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An upper bound solution to extrusion of circular billet to circular shape through cosine dies
Materials & Design Materials and Design 27 (2006) 411–415 www.elsevier.com/locate/matdes
Short communication
An upper bound solution to extrusion of circular billet to circular shape through cosine dies R. Narayanasamy b
a,*
, R. Ponalagusamy b, R. Venkatesan c, P. Srinivasan
c
a Department of Production Engineering, National Institute of Technology, Tiruchirappalli 620 015, Tamil Nadu, India Department of Mathematics and Computer Applications, National Institute of Technology, Tiruchirappalli 620 015, Tamil Nadu, India c School of Mechanical Engineering, SASTRA University, Tirumalaisamudram, Thanjavur 613 402, Tamil Nadu, India
Received 29 June 2004; accepted 16 November 2004 Available online 1 January 2005
Abstract Metal forming by extrusion is one of the widely used metal forming processes. Conventionally the extrusion is carried out using the shear faced dies. In these dies the metal is forced to go through abrupt change in cross-section. Hence, they suffer from the practical problems such as formation of dead metal zone, non-uniform flow of metal, more redundant work and designed based on empirical methods. Modifications have been done in the extrusion dies to incorporate gradual reduction in the area of cross-section in order to ensure smooth flow of metal and to dispense with the problems faced by the conventional dies. Such modified dies are known as streamlined extrusion dies. The profile of the streamlined extrusion dies is the crucial parameter to optimize the extrusion process. Many profiles such as third order polynomial equation, fifth order polynomial equation, Bezier curve, etc., have been suggested for the design of streamlined extrusion dies with the view to reduce the extrusion load for the given reduction ratio. In this work the extrusion die is assumed to have the cosine profile to extrude circular billet to circular shape. The plastic deformation work required to extrude circular billet to circular cross-section through cosine profile based is determined using upper bound solution. It has been proved that the die designed based on cosine profile is superior to the conventional shear dies and the straightly converging dies. It is also proposed to validate the results by the experiments. Ó 2004 Elsevier Ltd. All rights reserved. Keywords: Extrusion; Upper bound; Cosine die
1. Introduction Extrusion is the process by which a block of metal is reduced in cross-section by forcing it to flow through a die orifice under high pressure. Extrusion dies are used in the industries for high production rate and accuracy in the metal forming process. There are many factors that affect the extrusion process like die profile, friction factor, extrusion pressure and temperature. The extrusion process is carried out conventionally by shear faced *
Corresponding author. Tel.: +91 0431 25 00812/01801; fax: +91 0431 25 00133. E-mail address: [email protected] (R. Narayanasamy). 0261-3069/$ - see front matter Ó 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.matdes.2004.11.026
die, as shown in Fig. 1. But shear faced dies have many practical problems such as dead metal zone, breaking of whiskers, more redundant work and above all the design of shear die is done based on experience and made by trial and error methods. But these methods are approximate and time-consuming methods. The profile of the extrusion dies is the important parameter to optimize the extrusion pressure. In earlier work, Nagpal and Altan [1] used the dual stream functions to obtain upper bound solutions for the extrusion of an ellipse from cylindrical billets. Yang and Han [2] proposed an analytic method for estimating extrusion pressure for arbitrarily curved dies using upper bound solution. The three-dimensional approach
412
R. Narayanasamy et al. / Materials and Design 27 (2006) 411–415
Nomenclature R radius of the billet r radius of the extruded component f(z) function of variable z x, y, z Cartesian coordinate system c1, c2, c3, c4 constants g(z) function determined by the determinant J* total power consumption WI plastic deformation, WS die surface friction
Vx, Vy, Vz velocity components V0 velocity of billet a, /, w angles J Jacobian L length of the die Pave average pressure Rs the relative stress r0 yield stress of the material e:ij strain rate components
Deforming Zone
Y Dead Metal Zone
B
o A
Vo
C
Ram
Y'
Product
O B'
U
Fig. 1. Shear die.
for obtaining optimal die shape which produce minimal stress in the extrusion is explained elsewhere [3]. Yang et al. [4] analyzed the forward extrusion of composite rods through curved dies using flow function concept. An upper bound analysis for the extrusion of square sections from square billets through various curved dies is shown in [5]. Narayanasamy, et al. [6] proposed an analytical method for designing the streamlined extrusion dies. In this paper, the extrusion die is assumed to have the cosine profile and an upper bound analysis is proposed for the extrusion of circular section from circular billets. The material flow in the extrusion die does not remain on the same radial plane which contains the longitudinal axis, so that a three-dimensional approach is proposed in this paper. 2. Velocity functions The velocity field that has been derived from incompressibility conditions, and which satisfies the velocity boundary conditions, is a generalized kinematically admissible velocity field. The following assumptions are required to construct the kinematically admissible velocity field for the extrusion of circular sections from circular billets. The circular billet passing through the points OAB at the die entry goes through points O 0 A 0 B 0 at the die exit as shown in Fig. 2.
Ro
n
Die
A'
C'
X
Vf O' Z
r X'
Fig. 2. A general deformation zone for cosine die.
Any coordinate along line AA 0 as in Fig. 2 is formulated in a Cartesian coordinate system as follows: pz þ c2 ; 2L pz y ¼ c3 cos þ c4 ; 2L z ¼ z; x ¼ c1 cos
ð1Þ
where the constants c1, c2, c3 and c4 are determined by the following boundary conditions. Consider that the line does not produce any abrupt change of flow direction along the extrusion axis at the entry and the exit of die, the boundary conditions being given for Eq. (1) as ) x ¼ n sin / oy ¼0 oz at z ¼ 0; ð2aÞ ¼0 y ¼ n cos / oy oz ¼0 x ¼ n cos w oy oz ¼0 y ¼ n cos w oy oz
) at z ¼ L;
ð2bÞ
R. Narayanasamy et al. / Materials and Design 27 (2006) 411–415
where ÔnÕ is the distance from the axis to an arbitrary point ÔAÕ at the die entry, / and w are the sweep angles at entry and exit of the die, respectively, and L is the length of the die. Substituting of these boundary conditions into Eq. (1), the following equation is obtained: h i r pz r x ¼ n sin / sin w cos þ n sin w; R R h i 2L r pz r ð3Þ y ¼ n cos / cos w cos þ n cos w; R 2L R z ¼ z;
413
where R is the radius of the billet and r the radius of the extruded component. The domains of variation of n,/, z as follows:
Assuming the plastic zone to be bounded by entry and exit shear faces, the velocity field components can be obtained. Using volume constant principle, velocity of the billet along z-direction is V0 at the entrance and Vf at the exit of die. Then the velocity component of the deforming materials along x, y and z directions namely Vx, Vy and Vz could be obtained as per the following equations: n sin / 1 Rr f 0 ðzÞ Vx ¼ V 0; gðzÞ n cos / 1 Rr f 0 ðzÞ ð7Þ V 0; Vy ¼ gðzÞ 1 V0 Vz ¼ gðzÞ
0 6 n 6 R;
where
0 6 / 6 90 ; 0 6 w 6 90 ;
gðzÞ ¼
hr R
ð1 f Þ þ f
i2
:
0 6 z 6 L: Substituting f ðzÞ ¼ cos pz=2L in Eq. (3), the following is obtained: h ri r x ¼ n sin / 1 f ðzÞ þ n sin w; Ri R h r r ð4Þ y ¼ n cos / 1 f ðzÞ þ n cos w; R R z ¼ z: Function f(z) represented by cosine profile describes not only the coordinates inside the plastically deforming region but also the relationship between Cartesian and n, /, z coordinate system. It is to be noted that function f(z) can be any general function of ÔzÕ providing the function satisfies the boundary conditions of the problem. The Jacobian of Eq. (4) can be found as A þ Bf J ¼ C þ Df 0
nðA0 þ B0 Þf 0
0
nðC þ D Þf 0
nBf 0 nDf 1
0
2 W I ¼ pffiffiffi r0 3 2 ¼ pffiffiffi r0 3
:
Z V
Z
L
0
Z
1 1 : : 2 eij eij dV ; 2 1 Z /m Z R0 1 : : 2 eij eij j det J j dn du dz 2 0 0
jDV 3 j dS 3 ; S3
where
where ð1 f Þ þ f
where
e:ij i2
ð8Þ
Z L Z /m h i12 r0 V 2x þ V 2y þ V 2z ¼ m pffiffiffi n¼R0 3 0 0 1 dðx; zÞ d/ dz; cos a dðn; zÞ
det J ¼ ngðzÞ;
R
J ¼ W I þ W S;
r0 W S ¼ m pffiffiffi 3
Determinant of the Jacobian is there written as
gðzÞ ¼
The total power consumption (J*) required to deform circular billets to circular section through the cosine die is represented as the sum of the power consumed due to the plastic deformation (WI) and due to the die surface friction (WS). The cosine die produces no velocity discontinuities at the velocity boundaries. The total power consumption can be obtained as
ð5Þ
where r r A ¼ sin / A0 ¼ cos /; R hR h ri ri B ¼ 1 sin / B0 ¼ 1 cos /; R R r r 0 C ¼ cos / C ¼ sin /; Rh i h R ri r 0 D ¼ 1 cos / D ¼ sin /: R R
hr
3. Upper bound solution
ð6Þ
1 oV i oV j ¼ þ : 2 oxj oxi
ð9Þ
Knowing the velocity components, strain rate components and the coordinate transformation, the volume integration is numerically carried out by SimpsonÕs
414
R. Narayanasamy et al. / Materials and Design 27 (2006) 411–415
one-third rule for given values of yield stress of the material, r0 to obtain the plastic deformation work. The power computed can be converted to the average pressure (Pave) and the relative stress (Rs)
RS ¼
Total resistance, Ref [3]
WI ; pR2 V 0
ð10Þ
P ave : r0
ð11Þ
The die length (L) is also reduced to the relative die length RL ¼
L : R
ð12Þ
1.5 Relative stress ( Pave / σ0 )
P ave ¼
Reduction of Area : 60% Frictional factor m= 0.1
2.0
Total resistance, Cosine die 1.0
Ref [3]
Plastic deformation [WI]
Cosine die [Present work]
0.5 Die surface friction [WS] Ref [3]
4. Results and discussion Upper bound solution is theoretically applied for the predication of plastic deformation work (WI) for extrusion of circular billet to circular shape. Similar work was carried out by Gunasekera and Hoshino [3] for extruding polygonal section from circular billet, and by Maity et al. [5] to extrude square billet to square shape. Comparison is made between the performance of the extrusion dies designed based on straightly converging dies, concave circular with cosine dies under no frictional condition. Fig. 3 shows the variation of extrusion pressure with respect to reduction percentage. It is observed that under no friction condition the cosine profile based die consumes less extrusion pressure compared to straightly converging and concave circular profile based dies. Hence, it is noted that cosine profile based die is superior to straightly converging and concave circular dies. The power consumed due to plastic deformation (WI), die surface friction (WS) and total power consumption are computed for the cosine die as shown in Fig. 4. Comparison is also made between straightly converg-
Extrusion Pressure (Peva /σ0)
6.0 Concave circular (Maity[5])
5.0 4.0
Straight tapered die (Maity[5])
3.0
cosine die (Present work)
1.0
35
40
0.5
1.0
1.5
2.0
Relative die length (L / R0) Fig. 4. Variation of power components.
ing dies Gunasekera and Hoshino [3] and cosine profile based die to study the variation of relative extrusion pressure with respect to relative die length. It is further observed that the cosine profile based die needs lower total power consumption, plastic deformation and die surface friction for all the values of relative die length compared to straightly converging die.
5. Conclusion An upper bound solution has been developed for three-dimensional extrusion of circular sections from a circular bar through cosine die. It is concluded that the extrusion dies based on cosine profile is superior to straightly converging and concave circular profile. It is further observed that the cosine profile based die needs lower plastic deformation work, die surface friction and total power consumption compared to straightly converging die for all the values of relative die length compared to straightly converging die.
References
2.0
30
0
45
50
55
60
65
Reduction Percent
Fig. 3. Variation of the extrusion pressure with percentage reduction.
[1] Nagpal V, Altan T. Analysis of three-dimensional metal flow in extrusion shapes with the use of dual stream functions. In: Proceedings of the 3rd North American metal working research conference, Pitttsburgh; 1975. p. 26–40. [2] Yang DY, Han CH. A new formulation of generalized velocity field for axisymmetric forward extrusion through arbitrarily curved dies. Trans ASME J Eng Ind 1987;109:161–8.
R. Narayanasamy et al. / Materials and Design 27 (2006) 411–415 [3] Gunasekera JS, Hoshino S. Analysis of extrusion or drawing of polygonal sections through straightly converging dies. Trans ASME J Eng Ind 1982;104:38–44. [4] Yang DY, Kim YG, Lee CM. An upper-bound solution for axisymmetric extrusion of composite rods through curved dies. Int J Mach Tool Manu 1991;31(4):565–75.
415
[5] Maity KP, Kar PK, Das NS. A class of upper-bound solutions for the extrusions of square shapes from square billets through curved dies. J Mater Process Technol 1996;62:185–90. [6] Narayanasamy R, Srinivasan P, Venkatesan R. Computer aided design and manufacture of streamlined extrusion dies. J Mater Process Technol 2003;138:262–4.
Short communication
An upper bound solution to extrusion of circular billet to circular shape through cosine dies R. Narayanasamy b
a,*
, R. Ponalagusamy b, R. Venkatesan c, P. Srinivasan
c
a Department of Production Engineering, National Institute of Technology, Tiruchirappalli 620 015, Tamil Nadu, India Department of Mathematics and Computer Applications, National Institute of Technology, Tiruchirappalli 620 015, Tamil Nadu, India c School of Mechanical Engineering, SASTRA University, Tirumalaisamudram, Thanjavur 613 402, Tamil Nadu, India
Received 29 June 2004; accepted 16 November 2004 Available online 1 January 2005
Abstract Metal forming by extrusion is one of the widely used metal forming processes. Conventionally the extrusion is carried out using the shear faced dies. In these dies the metal is forced to go through abrupt change in cross-section. Hence, they suffer from the practical problems such as formation of dead metal zone, non-uniform flow of metal, more redundant work and designed based on empirical methods. Modifications have been done in the extrusion dies to incorporate gradual reduction in the area of cross-section in order to ensure smooth flow of metal and to dispense with the problems faced by the conventional dies. Such modified dies are known as streamlined extrusion dies. The profile of the streamlined extrusion dies is the crucial parameter to optimize the extrusion process. Many profiles such as third order polynomial equation, fifth order polynomial equation, Bezier curve, etc., have been suggested for the design of streamlined extrusion dies with the view to reduce the extrusion load for the given reduction ratio. In this work the extrusion die is assumed to have the cosine profile to extrude circular billet to circular shape. The plastic deformation work required to extrude circular billet to circular cross-section through cosine profile based is determined using upper bound solution. It has been proved that the die designed based on cosine profile is superior to the conventional shear dies and the straightly converging dies. It is also proposed to validate the results by the experiments. Ó 2004 Elsevier Ltd. All rights reserved. Keywords: Extrusion; Upper bound; Cosine die
1. Introduction Extrusion is the process by which a block of metal is reduced in cross-section by forcing it to flow through a die orifice under high pressure. Extrusion dies are used in the industries for high production rate and accuracy in the metal forming process. There are many factors that affect the extrusion process like die profile, friction factor, extrusion pressure and temperature. The extrusion process is carried out conventionally by shear faced *
Corresponding author. Tel.: +91 0431 25 00812/01801; fax: +91 0431 25 00133. E-mail address: [email protected] (R. Narayanasamy). 0261-3069/$ - see front matter Ó 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.matdes.2004.11.026
die, as shown in Fig. 1. But shear faced dies have many practical problems such as dead metal zone, breaking of whiskers, more redundant work and above all the design of shear die is done based on experience and made by trial and error methods. But these methods are approximate and time-consuming methods. The profile of the extrusion dies is the important parameter to optimize the extrusion pressure. In earlier work, Nagpal and Altan [1] used the dual stream functions to obtain upper bound solutions for the extrusion of an ellipse from cylindrical billets. Yang and Han [2] proposed an analytic method for estimating extrusion pressure for arbitrarily curved dies using upper bound solution. The three-dimensional approach
412
R. Narayanasamy et al. / Materials and Design 27 (2006) 411–415
Nomenclature R radius of the billet r radius of the extruded component f(z) function of variable z x, y, z Cartesian coordinate system c1, c2, c3, c4 constants g(z) function determined by the determinant J* total power consumption WI plastic deformation, WS die surface friction
Vx, Vy, Vz velocity components V0 velocity of billet a, /, w angles J Jacobian L length of the die Pave average pressure Rs the relative stress r0 yield stress of the material e:ij strain rate components
Deforming Zone
Y Dead Metal Zone
B
o A
Vo
C
Ram
Y'
Product
O B'
U
Fig. 1. Shear die.
for obtaining optimal die shape which produce minimal stress in the extrusion is explained elsewhere [3]. Yang et al. [4] analyzed the forward extrusion of composite rods through curved dies using flow function concept. An upper bound analysis for the extrusion of square sections from square billets through various curved dies is shown in [5]. Narayanasamy, et al. [6] proposed an analytical method for designing the streamlined extrusion dies. In this paper, the extrusion die is assumed to have the cosine profile and an upper bound analysis is proposed for the extrusion of circular section from circular billets. The material flow in the extrusion die does not remain on the same radial plane which contains the longitudinal axis, so that a three-dimensional approach is proposed in this paper. 2. Velocity functions The velocity field that has been derived from incompressibility conditions, and which satisfies the velocity boundary conditions, is a generalized kinematically admissible velocity field. The following assumptions are required to construct the kinematically admissible velocity field for the extrusion of circular sections from circular billets. The circular billet passing through the points OAB at the die entry goes through points O 0 A 0 B 0 at the die exit as shown in Fig. 2.
Ro
n
Die
A'
C'
X
Vf O' Z
r X'
Fig. 2. A general deformation zone for cosine die.
Any coordinate along line AA 0 as in Fig. 2 is formulated in a Cartesian coordinate system as follows: pz þ c2 ; 2L pz y ¼ c3 cos þ c4 ; 2L z ¼ z; x ¼ c1 cos
ð1Þ
where the constants c1, c2, c3 and c4 are determined by the following boundary conditions. Consider that the line does not produce any abrupt change of flow direction along the extrusion axis at the entry and the exit of die, the boundary conditions being given for Eq. (1) as ) x ¼ n sin / oy ¼0 oz at z ¼ 0; ð2aÞ ¼0 y ¼ n cos / oy oz ¼0 x ¼ n cos w oy oz ¼0 y ¼ n cos w oy oz
) at z ¼ L;
ð2bÞ
R. Narayanasamy et al. / Materials and Design 27 (2006) 411–415
where ÔnÕ is the distance from the axis to an arbitrary point ÔAÕ at the die entry, / and w are the sweep angles at entry and exit of the die, respectively, and L is the length of the die. Substituting of these boundary conditions into Eq. (1), the following equation is obtained: h i r pz r x ¼ n sin / sin w cos þ n sin w; R R h i 2L r pz r ð3Þ y ¼ n cos / cos w cos þ n cos w; R 2L R z ¼ z;
413
where R is the radius of the billet and r the radius of the extruded component. The domains of variation of n,/, z as follows:
Assuming the plastic zone to be bounded by entry and exit shear faces, the velocity field components can be obtained. Using volume constant principle, velocity of the billet along z-direction is V0 at the entrance and Vf at the exit of die. Then the velocity component of the deforming materials along x, y and z directions namely Vx, Vy and Vz could be obtained as per the following equations: n sin / 1 Rr f 0 ðzÞ Vx ¼ V 0; gðzÞ n cos / 1 Rr f 0 ðzÞ ð7Þ V 0; Vy ¼ gðzÞ 1 V0 Vz ¼ gðzÞ
0 6 n 6 R;
where
0 6 / 6 90 ; 0 6 w 6 90 ;
gðzÞ ¼
hr R
ð1 f Þ þ f
i2
:
0 6 z 6 L: Substituting f ðzÞ ¼ cos pz=2L in Eq. (3), the following is obtained: h ri r x ¼ n sin / 1 f ðzÞ þ n sin w; Ri R h r r ð4Þ y ¼ n cos / 1 f ðzÞ þ n cos w; R R z ¼ z: Function f(z) represented by cosine profile describes not only the coordinates inside the plastically deforming region but also the relationship between Cartesian and n, /, z coordinate system. It is to be noted that function f(z) can be any general function of ÔzÕ providing the function satisfies the boundary conditions of the problem. The Jacobian of Eq. (4) can be found as A þ Bf J ¼ C þ Df 0
nðA0 þ B0 Þf 0
0
nðC þ D Þf 0
nBf 0 nDf 1
0
2 W I ¼ pffiffiffi r0 3 2 ¼ pffiffiffi r0 3
:
Z V
Z
L
0
Z
1 1 : : 2 eij eij dV ; 2 1 Z /m Z R0 1 : : 2 eij eij j det J j dn du dz 2 0 0
jDV 3 j dS 3 ; S3
where
where ð1 f Þ þ f
where
e:ij i2
ð8Þ
Z L Z /m h i12 r0 V 2x þ V 2y þ V 2z ¼ m pffiffiffi n¼R0 3 0 0 1 dðx; zÞ d/ dz; cos a dðn; zÞ
det J ¼ ngðzÞ;
R
J ¼ W I þ W S;
r0 W S ¼ m pffiffiffi 3
Determinant of the Jacobian is there written as
gðzÞ ¼
The total power consumption (J*) required to deform circular billets to circular section through the cosine die is represented as the sum of the power consumed due to the plastic deformation (WI) and due to the die surface friction (WS). The cosine die produces no velocity discontinuities at the velocity boundaries. The total power consumption can be obtained as
ð5Þ
where r r A ¼ sin / A0 ¼ cos /; R hR h ri ri B ¼ 1 sin / B0 ¼ 1 cos /; R R r r 0 C ¼ cos / C ¼ sin /; Rh i h R ri r 0 D ¼ 1 cos / D ¼ sin /: R R
hr
3. Upper bound solution
ð6Þ
1 oV i oV j ¼ þ : 2 oxj oxi
ð9Þ
Knowing the velocity components, strain rate components and the coordinate transformation, the volume integration is numerically carried out by SimpsonÕs
414
R. Narayanasamy et al. / Materials and Design 27 (2006) 411–415
one-third rule for given values of yield stress of the material, r0 to obtain the plastic deformation work. The power computed can be converted to the average pressure (Pave) and the relative stress (Rs)
RS ¼
Total resistance, Ref [3]
WI ; pR2 V 0
ð10Þ
P ave : r0
ð11Þ
The die length (L) is also reduced to the relative die length RL ¼
L : R
ð12Þ
1.5 Relative stress ( Pave / σ0 )
P ave ¼
Reduction of Area : 60% Frictional factor m= 0.1
2.0
Total resistance, Cosine die 1.0
Ref [3]
Plastic deformation [WI]
Cosine die [Present work]
0.5 Die surface friction [WS] Ref [3]
4. Results and discussion Upper bound solution is theoretically applied for the predication of plastic deformation work (WI) for extrusion of circular billet to circular shape. Similar work was carried out by Gunasekera and Hoshino [3] for extruding polygonal section from circular billet, and by Maity et al. [5] to extrude square billet to square shape. Comparison is made between the performance of the extrusion dies designed based on straightly converging dies, concave circular with cosine dies under no frictional condition. Fig. 3 shows the variation of extrusion pressure with respect to reduction percentage. It is observed that under no friction condition the cosine profile based die consumes less extrusion pressure compared to straightly converging and concave circular profile based dies. Hence, it is noted that cosine profile based die is superior to straightly converging and concave circular dies. The power consumed due to plastic deformation (WI), die surface friction (WS) and total power consumption are computed for the cosine die as shown in Fig. 4. Comparison is also made between straightly converg-
Extrusion Pressure (Peva /σ0)
6.0 Concave circular (Maity[5])
5.0 4.0
Straight tapered die (Maity[5])
3.0
cosine die (Present work)
1.0
35
40
0.5
1.0
1.5
2.0
Relative die length (L / R0) Fig. 4. Variation of power components.
ing dies Gunasekera and Hoshino [3] and cosine profile based die to study the variation of relative extrusion pressure with respect to relative die length. It is further observed that the cosine profile based die needs lower total power consumption, plastic deformation and die surface friction for all the values of relative die length compared to straightly converging die.
5. Conclusion An upper bound solution has been developed for three-dimensional extrusion of circular sections from a circular bar through cosine die. It is concluded that the extrusion dies based on cosine profile is superior to straightly converging and concave circular profile. It is further observed that the cosine profile based die needs lower plastic deformation work, die surface friction and total power consumption compared to straightly converging die for all the values of relative die length compared to straightly converging die.
References
2.0
30
0
45
50
55
60
65
Reduction Percent
Fig. 3. Variation of the extrusion pressure with percentage reduction.
[1] Nagpal V, Altan T. Analysis of three-dimensional metal flow in extrusion shapes with the use of dual stream functions. In: Proceedings of the 3rd North American metal working research conference, Pitttsburgh; 1975. p. 26–40. [2] Yang DY, Han CH. A new formulation of generalized velocity field for axisymmetric forward extrusion through arbitrarily curved dies. Trans ASME J Eng Ind 1987;109:161–8.
R. Narayanasamy et al. / Materials and Design 27 (2006) 411–415 [3] Gunasekera JS, Hoshino S. Analysis of extrusion or drawing of polygonal sections through straightly converging dies. Trans ASME J Eng Ind 1982;104:38–44. [4] Yang DY, Kim YG, Lee CM. An upper-bound solution for axisymmetric extrusion of composite rods through curved dies. Int J Mach Tool Manu 1991;31(4):565–75.
415
[5] Maity KP, Kar PK, Das NS. A class of upper-bound solutions for the extrusions of square shapes from square billets through curved dies. J Mater Process Technol 1996;62:185–90. [6] Narayanasamy R, Srinivasan P, Venkatesan R. Computer aided design and manufacture of streamlined extrusion dies. J Mater Process Technol 2003;138:262–4.