# Upper bound analysis of wire drawing through a twin parabolic die

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## Abstract

According to the demand of stream function, a twin parabolic die as a streamline for wire drawing was designed and its kinematically admissible velocity field was then developed. Using the specific energy rate of MY (mean yield) criterion, the mathematical expression of internal deformation energy rate is obtained. Through surface integral of the proposed velocity field, the friction energy rate is also determined. Based on upper bound method, an analytical solution of drawing force is ultimately obtained. The effect of percentage reduction and die semi-angle on the drawing force are discussed. Comparison shows that the calculated drawing forces of the proposed die closely match its numerical drawing forces, and are the lowest compared with those based on conical die, hyperbolic die and elliptic die.

## Keywords

Twin parabolic die MY criterion Drawing force Surface integral Analytical solution## List of symbols

*J*^{*}Externally supplied energy rate

*k*Shear yield stress

*L*Length of land of die

*l*Length of contact arc

*m*Friction factor

- \(n_{\sigma }\)
Stress state coefficient

- \(R_{0} ,R,R_{1}\)
Radii at entry, in deformation zone, and at exit

*U*Flow volume per second of the deformation part

- \(v_{0} ,v_{1}\)
Horizontal velocities at entry and exit

- \(v_{z} ,v_{r} ,v_{\theta }\)
Velocity components in

*z, r, θ*directions*v*_{t}Velocity of a particle at a point

- \(\varDelta v_{t}\)
Tangential discontinuity along die surface

- \(\left( {v_{r} } \right)_{L} ,\;\left( {v_{r} } \right)_{R}\)
Radial velocity components on the left and right sides of the

*BE*section- \(\dot{W}_{f}\)
Friction energy rate

- \(\dot{W}_{f1} ,\dot{W}_{f2} ,\dot{W}_{f}\)
Friction energy rates in zone I and zone II, and in the total zone

- \(\dot{W}_{s1}\)
Shear energy rate on the entry and exit section

- \(\dot{W}_{s2}\)
Shear energy rate on the conjunction section

- \(\dot{W}_{s}\)
Total shear energy rate

- \(\dot{W}_{i1} ,\dot{W}_{i2} ,\dot{W}_{i}\)
Internal deformation energy rate in zone I, II and in the total zone

*α*Die semi-angle, \(\alpha = \arctan \left( {{{\varDelta R} \mathord{\left/ {\vphantom {{\varDelta R} l}} \right. \kern-0pt} l}} \right)\)

- \(\alpha_{opt}\)
Optimal die semi-angle

- \(\varDelta R\)
Absolute radius reduction,\(\varDelta R = R_{0} - R_{1}\)

*λ*Percentage reduction, \(\lambda = {{\varDelta R} \mathord{\left/ {\vphantom {{\varDelta R} {R_{0} }}} \right. \kern-0pt} {R_{0} }}\)

- \(\dot{\varepsilon }_{z} ,\dot{\varepsilon }_{r} ,\dot{\varepsilon }_{\theta }\)
Strain rate components in

*z, r, θ*directions- \(\dot{\varepsilon }_{\hbox{max} } ,\dot{\varepsilon }_{\hbox{min} }\)
Maximum and minimum strain rate components

- \(\eta\)
Elongation coefficient of drawing, \(\eta = \left( {{{R_{0} } \mathord{\left/ {\vphantom {{R_{0} } {R_{1} }}} \right. \kern-0pt} {R_{1} }}} \right)^{2}\)

- \(\sigma_{s} ,\;D\left( {\dot{\varepsilon }_{ij} } \right)\)
Yield stress and specific plastic energy rate

- \(\sigma_{f}\)
Externally supplied drawing stress

- \(\sigma_{zb}\)
Back tension stress

- \(\sigma_{\hbox{max} }\)
Maximum of Mises equivalent stress

- \(\tau_{f}\)
Frictional shear stress between the die and the workpiece

## Notes

### Acknowledgments

The authors wish to acknowledge the National Natural Science Foundation of China (Grant No. 51504156), the Basic Research Program of Jiangsu Province (Grant No. BK20140334), the Natural Science Foundation of the Jiangsu Higher Education Institutions of China (Grant No. 14KJB460024) and the Project Funded by China Postdoctoral Science Foundation (Grant No. 2014M561707). The authors also wish to acknowledge valuable suggestions from reviewers.

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