Two-parameter Rigid Block Approach to Upper Bound Analysis of Equal Channel Angular Extrusion Through a Segal 2θ-die

Perig, Alexander

doi:10.1590/1516-1439.004215

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Articles • Mater. Res. 18 (3) • May-Jun 2015 • https://doi.org/10.1590/1516-1439.004215 copy

Two-parameter Rigid Block Approach to Upper Bound Analysis of Equal Channel Angular Extrusion Through a Segal 2θ-die

Authorship SCIMAGO INSTITUTIONS RANKINGS

This article deals with a phenomenological description of experimentally determined complex geometric shape of material dead zone during Equal Channel Angular Extrusion (ECAE) through a Segal 2θ-die with a channel intersection angle of 2θ>0°and 2θ<180°. Taking into account the complex dead zone geometry in a 2θ-die, a two-parameter Rigid Block Method (RBM) approach to a two-parameter Upper Bound Method (UBM) has been introduced with Discontinuous Velocity Field (DVF) for planar flow of plastic incompressible continua. The two-parameter UBM has allowed us to derive the numerical estimations for such energy-power parameters of ECAE as punching pressure and accumulated plastic strain for 2θ-dies. The obtained computational data have been compared with the one-parameter analytic UBM solution. Good agreement between the two computational results has been found.

ECAE; upper bound method; rigid block method; discontinuous velocity field; dead zone; strain

Velocity discontinuity lines $i - j$	$l_{i - j}$	$[V_{i - j}]$
$1 - 2$	$a \sqrt{1 + {(\cot θ - y)}^{2}}$	$\frac{V_{1} x \sqrt{1 + {(\cot θ - y)}^{2}}}{y (1 - x)}$
$2 - 3$	$\frac{a}{\sin θ} (1 - x)$	$\frac{2 V_{1}}{y (1 - x)} (y \cdot \cos θ - x \cdot (\cos θ \cdot \cot θ + \sin θ))$
$3 - 4$	$a \sqrt{1 + {(\cot θ - y)}^{2}}$	$\frac{V_{1} x \sqrt{1 + {(\cot θ - y)}^{2}}}{y (1 - x)}$
$2 - 5$	$a \sqrt{y^{2} + x^{2} (1 + {(\cot θ)}^{2}) - 2 x y \cdot cot θ}$	$\frac{V_{1} \sqrt{y^{2} + x^{2} (1 + {(\cot θ)}^{2}) - 2 x y \cdot cot θ}}{y (1 - x)}$
$3 - 5$	$a \sqrt{y^{2} + x^{2} (1 + {(\cot θ)}^{2}) - 2 x y \cdot cot θ}$	$\frac{V_{1} \sqrt{y^{2} + x^{2} (1 + {(\cot θ)}^{2}) - 2 x y \cdot cot θ}}{y (1 - x)}$

Velocity discontinuity lines $i - j$	$V_{i - j}^{n}$
$1 - 2$	$\frac{V_{1}}{\sqrt{1 + {(\cot θ - y)}^{2}}}$
$2 - 3$	$\frac{V_{1} \sin θ}{(1 - x)}$
$3 - 4$	$\frac{V_{1}}{\sqrt{1 + {(\cot θ - y)}^{2}}}$
$2 - 5$	$0$
$3 - 5$	$0$

Velocity discontinuity lines $i - j$	$l_{i - j}$	$[V_{i - j}]$	$V_{i - j}^{n}$
$1 - 2$	$a \sqrt{1 + {(1 - y)}^{2}}$	$\frac{V_{1} \cdot x \sqrt{1 + {(1 - y)}^{2}}}{y \cdot (1 - x)}$	$\frac{V_{1}}{\sqrt{1 + {(1 - y)}^{2}}}$
$2 - 3$	$\sqrt{2} \cdot a \cdot (1 - x)$	$\frac{\sqrt{2} \cdot V_{1} \cdot (y - 2 x)}{y \cdot (1 - x)}$	$\frac{V_{1}}{\sqrt{2} \cdot (1 - x)}$
$3 - 4$	$a \sqrt{1 + {(1 - y)}^{2}}$	$\frac{V_{1} \cdot x \sqrt{1 + {(1 - y)}^{2}}}{y \cdot (1 - x)}$	$\frac{V_{1}}{\sqrt{1 + {(1 - y)}^{2}}}$
$2 - 5$	$a \sqrt{x^{2} + {(y - x)}^{2}}$	$\frac{V_{1} \cdot \sqrt{x^{2} + {(y - x)}^{2}}}{y \cdot (1 - x)}$	$0$
$3 - 5$	$a \sqrt{x^{2} + {(y - x)}^{2}}$	$\frac{V_{1} \cdot \sqrt{x^{2} + {(y - x)}^{2}}}{y \cdot (1 - x)}$	$0$

δ	2θ=75º	2θ=90º	2θ=105º	2θ=120º	2θ=135º
2-param UBM vs 1-param UBM	δ(p)=2.852%	δ(p)=3.012%	δ(p)=1.233%	δ(p)=1.568%	δ(p)=17.849%
2-param UBM vs 1-param UBM	δ(γ_S)=3.426%	δ(γ_S)=3.012%	δ(γ_S)=1.434%	δ(γ_S)=2.647%	δ(γ_S)=8.496%

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[1] *e-mail: olexander.perig@gmail.com