12 A feature based analysis of tube extrusion.pdf

Journal of Materials Processing Technology 190 (2007) 363–374

A feature based analysis of tube extrusion

M. Malpani, S. Kumar ∗

Department of Mechanical Engineering, Institute of Technology, Banaras Hindu University, Varanasi 221005, India

Received 20 April 2005; received in revised form 29 September 2006; accepted 2 February 2007

Abstract

The paper presents a feature based upper-bound model to analyze three dimensional complex geometry tube extrusion process having generalized

die and mandrel proﬁles. The analysis is based on kinematically admissible velocity ﬁeld to obtain the optimal extrusion pressure by optimizing the

die length using Variable Interval Golden Section Method. Various parameters such as product proﬁle, shape complexity factor, die and mandrel

proﬁles, reduction, friction, ram velocity and die length, etc., have been chosen for the process and parametric study on extrusion. The experimental

and the proposed analytical results have been compared with experimental work by authors and analytical works of other researchers to validate

the model that shows good agreement.

Keywords: Extrusion; Feature; Upper bound technique

1. Introduction

et al. [17] , Kim et al. [18] , Muller [19] and Wu and Hsu [20] .

Chitkara and Aleen [21] proposed a generalized upper-bound

model to calculate the working pressure in extrusion-piercing

of hollow tubes using a solid circular billets through different

die-mandrel combinations of different ID/OD geometries. They

conducted experiments to validate their results that show good

agreement. Whereas Santosh et al. [22] proposed a feature based

generalized upper bound analysis for solid shapes to be extruded

through different die proﬁles.

Udagawa et al. [23] analyzed axisymmetric tube extrusion

using FEM approach through a ﬂat die using a large plastic

incremental deformation analysis. They observed that under cer-

tain processes conditions, the tensile stress along the die–work

and mandrel-work interfaces leads to surface defects. Reddy et

al. [24] reports a comprehensive investigation of axisymmetric

steady state tube extrusion thorough streamlined die using mixed

(pressure-velocity) FEM formulation and studied the inﬂuence

of process variations like reduction, coefﬁcient of friction, radius

of the mandrel and hardening of the material, etc., on tool design

and ﬁnal product quality for strain hardening materials.

Recently HyperXtrude [25] , an implicit and hp-adaptive FE

based software has been used to analyses considering the extru-

sion process as a ﬂuid ﬂow and heat transfer problem. Pradhan

et al. [26] have proposed use of an Elurarian approach where

in FE software PROCAST is used to simulate plastecine extru-

sion of tubes using porthole die and to predict extrusion process

parameters in a very good manner.

A long, straight and hollow seamless tubular product is

directly extruded from a solid billet using welding chamber dies

such as porthole, bridge or spider. Several other techniques such

as rotary piercing, rolling, forward and backward extrusion as

well as radial extrusion, etc., are also used to produce tubes. Man-

drel is generally ﬁxed to the die and forms internal (ID) geometry

of the product whereas outer (OD) geometry is formed due to

the die input geometry. Die and mandrel proﬁles (conical, ﬂat,

streamlined, hyperbolic, cosine, etc.) are very important to pro-

vide the deformation and ﬂow of the billet material to follow the

path accordingly. Depending upon the ID and OD geometries

and die-mandrel surface proﬁles, an optimum die for extrusion

can be designed as well as manufactured requiring minimum

energy for extruding an alloy.

Several notable attempts have been made to analyze the extru-

sion of tube process using the slab method, the slip line method,

the upper bound method and the Finite Element method by

Sachs and Baldwin [1] , Johnson and Mellor [2] , Kaftanoglu

[3] , Collins and Williams [4] , Avitzur [5] , Chitkara and Butt [6] ,

Mehta et al. [7] , Chang and Choi [8] , Hartley [9] , Vaidyanathan

and Blazynski [10] , Altan [11] , Yang and Han [12] , Yang et al.

[13] , Prakashi and Juneja [14] , Kiuchi [15] , Bae [16] , Pihlainen

∗ Corresponding author.

E-mail address: santoshk itbhuv@yahoo.com (S. Kumar).

doi: 10.1016/j.jmatprotec.2007.02.003

364

M. Malpani, S. Kumar / Journal of Materials Processing Technology 190 (2007) 363–374

From the above review, it appears that there is no general-

ized methodology available for evaluating the extrusion power

and related die parameters for die design of general class of hol-

low extruded shapes. The purpose of present two step model

is to (i) develop an automatic geometry based recognition pro-

cess by ﬁrst drawing the ID and OD geometry of the product

in AutoCAD and recognizing them, generating pseudo vertices

according to the requirements of incompressibility condition of

the material ﬂow and (ii) using the data generated in the ﬁrst step,

to develop an upper bound model for determination of average

ram pressure and to select a press capacity, type of die includ-

ing the die proﬁle, die and mandrel dimensions, and optimum

speed required for tube extrusion using the concept proposed by

Santosh et al. [22] .

Optimum die length has been the major factor in extrusion

die design. Die and mandrel proﬁles are another deciding factor

in selection of hollow die for extruding tubes. The validity of the

proposed two step method having complex ID and OD geometry

as well as die proﬁle has been tested by carrying out experiments

and comparing the results of average extrusion pressure with

the available experimental as well as analytical results by other

authors.

Fig. 1. (a) Initial ID and OD geometries before division and (b) after division

of the entities.

as follows:

Step (1): Obtain the dimensions of the geometry of outer (OD)

and inner proﬁle (ID) of product from user and draw the proﬁle

in Auto CAD using suitable entities.

Step (2): Divide both the proﬁle in more then 100 point entities.

Delete original geometry and save the geometry points of both

proﬁles in separate ﬁle of the AutoCAD in DXF format. For

higher accuracy and good results, division of the geometry in

made in larger number of points.

Step (3): Read the coordinates from entity section of DXF ﬁle

generated in the step (2) for both ID and OD proﬁles. Calculate

their C.G. and subtract it from vertices to get coordinates of

each proﬁle with respect to the origin.

Step (4): Multiply coordinates to the zoom factor to get vertices

of the original product. Divide the outer and inner proﬁle in

equal number of elementals by using the concept of pseudo

coordinate generation ( [22] and [28] ).

Step (5): Use the pseudo vertices generated in step (4) and

again divide the whole geometry in equal elemental area of

hollow product by comparing it with required small elemental

area using generated pseudo vertexes that satisfy Eqs. (1) and

(2) . Save the data in different DATA ﬁle.

2. Data pre-processing and generation

The input geometrical data corresponding to OD and ID

geometries (say DATA.O and DATA.I) are having several ver-

tices ( x and y coordinates) from line entities connected in a closed

sequence. The entities are further required to division due to the

following reasons: (i) during initial division of the geometry, it is

not necessary that points obtained corresponding to both OD and

ID vertices starts from the same position, (ii) it is not necessary

that vertices data written in DATA.O ﬁle is equal to the vertices

data listed in DATA.I ﬁle. It is also possible that both coordi-

nates do not coincide at the line passing through the C.G. of the

ﬁnal product shape and (iii) for the purpose of extrusion incom-

pressibility conditions also, the deformation zone is required to

be divided further into a large number of smaller line entities

to satisfy the equal area requirement at the exit velocity proﬁle.

This is meat using the pseudo vertices generation concept (San-

tosh et al. [22] ) to be applied on the ID and OD geometries of

the product.

In the ﬁrst step, 2D based geometry recognition methodology

is used considering the cross sections of extruded components

to be uniform. The initial data processing of the cross section

requires geometrical and topological information of the com-

ponent in wire frame model derived from a neutral AutoCAD

format such as DXF ﬁle. Fig. 1 (a) shows the initial ID and OD

geometry and Fig. 1 (b) shows after generation of more pseudo

vertices to meet the requirements of velocity and incompressibil-

ity conditions. For more details on the used algorithm (pseudo

vertices generation and implementation), work carried by Mal-

pani [28] may be referred. Fig. 2 shows the ID and OD of

one such case to implement the newly generated pseudo ver-

texes on OD geometry are 5, 6, 7 and on ID geometry are e ,

f and g for implementing the incompressibility condition. The

methodology of data processing and generation is given below

tan − 1 y −

θ l =

(1)

x −

tan − 1 y 2 − y 1

x 2 − x 1

φ l =

(2)

Fig. 2. Implementation of pseudo vertex generation on OD and ID geometries

[28] .

M. Malpani, S. Kumar / Journal of Materials Processing Technology 190 (2007) 363–374

365

Step (6): Reopen the DATA ﬁle and calculate product area,

periphery of the outer proﬁle, minimum inscribing circle diam-

eter, shape complexity factor and total number of elements

generated based on the percentage reduction and velocity of

extrusion. Calculate the ﬁnal product shape and calculate the

billet and mandrel radius.

Step (7): Print the data calculated in step (6) in a ﬁle (say

ARPERI.F). The ﬁle so generated is used to calculate total

power of deformation as explained below.

3. Formulation

Fig. 3. Deformation zone typical streamline surfaces in extrusion.

3.1. Constitutive equations

one that minimizes the total power ϕ T is the actual velocity ﬁeld.

Neglecting the elastic deformation, the strain rate tensor

σ ij

∈ ij

τ | v t | S i d S i

∂v i

∂x j

ϕ T =

d Ω +

(10)

∂v j

∂x i

S i

∈ ij

(3)

where Ω is the plastic deformation zone, τ the shear stress on

velocity discontinuity surfaces S i , i =1,2,3( Fig. 3 ) and are

the tangential velocity discontinuity on S i . Asterisk ( * ) indicates

that the values of stress, strain rate and velocity discontinuity

have been obtained from an assumed kinematically admissible

velocity ﬁeld. The ﬁrst term expresses the internal power of

deformation over the volume of the deformation zone, while

the second term represents the power dissipated in shearing

the material over the velocity discontinuity surfaces and at the

tool–work interface (i.e. frictional power).

where v i and v j represent the velocity components along x i and

x j are the variables in the Cartesian coordinate system. The

constitutive law for rigid plastic/viscoplastic material relating

the deviatoric stress tensor σ ij

∈ ij

and the strain rate tensor

expressed as

σ ij =

2 μ

∈ ij

(4)

where μ is the Levy-Mises coefﬁcient. For a material yielding

according to Von-Mises criterion, the Levy-Mises coefﬁcient μ

is given by

3.3. Deformation zone and velocity boundary conditions

3 ¯

μ =

(5)

The upper bound solution obtained by earlier researchers

using straight and arbitrarily shaped plastic boundaries indicate

that there is little effect of the shapes of surfaces S 1 and S 2

( Fig. 3 ) on the overall solution. Hence, in the present work, the

deformation zone Ω assumed to be bounded by straight plastic

boundaries at the end sections of the die. This assumption sim-

pliﬁes the mathematical treatment of the problem signiﬁcantly

without compromising much on accuracy and provides greater

ﬂexibility. Material is assumed rigid outside the entry and exit

sections of the die. Therefore, the axial velocity ( v z ) at the entry

( v o ) and exit ( v f ) sections of the die should be uniform. These

conditions are given by:

∈

where the generalized yield stress σ and the generalized strain

rate ¯

∈

are deﬁned as

2 σ ij σ ij

σ =

(6)

and

∈=

∈ ij

(7)

The generalized strain ¯

∈

is therefore, deﬁned as

v z = v o on S 1

and

v z = v o on S 2

(11)

∈=

∈

d t

(8)

At cross-section S 2 , different points have different velocities.

A point at the center has maximum velocity and a point on the

periphery has minimum velocity. Corresponding to N different

points, the common velocity of extrudate, v f

where the integration is to be carried along the particle path.

In general σ depends on ¯

, ¯

∈

and temperature T .

is deﬁned as

σ = F ( ¯

∈ ,

∈ ,T )

(9)

i = N

i =

1 v i

In case of cold extrusion, the effect of temperature can be

neglected on generalized stress σ .

v f =

(12)

There should not be any metal ﬂow across boundary S 3 and

the axis of symmetry and boundary S 4 and at the axis of sym-

metry. This condition on the boundaries can be expressed as

v n (normal velocity)

3.2. Upper bound formulation

The upper bound theorem (Prager and Hodge [27] ) states that

among all possible kinematically admissible velocity ﬁelds, the

0, on die surface S 3 and mandrel surface

S 4 and at the axis of symmetry.

366

M. Malpani, S. Kumar / Journal of Materials Processing Technology 190 (2007) 363–374

try, (viii) friction factor between the die and work piece material

and mandrel and work piece material is assumed to be indepen-

dent of slip and (ix) deformation takes under place homogeneous

and steady state conditions.

Assumptions (i) and (ix) state that the reduction ratio is kept

constant while the subdivision of the domain into small ele-

ments. Analyzing one such power element A B BACDD C A

area of rectangle A B C D

cross-section area on product side

area of rectangle ABCD

cross-section area on billet side

(14)

If the coordinates of points C and D and A and B are ( x 1 ,

y 1 ), ( x 2 , y 2 ), ( x 3 , y 3 ) and ( x 4 , y 4 ), respectively, and A a is the

cross-sectional area of the extruded product shape.

Then,

Fig. 4. Geometry of extrusion die and streamlines having generalized section.

The ﬂow of metal has to satisfy the incompressibility condi-

tion as:

( x

O C = a 1 =

1 ) ,O D = a 2 =

1 + y

2 + y

2 ) .

∈=

( ˙

∈ xx +

∈ yy +

∈ zz )

i . e .

(13)

( x

O A = a 3 =

3 ) ,O B = a 4 =

3 + y

4 + y

4 )

3.4. Kinematically admissible velocity and strain rate ﬁelds

Since deformation is under certain assumptions and stream-

lines are the basic ﬂow path during the deformation process,

it is necessary to build the proposed kinematically admissible

velocity using streamlines concept. The geometry of die and the

velocity ﬁeld in terms of streamlines in the steady ﬂow process

of extrusion is shown in Fig. 4 .

One power element A B BACDD C A is chosen to demon-

strate the analytical construction of the velocity and strain rate

ﬁelds. In Fig. 4 , billet with initial radius is given by ( R o ) at entry,

is extruded through a shaped die (continuous) constructed with

help of pre-deﬁned streamlines that represents the die surface, to

the ﬁnal generalized product shape at the exit. The mandrel with

initial radius ( R i ) at entry may be constructed ﬂat, conical by a

number of predeﬁned streamlines representing the surface of the

inner hole. An arbitrary point E on the die surface at the entrance

of the die can be combined with the corresponding point E on

the die surface at the exit. The surface deﬁned by the points O ,

E , E and O becomes a three-dimensional stream surface.

The Upper bound solution is based on the assumptions that

(i) the material of billet passes through the sector ABCD at the

entry and goes through section A B C D at the exit preserving

the extrusion ratio, (ii) stream surface of die consist of ODD O

and mandrel OBB O consists of a number of streamlines (curved

or straight) starting from a point (say E ) at the entry and end-

ing at the corresponding point (say E ) at the exit, maintaining

the proportionality of the position, (iii) the material is incom-

pressible, rigid, and perfectly plastic which follows a particular

strain-hardening curve, (iv) the deformation zone is bounded by

straight plastic boundaries at entry and exit sections of the die,

(v) the neutral line is the line joining C.G. of billet (i.e. point O )

and C.G. of the extruded product cross-section (i.e. point O ),

(vi) the elastic strain is small and can be neglected, (vii) the

C.G. of the OD geometry coincides with the C.G. of ID geome-

a 1 × a 2 =

b 1

a 3 × a 4 , where b =

cos ψ =

( x 1 x 2 + y 1 y 2 )

and

b 1 =

( x 3 x 4 + y 3 y 4 )

from Eq. (14) ,

((1 / 2) a 1 a 2 sin ψ −

(1 / 2) a 3 a 4 sin ψ )

( φ/ 2) R

o −

( φ/ 2) R

A a

πR

o − πR

(15)

and

φA a

a 1 a 2 − a 3 a 4 .

sin ψ =

Let streamlines on the stream surface be represented by a

fourth order polynomial to satisfy the smooth entry and exit of

the material ﬂow. Any coordinate along the streamline EE

formulated in a Cartesian coordinate system as follows:

⎫

⎬

x = f 1 ( z )

= b 1 z

+ b 2 z

+ b 3 z

+ b 4 z + b 5

y = f 2 ( z )

= c 1 z

+ c 2 z

+ c 3 z

+ c 4 z + c 5

(16)

⎭

z = z

where b i and c i (for i = 1, 2, 3, 4 and 5) are constants, determined

by the boundary conditions. Since the streamline does not pro-

duce any abrupt change in ﬂow direction along the extrusion

axis at entry and exit, the boundary conditions can be written as

⎫

⎬

∂x

∂z

∂

x =

( R i + n ) sin φ ;

∂z

⎭

∂y

∂z

∂

y =

( R i + n ) cos φ ;

∂z

at billet side ( z =

(17)

M. Malpani, S. Kumar / Journal of Materials Processing Technology 190 (2007) 363–374

367

⎫

⎬

where

( R i + n ) a 2 sin ψ

R o

∂x

∂z

x =

;

a 2 con φ

R o

at product side ( z = L )

n e = R i + n ;

A =

sin φ ;

B =

− A ;

( R i + n ) a 2 cos ψ

R o

∂y

∂z

⎭

y =

;

π ( a 1 a 2 − a 3 a 4 ) ;

(18)

con

C =

cos φ

and

where R o is the billet radius, φ and ψ are the angles between the

plane of symmetry and the stream surface at entry and exit of

the die, respectively, and L is the die length, R i is the mandrel

radius at entry.

Substituting these boundary conditions from Eqs. (17) and

(18) into Eq. (16) gives

a 2 cos ψ

R o

D =

− C

(25)

Hence

⎡

⎤

A + Bf

C + Df

⎣

⎦

n e A + n e B f e C + n e D f

J =

(26)

⎫

⎬

a 2 sin ψ

R o

n e Bf

n e Df

x = n e

sin φ + n e

−

sin φ

f ( z )

a 2 cos ψ

R o

Determinant of the Jacobin is therefore written as

(19)

y = n e

cos φ + n e

−

cos φ

f ( z )

⎭

det J = n e g ( φ, z ) ,

(27)

z = z

where

[( A + Bf )( C + D f )

( A + B f )( C + Df )]

g ( φ, z )

−

f ( z )

0at z =

0 and

f ( z )

1at z = L

and

n e = R i + n

(20)

(28)

Strain rate components ε ij

are represented by,

∂v i

∂x j

In the present analysis, function f is represented by the fol-

lowing fourth order curve.

∂v j

∂x i

∈ ij =

4 z

3 z

where v i and v j represent the velocity components along x i and

x j axes in Cartesian coordinate system, respectively. The partial

derivatives of the above equations are obtained with the help of

coordinate transformation as:

= f ( z )

−

(21)

Eq. (19) describes coordinates along streamlines inside the

plastically deforming region but the relationship between the

Cartesian and n , φ , z coordinate systems also. Although the

present analysis employs a fourth order curve represented by Eq.

(21) for the description of die and mandrel proﬁle and for the

assumed streamlines of particles, it is to be noted that function f

in Eq. (21) can be any general function of z provided the function

satisﬁes the given boundary conditions as given in Eq. (20) .

In general, mandrel is used as ﬂat shaped and therefore, x and

y coordinates in the function are independent of axial coordinate

z . Hence Eq. (21) for ﬂat mandrel (the inner proﬁle is given with

C 1 = 0):

∂v i

∂x j

∂v i

∂u k

∂x j

(29)

k = 1

Assuming that the plastic zone is bounded by entry and exit

shear surfaces, the velocity ﬁeld components can be obtained.

Because of volume continuity, the velocity component along z -

direction in the Cartesian coordinate system ( v z ) should be v o at

the entrance, and v p at the exit of die. v o and v p are the speeds

of the billet and the outgoing product, respectively. v p

can be

described in terms of v o as

C 1 z

)( φ/ 2 π )

(1 / 2)( a 1 a 2 − a 3 a 4 ) sin ψ v o

π ( R

o − R

f = f ( z )

(22)

v p =

(30)

The Jacobian of Eq. (19) can be found as

These requirements are satisﬁed using Eqs. (20) and (21) .

Therefore, other velocity components for incompressible mate-

rial are determined as

⎡

⎤

∂x

∂n

∂y

∂n

∂z

∂n

⎣

⎦

n e Bf

g ( φ, z ) v o

∂x

∂φ

∂y

∂φ

∂z

∂φ

v x =

J =

(23)

n e Df

g ( φ, z ) v o

∂x

∂z

∂y

∂z

(31)

v y =

g ( φ, z ) v o

Here

x = n e A + n e Bf

y = n e C + n e Df

z = z

v z =

Using the velocity ﬁelds obtained by the above equations it

has been found that the velocity boundaries conditions as given

in Eqs. (11) and (13) are satisﬁed. Since the proposed velocity

(24)

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