, Volume 51, Issue 9, pp 2099–2110 | Cite as

Upper bound analysis of wire drawing through a twin parabolic die

  • Shun Hu Zhang
  • Xiao Dong Chen
  • Jian Zhou
  • De Wen Zhao


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.


Twin parabolic die MY criterion Drawing force Surface integral Analytical solution 

List of symbols


Externally supplied energy rate


Shear yield stress


Length of land of die


Length of contact arc


Friction factor

\(n_{\sigma }\)

Stress state coefficient

\(R_{0} ,R,R_{1}\)

Radii at entry, in deformation zone, and at exit


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


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


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


Shear energy rate on the entry and exit section


Shear energy rate on the conjunction section


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


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


Optimal die semi-angle

\(\varDelta R\)

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


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


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


Externally supplied drawing stress


Back tension stress

\(\sigma_{\hbox{max} }\)

Maximum of Mises equivalent stress


Frictional shear stress between the die and the workpiece



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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Copyright information

© Springer Science+Business Media Dordrecht 2016

Authors and Affiliations

  • Shun Hu Zhang
    • 1
    • 2
  • Xiao Dong Chen
    • 2
  • Jian Zhou
    • 1
  • De Wen Zhao
    • 3
  1. 1.Shagang School of Iron and SteelSoochow UniversitySuzhouChina
  2. 2.School of Chemical and Environmental EngineeringSoochow UniversitySuzhouChina
  3. 3.State Key Lab of Rolling and AutomationNortheastern UniversityShenyangChina

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