a
rigid body with velocity vi. In zones V and VI, the extruded bimetallic rod is assumed to flow
horizontally as a rigid body with velocity vf. Zones I to IV are the deformation regions, where
the velocity is complex. Zone I is surroundedby surfaces S1,S2, the die surface and the interface
surface. Zone II is surrounded by surfaces S3, S4 and the interface surface.
The material inside the container along the total length L is divided into two segments.

Within the length l, the bimetallic rod is twisted plastically inside the container and the regions



Upper bound analysis of bimetallic rod extrusion process ... 629

Fig. 1. Schematic diagram of bimetallic rod extrusion process through a rotating conical die, geometric
parameters and its deformation zones

enclosed are denoted as zones III and IV. A cylindrical coordinate system (r,θ,y) is used to
describe the velocity field indeformation zones III and IVwhere the axial coordinate y is parallel
to the extruding direction. The bimetallic rod in the remaining length (L− l) is designated by
zones VII and VIII. In these zones, the incoming material is assumed to flow horizontally as a
rigid body with velocity vo.
The surfaces S1 and S3 are located atdistance ri fromtheorigin, and the surfaces S2 and S4

are located at distance rf from the origin. The mathematical equations for radial positions of
the four velocity discontinuity surfaces S1, S3 and S2, S4 are given by

ri =
R1i
sinα

rf =
R1f
sinα

(2.2)

where α is the semi die angle.
In addition to these surfaces, there are three frictional surfaces between the die wall and

sleeve S5, between the twisted surface of the sleeve material inside the container and container
surface S6, and between the container and sleeve S7.
The interface surface between the inner and the outer materials is defined by the angle β,

shown in Fig. 1, defined as

sinβ=
R1i
R2i
sinα (2.3)

The first step inmodeling and analysing ametal forming process by the use of the upper bound
approach is to select a suitable velocity field for the material which is deformed plastically.

2.1. Velocity fields and power terms for deformation zones I and II

For deformation zones I and II, the same velocity field proposed by Ma et al. (2004b) for
mono-metal rod extrusion process through a rotating conical die is extended here for flow of
the bimetallic rod extrusion through rotating conical dies. So the velocity field is described by
following spherical components

U̇r =−vf
(rf
r

)2

cosθ U̇θ =0 U̇ϕ = η1ω̇d sinθ
r3f
r2

(2.4)

where ω̇d is the angular velocity of the die and η is the slippage parameter (angular velocity
ratio of the bimetallic rod at the exit of the conical die to the rotating die).
For the present work, the bonding condition between the core and the sleeve is assumed to

be sticky and there is no slippage between the core and sleeve materials.



630 H. Haghighat, M.M.Mahdavi

The strain rates in spherical coordinates are defined as

ε̇rr =
∂U̇r
∂r
=2vfr

2
f

cosθ

r3
ε̇θθ =

1

r

∂U̇θ
∂θ
+
U̇r
r
=−vfr2f

cosθ

r3

ε̇ϕϕ =
1

r sinθ

∂U̇ϕ
∂ϕ
+
U̇r
r
+
U̇θ
r
cotθ=−vfr2f

cosθ

r3

ε̇rθ =
1

2

(∂U̇θ
∂r
−
U̇θ
r
+
1

r

∂U̇r
∂θ

)

=
1

2
vfr
2
f

sinθ

r3

ε̇θϕ =
1

2

( 1

r sinθ

∂U̇θ
∂ϕ
+
1

r

∂U̇ϕ
∂θ
− cotθ
r
U̇ϕ
)

=0

ε̇ϕr =
1

2

(∂U̇ϕ
∂r
−
U̇ϕ
r
+
1

rsinθ

∂U̇r
∂ϕ

)

=−
1

2
η1ω̇dr

3
f

3sinθ

r3

(2.5)

where ε̇ii (with i= j) is the shear strain rate component.
With the strain rate tensor and the velocity field, the standard upper bound method can

be implemented. This method involves calculating the internal power of deformation over the
deformation zones volume, the shear power losses over two surfaces of velocity discontinuity, and
the frictional power losses between the workpiece and tool.
The internal power dissipated in the deformation zone is given by

Ẇi =
2σo√
3

∫

V

√

1

2
ε̇ijε̇ij dV (2.6)

For deformation zone I that is surrounded by two velocity discontinuity surfaces S1, S2, the
interface surface and the die surface, substituting the strain rate tensor fromEqs. (2.5) into Eq.
(2.6) and noting that dV =2πr2 sinθdrdθ, the internal power of deformation becomes

Ẇi1 =2π
2σs√
3

ro
∫

rf

α
∫

β

√

1

2
ε̇2rr +

1

2
ε̇2
θθ
+
1

2
ε̇2ϕϕ+ ε̇

2
rθ
+ ε̇2ϕr r

2 sinθ dθ dr (2.7)

where σs is the mean flow stress of the sleeve, which is determined by

σs =
1

ε

ε
∫

0

σ dε ε= ln
R21i−R22i
R2
1f
−R2

2f

(2.8)

The internal power of deformation in zone II becomes

Ẇi2 =2π
2σc√
3

ro
∫

rf

β
∫

0

√

1

2
ε̇2rr +

1

2
ε̇2
θθ
+
1

2
ε̇2ϕϕ+ ε̇

2
rθ
+ ε̇2
θϕ
+ ε̇2ϕr r

2 sinθ dθ dr (2.9)

where σc is the mean flow stress of the core, given by

σc =
1

ε

ε
∫

0

σ dε ε= ln
R22i
R2
2f

(2.10)

The general equation for the power losses along the shear surface of velocity discontinuity in the
upper boundmodel is

ẆS =
σo√
3

∫

S

|∆V | dS (2.11)



Upper bound analysis of bimetallic rod extrusion process ... 631

where for the velocity discontinuity surfaces S1 and S3

∆V1 = vo sinθ dS1 =2πr
2
o sinθdθ (2.12)

For the velocity discontinuity surfaces S2 and S4

∆V2 = vf sinθ dS2 =2πr
2
f sinθdθ (2.13)

InsertingEqs. (2.12) and (2.13) intoEq. (2.11), thepowerdissipatedon thevelocity discontinuity
surfaces S1, S2, S3 and S4 are determined as

ẆS1 =π
σs√
3
vor
2
o

(

α−β− sin2α− sin2β
2

)

ẆS3 =π
σc√
3
vor
2
o

(

β− sin2β
2

)

ẆS2 =π
σs√
3
vfr
2
f

(

α−β− sin2α− sin2β
2

)

ẆS4 =π
σc√
3
vfr
2
f

(

β− sin2β
2

)

(2.14)

The general equation for the frictional power losses along the surface with a constant friction
factor m is

Ẇf =m
σo√
3

∫

Sf

|∆V | dS (2.15)

For the conical surface of the die, the frictional surface S5, the magnitude of the velocity diffe-
rence and the differential surface are

∆V3 =
√

∆V 2r +∆V
2
ϕ dS3 =2πr sinαdr (2.16)

where

∆Vr = U̇r
∣

∣

∣

θ=α
= vfr

2
f

cosα

r2
∆Vϕ = rsinαω̇d− U̇ϕ

∣

∣

∣

θ=α
= r sinαω̇d

(

1−
ηr3f
r3

)

(2.17)

Inserting Eqs. (2.17) into Eq. (2.16) and then placing into Eq. (2.15), gives the frictional power
losses along the conical surface of the die as

Ẇf3 =2π
mdσc√
3

ro
∫

rf

√

(

vfr
2
f

cosα

r2

)2

+
[

r sinαω̇d
(

1−η1
r3
f

r3

)]2

r sinαdr (2.18)

where md is the constant friction factor between the sleeve material and the die.

2.2. Velocity fields and power terms for deformation zone III

For deformation zone III and IV, using the cylindrical coordinate system (r,θ,y) in Fig. 1,
the same components of the velocity fieldwere employed byMa et al. (2004b) in the deformation
zone III to analyze the mono-metal rod extrusion process through rotating conical dies

U̇y = vo U̇r =0 U̇θ = η2ω̇dr
y

l
with η2 = η1

(rf
ro

)3

(2.19)

where η2 is the slippage parameter (angular velocity ratio of the bimetallic rod at the entrance
of the conical die to the rotating die).



632 H. Haghighat, M.M.Mahdavi

The strain rates in cylindrical coordinates are defined as

ε̇rr =
∂U̇r
∂r
=0 ε̇θθ =

1

r

∂U̇θ
∂θ
+
U̇r
r
=0

ε̇zz =
∂U̇y
∂y
=0 ε̇rθ =

1

2

(∂U̇θ
∂r
+
1

r

∂U̇r
∂θ
− U̇θ
r

)

=0

ε̇θy =
1

2

(∂U̇θ
∂y
+
1

r

∂U̇y
∂θ

)

=0 ε̇yr =
1

2

(∂U̇r
∂y
+
∂U̇y
∂r

)

=0

ε̇yθ =
1

2

(∂U̇θ
∂y
+
1

r

∂U̇y
∂θ

)

=
1

2
η2ω̇d
r

l

(2.20)

Placing Eqs. (2.20) into Eq. (2.6) and noting that dV = 2πrdrdy, the internal power of defor-
mation in zone III is determined as

Ẇi3 =
2σs√
3

R2i
∫

R1i

l
∫

0

1

2
η2ω̇d
r

l
2πr dr dy=

2π

3
√
3
σsη2ω̇d(R

3
1i−R32i) (2.21)

The internal power of deformation in zone IV is determined as

Ẇi4 =
2σc√
3

R2i
∫

0

l
∫

0

1

2
η2ω̇d
r

l
2πr dr dy=

2π

3
√
3
σcη2ω̇dR

3
2i (2.22)

Thepower dissipated on the frictional surfaces S6 and S7 are also can be obtained byEq. (2.15).
The velocity discontinuity on the surface S6 and its area become

∆V6 =
√

∆V 2y +∆V
2
θ
=

√

v2o +
(

β2ω̇dR1i
y

l

)2

dS7 =2πR1idy (2.23)

The frictional power losses along the surface S6 can be given by

Ẇf6 =
mcσs√
3

l
∫

0

√

v2o +
(

η2ω̇dR1i
y

l

)2

2πR1i dy=2π
mcσs√
3
R21iη2ω̇dA (2.24)

where

A=
1

2

√

( vo
β2ω̇dR1i

)2

+1+
l

2

( vo
β2ω̇dR1i

)2

ln
β2ω̇dR1i+

√

v2o +(β2ω̇dR1i)
2

vo
(2.25)

where mc is the constant friction factor between the sleeve material and the container.

Finally, for the frictional surface S7

Ẇf7 =
mcσys√
3

L−l
∫

0

2πR1ivo dy=2π
mσys√
3
R1ivo(L− l) (2.26)

where σys is the mean flow stress of the sleeve prior to any deformation and L is length of the
rod in the container.



Upper bound analysis of bimetallic rod extrusion process ... 633

2.3. Total power

Based on the upper bound model, the total power needed for the extrusion process can be
obtained by summing the internal power and the power dissipated on all frictional and velocity
discontinuity surfaces as

J∗ = Ẇi1+Ẇi2+Ẇi3+Ẇi4+ẆS1+ẆS2+ẆS3+ẆS4+Ẇf5+Ẇf6+Ẇf7 (2.27)

Therefore, the total upper bound solution for the relative extrusion pressure Pave/σo is given
by

Pave
σs
=
J∗+Mdω̇d
πvoR

2
1iσs

(2.28)

All integrals that are presented in the power terms are evaluated by numerical integration. For
given extrusion conditions, the total power in the equation above is function of the slippage
parameter η1 and twisting length of the material inside the container l.
As the balance among twisting moments must be maintained, the moment applied by the

rotary die is balanced with summing up the moment caused by the circumferential friction in
the container. In addition to the power applied by the punch, a twist moment Md is supplied
by the rotating die, and this moment can be calculated as

Md =2πmd
σs√
3

ro
∫

rf

cosγ3(r sinα)
2 dr cosγ3 =

∆Vϕ3
∆V3

(2.29)

The twist moment within the container is given as

M ′7 =2πmc
σys√
3

l
∫

0

cosγ7R
2
o dy cosγ7 =

∆Vθ7
∆V7

(2.30)

The balance of the couples gives

Md =M
′

7 (2.31)

The twisting length l can be determined by satisfying Eq. (2.32) with a given η1

l=

mdσs
ro
∫

rf

cosγ3(r sinα)
2 dr

mcσysR
2
1i

[

− vo
η2ω̇dR2i

+

√

(

vo
η2ω̇dR2i

)2

+1
]

(2.32)

3. Results and discussion

Tomake a comparison with the developedmodel, a bi-metal rod composed of aluminium as the
core layer and copper as the sleeve layer, was used. The configuration of the sleeve and core
layers is shown in Fig. 2. The flow stresses for copper and aluminium in room temperature were
obtained as (Hwang and Hwang, 2002)

σAl =189.2ε
0.239MPa σcu =335.2ε

0.113MPa (3.1)

The extrusion process is simulated by using the finite element code ABAQUS. A three-
-dimensionalmodel is used forFEManalyses. Thebilletmodel ismeshedwithC3D8R elements.



634 H. Haghighat, M.M.Mahdavi

Fig. 2. Configuration of the bimetallic rod before extrusion (dimensions are in mm)

The punch and die are assumed as rigidmodels. Since the analytical rigid option is used for the
rigid bodies, they are not meshed.

The punch model is loaded by specifying displacement in the axial direction, and the die
model is able to rotate along the die axis and it is fixed in other directions by applying displace-
ment constraints on its nodes. Figure 3a illustrates themesh used to analyze the deformation in
extrusion of thebimetallic rodwith the configuration shown inFig. 2 and α=20◦, vo =1mm/s,
md =0.2, mc =0.2. The deformedmodels of the sleeve and core are shown in Fig. 3b.

Fig. 3. (a) The finite element mesh and (b) the deformedmesh, in extrusion process of a bimetallic rod
through a rotating die

In Fig. 4, the extrusion pressure variation obtained from the upper bound solution is com-
pared with the FEM. The results show a good agreement between the upper bound data and
theFEMresults. As shown in Fig. 4, the analytically predicted pressure is higher than theFEM
result, which is due to the nature of the upper bound theory.

This figure also shows that with the increasing of the angular velocity of the die the relative
extrusion pressure is decreased, but this reduction saturates at a high die angular velocity. It
can be seen that the relative extrusion pressure is decreased by about 6% by the die rotation.

In Fig. 5, the relative extrusion pressure for different semi-die angles obtained from the
upper bound solution is compared with the FEM simulation results. The results show a good
agreement between the upper bound data and the FEM results. It is observed that there is an
optimal die angle which minimizes the extrusion force.

The effect of angular velocity on the relative extrusion pressure for different values of die
friction factors is shown inFig. 6. It is observed that theextrusionpressure isdecreasedbygrowth
of the die angular velocity and drop of the die friction factor, but this reduction saturates at a
high die angular velocity.



Upper bound analysis of bimetallic rod extrusion process ... 635

Fig. 4. Comparison of analytical relative extrusion pressure with FEMdata for different angular
velocities of the die for α=20◦, vo =1mm/s,R1o =15mm,R1f =13mm,R2o =9mm,md =0.2,

mc =0.2

Fig. 5. Comparison of analytical relative extrusion pressure with FEM data for different semi-die angles
for vo =1mm/s,R1o =15mm,R1f =13mm,R2o =9mm, ω=0.1rad/s,md =0.2,mc =0.2

Fig. 6. Effect of angular velocity of the die on the relative extrusion pressure for different friction factors
of the die for α=20◦, vo =1mm/s,R1o =15mm,R1f =13mm,R2o =9mm,mc =0.2

The effect of the die angle on the relative extrusion pressure for different values of die friction
factors is shown in Fig. 7. As it is expected, for a given value of die friction factor, there is an
optimal die angle whichminimizes the extrusion pressure, and the optimum die angle increases
when the friction factor increases. This figure also shows that an increase in the friction factor
of the die tends to increase the extrusion pressure.

The effect of angular velocity on the relative extrusion pressure for different values of the
tube entrance speed is shown in Fig. 8. It is observed that the extrusion pressure is decreased
by a drop in the die angular velocity and a drop in the entrance speed, but this reduction is low
at high entrance speeds.



636 H. Haghighat, M.M.Mahdavi

Fig. 7. Effect of semi-die angle on the relative extrusion pressure for different friction factors of the die
for vo =1mm/s,R1o =15mm,R1f =13mm,R2o =9mm, ω=0.1rad/s,mc =0.2

Fig. 8. Effect of angular velocity of the die on reduction of the extrusion pressure for different extruding
speeds for α=20◦,R1o =15mm, R1f =13mm,R2o =9mm, md =0.2,mc =0.2

4. Conclusions

In this study, an upper boundmodel for analysis of the bimetallic rod extrusion process through
rotating conical dies was developed. Derivations for three main components of the consumed
power during theprocess includingdeformation, discontinuity, frictionpower and the relation for
calculating the relative extrusion pressurewere presented. The results showed a good agreement
between the analytical solution and FEM simulation. The developed upper bound solution can
be very beneficial in studying the influence of multiple variables on the bimetallic rod extrusion
process through rotating conical dies and for a given process parameters. It can be used for
finding the optimum die angle which minimizes the extrusion pressure.

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Manuscript received July 14, 2012; accepted for print September 9, 2012