Abstract
In this study, a methodology has been presented for radial positioning of the die holes in multi-hole extrusion of non-symmetric sections. The objective of radial positioning is to minimize exit profile curvature. For this purpose, a two-hole die with non-symmetric T-shaped holes has been investigated. A kinematically admissible velocity field at deformation zone has been developed. The deformation region includes the dead metal zone (DMZ) which is assumed to be linear. The DMZ length was obtained by energy minimization through the upper bound method. To predict the exit profile curvature a deviation function has been suggested. Using the proposed function, the velocity field has been used for prediction of the exit profile curvature and thereby positioning of the die holes. The predictions of the proposed methodology were validated with physical modeling. With little development, this methodology could be applied to dies, which have more holes and/or generate profiles that are more complex. This will help die designers enhance the quality of extrusion process.
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Abbreviations
- e1, e2:
-
Off-centricity values of exit section in X and Y directions, respectively
- \( (e3,e4),(e5,e6) \) :
-
Co-ordinates of center points at exit and at entry
- \( f,g,h \) :
-
Position vector components of any particle inside the deformation zone in X, Y and Z directions, respectively
- u, q, t :
-
Parameters changing between 0 and 1
- \( F_{1} ,F_{2} \) :
-
Conformal functions
- \( R \) :
-
Initial billet radius
- L :
-
DMZ length
- L c :
-
Billet length outside the assumed deformation zone
- V 0 :
-
Entry velocity
- A 0 :
-
Billet cross sectional area
- RA :
-
Reduction in area
- y c :
-
Centroid of semi-circle
- \( \dot{W}_{tot} \) :
-
Upper bound on extrusion power
- \( \dot{W}_{d} \) :
-
Power of deformation in the assumed converging deformation region
- \( \dot{W}_{i} ,\dot{W}_{e} /\dot{W}_{f} ,\dot{W}_{s} ,\dot{W}_{fc} \) :
-
Power consumptions due to plastic deformation, shear (entry/exit), friction at the surface of DMZ and friction at the billet–container interface, respectively
- m c :
-
Constant of friction factor at the billet–container interface
- Y :
-
Yield stress of billet
- V x , V y , V z :
-
Velocity components in X, Y and Z directions, respectively
- G(u, q, t):
-
Improved velocity function
- M 3 :
-
Friction function
- b 1, b 2, b 3 :
-
Constants of friction function
- J :
-
Jacobian
- \( \det J \) :
-
Determinant of Jacobian
- \( P_{nor} \) :
-
Normalized extrusion pressure
- \( \Delta V_{e} ,S_{e} \) :
-
Velocity discontinuity and surface of velocity discontinuity at entry
- \( \Delta V_{f} ,S_{f} \) :
-
Velocity discontinuity and surface of velocity discontinuity at exit
- \( K_{xz} ,K_{yz} \) :
-
Exit profile curvature in XZ and YZ planes, respectively
- \( \phi ,\theta \) :
-
Angle of the deforming region at the entry and at the exit of the assumed converging deformation zone
- \( \rho_{0} \) :
-
Sweep radius at the entry of deformation zone
- \( \rho_{1} ,\rho_{2} \) :
-
Radius of the deforming region at the entry and at the exit, respectively
- \( \gamma_{r} ,\gamma_{d} \) :
-
Angle of inclination of a surface element within the deformation zone in radians and degree respectively
- \( \kappa_{xz} ,\kappa_{yz} \) :
-
Curvature of the extruded profile from triangular elements in XZ and YZ planes
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The authors would like to thank Mr. Vahid Aligholizadeh for his helps and comments.
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Nazari Onlaghi, S., Assempour, A. On the minimization of the exit profile curvature in extrusion through multi-hole dies: a methodology and some verifications. Meccanica 50, 1249–1261 (2015). https://doi.org/10.1007/s11012-014-0080-1
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DOI: https://doi.org/10.1007/s11012-014-0080-1