In this paper, plane strain extrusion through arbitrarily curved dies is investigated analytically, numerically, and experimentally. Two kinematically admissible velocity fields based on assuming proportional angles, angular velocity field, and proportional distances from the midline in the deformation zone, sine velocity field, are developed for use in upper bound models. The relative average extrusion pressures for the two velocity fields are compared to each other and also with the velocity field of a reference for extrusion through a curved die. The results demonstrate that the angular velocity field is the best. Then, by using the developed analytical model, optimum die lengths which minimize the extrusion loads are determined for a streamlined die and also for a wedge shaped die. The corresponding results for those two die shapes are also determined by using the finite element code and by doing some experiments and are compared with upper bound results. These comparisons show a good agreement.

References

References
1.
Avitzur
,
B.
, 1963, “
Analysis of Wire Drawing and Extrusion Through Conical Dies of Small Cone Angle
,”
ASME J. Eng. Ind.
,
85
, pp.
89
96
.
2.
Avitzur
,
B.
, 1964, “
Analysis of Wire Drawing and Extrusion Through Conical Dies of Large Cone Angle
,”
ASME J. Eng. Ind.
,
86
, pp.
305
316
.
3.
Avitzur
,
B.
, 1966, “
Flow Characteristics Through Conical Converging Dies
,”
ASME J. Eng. Ind.
,
88
, pp.
410
420
.
4.
Avitzur
,
B.
, 1967, “
Strain-Hardening and Strain-Rate Effects in Plastic Flow Through Conical Converging Dies
,”
ASME J. Eng. Ind.
,
89
, pp.
556
562
.
5.
Chen
,
C.-T.
, and
Ling
,
F.-F.
, 1968, “
Upper Bound Solutions to Axisymmetric Extrusion Problems
,”
Int. J. Mech. Sci.
,
10
, pp.
863
879
.
6.
Zimmerman
,
Z.
, and
Avitzur
,
B.
, 1970, “
Metal Flow Through Conical Converging Dies—A Lower Upper Bound Approach Using Generalized Boundaries of the Plastic Zone
,”
ASME J. Eng. Ind.
,
92
, pp.
119
129
.
7.
Yang
,
D.-Y.
,
Han
,
C.-H.
, and
Lee
,
B.-C.
, 1985, “
The Use of Generalized Deformation Boundaries for the Analysis of Axisymmetric Extrusion Through Curved Dies
,”
Int. J. Mech. Sci.
,
27
, pp.
653
663
.
8.
Yang
,
D.-Y.
, and
Han
,
C.-H.
, 1987, “
A New Formulation of Generalized Velocity Field for Axisymmetric Forward Extrusion Through Arbitrarily Curved Dies
,”
ASME J. Eng. Ind.
,
109
, pp.
161
168
.
9.
Chen
,
C.-T.
, and
Ling
,
F.-F.
, 1968, “
Upper-Bound Solutions to Axisymmetric Extrusion Problems
,”
Int. J. Mech. Sci.
,
10
, pp.
863
879
.
10.
Nagpal
,
V.
, 1974, “
General Kinematically Admissible Velocity Fields for Some Axisymmetric Metal Forming Problems
,”
ASME J. Eng. Ind.
,
96
, pp.
1197
1201
.
11.
Chen
,
C.-C.
,
Oh
,
S.-I.
, and
Kobayashi
,
S.
, 1979, “
Ductile Fracture in Axisymmetric Extrusion and Drawing—Part I: Deformation Mechanics of Extrusion and Drawing Metal
,”
ASME J. Eng. Ind.
,
101
, pp.
23
35
.
12.
Liu
,
T.-S.
, and
Chung
,
N.-L.
, 1990, “
Extrusion Analysis and Workability Prediction Using Finite Element Method
,”
Comput. Struct.
,
36
, pp.
369
377
.
13.
Kim
,
N.-H.
,
Kang
,
C.-G.
, and
Kim
,
B. M.
, 2001, “
Die Design Optimization for Axisymmetric Hot Extrusion of Metal Matrix Composites
,”
Int. J. Mech. Sci.
,
43
, pp.
1507
1520
.
14.
Lin
,
Z.
,
Juchen
,
X.
,
Xinyun
,
W.
, and
Guoan
,
H.
, 2003, “
Optimization of Die Profile for Improving Die Life in the Hot Extrusion Process
,”
J. Mater. Process. Technol.
,
142
, pp.
659
664
.
15.
Gordon
,
W. -A
,
Van Tyne
,
C.-J.
, and
Moon
,
Y.-H.
, 2007, “
Axisymmetric Extrusion Through Adaptable Dies—Part I: Flexible Velocity Fields and Power Terms
,”
Int. J. Mech. Sci.
,
49
, pp.
86
95
.
16.
Gordon
,
W. -A
,
Van Tyne
,
C.-J.
, and
Moon
,
Y.-H.
, 2007, “
Axisymmetric Extrusion Through Adaptable Dies—Part II: Comparison of Velocity Fields
,”
Int. J. Mech. Sci.
,
49
, pp.
96
103
.
17.
Gordon
,
W. -A
,
Van Tyne
,
C.-J.
, and
Moon
,
Y.-H.
, 2007, “
Axisymmetric Extrusion Through Adaptable Dies—Part III: Minimum Pressure Streamlined Die Shapes
,”
Int. J. Mech. Sci.
,
49
, pp.
104
115
.
18.
Gordon
,
W. -A
,
Van Tyne
,
C.-J.
, and
Moon
,
Y.-H.
, 2007, “
Overview of Adaptable Die Design for Extrusions
,”
Int. J. Mech. Sci.
,
49
, pp.
96
103
.
19.
Hillier
,
M.-J.
, and
Johnson
,
W.
, 1963, “
Plane Strain Extrusion Through Partially Rough Curved Dies
,”
Int. J. Mech. Sci.
,
5
, pp.
191
201
.
20.
Avitzur
,
B.
, 1968,
Metal Forming: Processes and Analysis
,
McGraw-Hill
,
New York
.
21.
Barata Marques
,
M. -J.-M.
, and
Martins
,
P. -A.-F.
, 1989, “
A Solution to Plane Strain Extrusion by the Upper Bound Approach and the Weighted Residuals Method
,”
Int. J. Mech. Sci.
,
31
, pp.
395
406
.
22.
Dewen
,
Z.
,
Hongjin
,
Z.
, and
Guodong
,
W.
, 1995, “
Curvilinear Integral of the Velocity Field of Drawing and Extrusion Through Elliptic Die Profile
,”
Trans. Nonferrous Met. Soc. China
,
5
, pp.
79
83
.
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