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Physical modelling and analytical evaluation of deformation zones in the extrusion of non-axisymmetric profiles Cover

Physical modelling and analytical evaluation of deformation zones in the extrusion of non-axisymmetric profiles

Open Access
|Dec 2025

Figures & Tables

Figure 1

Configuration of plastic zones during extrusion of (a) circular, (b) square, (c) triangle, and (d) rectangle cross-sections.
Configuration of plastic zones during extrusion of (a) circular, (b) square, (c) triangle, and (d) rectangle cross-sections.

Figure 2

Test stand schematic diagram.
Test stand schematic diagram.

Figure 3

Tool set for extrusion testing of model material (1 – container base, 2 – support plate, 3 – die, 4 – container, 5 – billet, 6 – embosser, and 7 – punch).
Tool set for extrusion testing of model material (1 – container base, 2 – support plate, 3 – die, 4 – container, 5 – billet, 6 – embosser, and 7 – punch).

Figure 4

Extrusion test dies: (a) Circular die opening and (b) triangular die opening.
Extrusion test dies: (a) Circular die opening and (b) triangular die opening.

Figure 5

Extrusion test dies: (a) Square-shaped die opening and (b) rectangular die opening.
Extrusion test dies: (a) Square-shaped die opening and (b) rectangular die opening.

Figure 6

Stress–strain relationship obtained experimentally in the upsetting test for Pb1.
Stress–strain relationship obtained experimentally in the upsetting test for Pb1.

Figure 7

Preparation of split ingots for testing using the visioplasticity method: (a) Split ingot, (b) half of ingot with grid applied, and (c) assembled ingot prepared for testing.
Preparation of split ingots for testing using the visioplasticity method: (a) Split ingot, (b) half of ingot with grid applied, and (c) assembled ingot prepared for testing.

Figure 8

Grid deformation during extrusion of lead through flat dies with various orifice shapes (λ = 12).
Grid deformation during extrusion of lead through flat dies with various orifice shapes (λ = 12).

Figure 9

Scheme of measuring the depth of the plastic zone L
                  pz, the dead zone L
                  dz, and the dead zone angle α
                  dz.
Scheme of measuring the depth of the plastic zone L pz, the dead zone L dz, and the dead zone angle α dz.

Figure 10

Relationship between the plastic zone depth (L
                  pz) and the cross-sectional shape of the extruded product.
Relationship between the plastic zone depth (L pz) and the cross-sectional shape of the extruded product.

Figure 11

Relationship between the height of the dead zone (L
                  dz) and the cross-sectional shape of the extruded product.
Relationship between the height of the dead zone (L dz) and the cross-sectional shape of the extruded product.

Figure 12

Relationship between the dead zone angle (α
                  dz) and the cross-sectional shape of the extruded product.
Relationship between the dead zone angle (α dz) and the cross-sectional shape of the extruded product.

Figure 13

Relationship of maximum extrusion force (F
                  max) to plastic zone depth (L
                  pz), cross-sectional shape of the extruded product, and extrusion ratio (λ).
Relationship of maximum extrusion force (F max) to plastic zone depth (L pz), cross-sectional shape of the extruded product, and extrusion ratio (λ).

Figure 14

Relationship of maximum extrusion force (F
                  max) to dead zone height (L
                  dz), cross-sectional shape of the extruded product, and extrusion ratio (λ).
Relationship of maximum extrusion force (F max) to dead zone height (L dz), cross-sectional shape of the extruded product, and extrusion ratio (λ).

Figure 15

Relationship of maximum extrusion force (F
                  max) to dead zone angle (α
                  dz), cross-sectional shape of the extruded product, and extrusion ratio (λ).
Relationship of maximum extrusion force (F max) to dead zone angle (α dz), cross-sectional shape of the extruded product, and extrusion ratio (λ).

Figure 16

Axial strains (ε
                  z) and radial strains (ε
                  r) as functions of ingot length for extruded cross-sections at λ = 12.
Axial strains (ε z) and radial strains (ε r) as functions of ingot length for extruded cross-sections at λ = 12.

Figure 17

Velocity distribution in the die orifice region (λ = 12).
Velocity distribution in the die orifice region (λ = 12).

Figure 18

Division of the plastic zone volume into three components: V
                  1, V
                  2, and V
                  3.
Division of the plastic zone volume into three components: V 1, V 2, and V 3.

Figure 19

Method for Selecting the Point that Defines the Boundary of the Plastic Zone.
Method for Selecting the Point that Defines the Boundary of the Plastic Zone.

Figure 20

Relationship between volume of plastic zone V
                  pz, shape of an die orifice, and extrusion ratio λ.
Relationship between volume of plastic zone V pz, shape of an die orifice, and extrusion ratio λ.

Figure 21

Relationship between extrusion force (F
                  max) obtained from experimental measurements and analytically evaluated deformation zone parameters (L
                  pz, L
                  dz, α
                  dz, V
                  pz) for non-axisymmetric profiles.
Relationship between extrusion force (F max) obtained from experimental measurements and analytically evaluated deformation zone parameters (L pz, L dz, α dz, V pz) for non-axisymmetric profiles.

Figure 22

Plastic flow of material during the transition from a circular billet cross-section to product cross-sections: square, triangular (A-A), triangular (B-B), and rectangular (A-A).
Plastic flow of material during the transition from a circular billet cross-section to product cross-sections: square, triangular (A-A), triangular (B-B), and rectangular (A-A).

Chemical composition of the material used in the study (Pb1) – weight%_

PbAgAsBiFeCuZnSbSn
99,980.00050.00050.01000.00060.00050.00050.00050.0005

Extrusion parameters_

Extrusion speed (mm/s)1
Extrusion ratio λ 3; 12; 60

Basic mechanical properties of the material used in the study (Pb1)_

Unit
Tensile strength R m MPa17
Yield strength R e MPa5
Elongation A %45
DOI: https://doi.org/10.2478/msp-2025-0040 | Journal eISSN: 2083-134X | Journal ISSN: 2083-1331
Language: English
Page range: 17 - 37
Submitted on: Jul 17, 2025
Accepted on: Nov 6, 2025
Published on: Dec 2, 2025
Published by: Wroclaw University of Science and Technology
In partnership with: Paradigm Publishing Services
Publication frequency: 4 issues per year

© 2025 Beata Pawłowska, published by Wroclaw University of Science and Technology
This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.