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An analysis for magnesium alloy curvature products formed by staggered extrusion (SE) based on the upper bound method

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Abstract

Staggered extrusion (SE) is a new method to solve the bottleneck of traditional curvature products, such as long manufacturing cycle, many forming processes and difficult quality control. How to quantitatively control the curvature of extruded products is the key to implement this method. Herein, the upper bound method is used to calculate and analyze the power consumption of each characteristic zone in the SE process. The theoretical model of extrusion load and curvature is established. The bending radius of profile Rc under different staggered distance h and the extrusion ratio λ is calculated. The reciprocal relationship between curvature radius Rc and curvature κ is used to obtain the curvature change relationship. The results show that the staggered distance h has an important influence on the curvature κ. When the staggered distance h increases from 8 to 24 mm and other conditions remain unchanged, the curvature κ increases from 0.0546 to 0.1607. Any combination of the staggered distance h and the extrusion ratio λ corresponds to an eccentricity ratio ξ. The eccentricity ratio ξ decreases with the increase of the staggered distance h or the extrusion ratio λ. By comparison, it can be seen that the variation trend of the theoretical predicted value and the FE modeling in the steady-state extrusion stage is consistent. The experimental results are in good agreement with the curvature theory prediction model. These results provide a scientific basis for the formulation of the SE process and precisely controlling magnesium alloy curvature products.

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Availability of data and materials

The data obtained in the framework of this study are available to the journal upon request.

Abbreviations

D 1 :

Diameter of the billet

D 2 :

Diameter of the extruded profile

S 0 :

Cross-sectional area of the billet

S 1 :

Cross-sectional area of the extruded profile

S 2, S 3, S 4, S 5 :

Cross-sectional area of the related profile

h :

Staggered distance

Δv, Δv 1, Δv 2, Δv 3, and Δv 4 :

Amount of velocity discontinuity

v 1 :

Extrusion velocities of the stem

v 2, v 3 :

Velocity at the mass center of the related profile

v o :

Velocity at the center of the extruded profile

v I, v II :

Velocity in zones I and II

v 1e, v 2e :

Maximum and minimum material flow velocity across the die exit

v z :

Material flow velocity at a certain point of the die

F 1 :

Extrusion force of stem

Ẇ, Ẇ 0 :

Internal and external plastic dissipation power

Ẇ i, Ẇ Γ, Ẇ S :

Plastic dissipation power due to plastic deformation, velocity discontinuity surface, and frictional force

͞y i :

Coordinates of the mass center of the related profile

V i :

Volume of the related profile

O2, O3 :

Mass

Δt :

A very short time

Δd 2, Δd 3 :

The distance traveled in a very short time

R c :

Bending radius of profile

α,  β,  θ :

Angles in Fig. 2

l 0 :

Distance between container and die

l 1 :

Die land length

l 2 :

Distance between point B and die exit

l 3 :

Distance between the first-touched billet part and die

λ :

Extrusion ratio

ξ :

Eccentricity ratio

τ :

Shear stress

μ :

Constant coefficient of friction

ε ij*:

Strain rate tensor

σ ij*:

Strain tensor

κ :

Curvature

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Funding

This project is supported by the National Natural Science Foundation of China (No. 51975166).

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Authors and Affiliations

  1. School of Materials Science and Chemical Engineering, Harbin University of Science and Technology, Harbin, 150040, People’s Republic of China

    Xi Jie Zhang, Feng Li, Ye Wang & Zi Yu Chen

  2. Key Laboratory of Advanced Manufacturing and Intelligent Technology, Ministry of Education, Harbin University of Science and Technology, Harbin, 150080, People’s Republic of China

    Feng Li

Authors
  1. Xi Jie Zhang
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  2. Feng Li
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  3. Ye Wang
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  4. Zi Yu Chen
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Contributions

Staggered extrusion process is a new method for bending products forming integrated. The theoretical model established by upper bound method lays a foundation for controlling the bending products curvature.

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Correspondence to Feng Li or Ye Wang.

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Zhang, X.J., Li, F., Wang, Y. et al. An analysis for magnesium alloy curvature products formed by staggered extrusion (SE) based on the upper bound method. Int J Adv Manuf Technol 119, 303–313 (2022). https://doi.org/10.1007/s00170-021-08167-z

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