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Prediction of exit profile distortion in forward extrusion process using Riemann mapping theorem and upper bound method

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Abstract

One of the substantial topics in the forward extrusion process is using theoretical methods to predict the exit profile distortion due to the non-symmetry of sections or the off-centricity of die cavity. In this paper, a new approach based on the Riemann mapping theorem and upper bound method is developed to obtain the velocity field and the strain distribution. Then, the strain distribution is determined for the die with off-centered square sections. Afterwards, the exit profile curvature is calculated using an approach based on elastic–plastic bending of beams. Theoretical results are compared to the results yielded through experimental and finite element simulation and validated. Subsequently, the effect of various parameters such as relative die length and friction factor is examined on the amount of the exit profile distortion. The results show that the obtained theory can be used to predict the strain distribution and the exit profile distortion in addition to the process pressure.

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Abbreviations

Bz, Dz :

Arbitrary curve boundary and its area

Bw, Dw :

Unite circle boundary and its area

E:

Elastic modulus

J:

Total power consumption

K:

Strength coefficient

Kxz, Kyz :

Exit profile curvature in XZ and YZ planes

L:

Die length

L/R:

Relative die length

M(u,q,t):

Function defining the velocity in the z direction

Pave :

Average extrusion pressure

R:

Radius of the billet

T:

Time

Wi :

Power due to internal deformation

Wx :

Power due to exit surface of velocity discontinuities

We :

Power due to entry surface of velocity discontinuities

Wf :

Power due to die–material interface friction

a:

Half of the square length

e1, e2 :

Off-centric positions in x and y directions

f(z):

Mapping power series

f, g, h:

Functions of u, q, and t, which representing position of control points in Cartesian coordinates

m:

Frictional factor

n:

Strain hardening exponent

u, q, t:

Dimensionless Parameters changing between 0 and 1

ye :

Distance of the yield point to the neutral axis

\(\varepsilon_{e}\) :

Strain at the yield point

\({\dot{\upvarepsilon }}_{\text{ij }}\) :

Components of strain rate

\(\varepsilon_{\text{x}}\), \(\varepsilon_{\text{y}}\), \(\varepsilon_{\text{z}}\) :

Strain components

\(\vartheta\) :

Poisson ratio

\(\rho\) :

Density

\(\rho_{{f,{\text{xz}}}}\), \(\rho_{{f,{\text{yz}}}}\) :

Final radius of profile curvature

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Funding

This study was funded by Babol Noshirvani University of Technology (grant number BNUT/370203/97).

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Authors and Affiliations

  1. Advanced Materials Forming Research Center, Babol Noshirvani University of Technology, P.O.B. 484, Babol, Iran

    Majid Sheikhpour, Seyed Jamal Hosseinipour & Mohammad Javad Mirnia

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  1. Majid Sheikhpour
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  2. Seyed Jamal Hosseinipour
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  3. Mohammad Javad Mirnia
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Correspondence to Seyed Jamal Hosseinipour.

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Sheikhpour, M., Hosseinipour, S.J. & Mirnia, M.J. Prediction of exit profile distortion in forward extrusion process using Riemann mapping theorem and upper bound method. Meccanica 55, 1099–1118 (2020). https://doi.org/10.1007/s11012-020-01141-1

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  • DOI: https://doi.org/10.1007/s11012-020-01141-1

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