MSMechanical SciencesMSMech. Sci.2191-916XCopernicus GmbHGöttingen, Germany10.5194/ms-6-41-2015ECAP process improvement based on the design of rational inclined punch shapes for the acute-angled Segal 2θ-dies: CFD 2-D description of dead zone reductionPerigA. V.olexander.perig@gmail.comhttps://orcid.org/0000-0002-6923-6797GolodenkoN. N.Manufacturing Processes and Automation Engineering Department, Donbass State Engineering Academy, Shkadinova Str. 72, 84313 Kramatorsk, UkraineDepartment of Water Supply, Water Disposal and Water Resources Protection, Donbass National Academy of Civil Engineering and Architecture, Derzhavin Str. 2, 86123 Makeyevka, UkraineA. V. Perig (olexander.perig@gmail.com)14April201561414927July201422March20155April2015This work is licensed under a Creative Commons Attribution 3.0 Unported License. To view a copy of this license, visit http://creativecommons.org/licenses/by/3.0/This article is available from https://ms.copernicus.org/articles/6/41/2015/ms-6-41-2015.htmlThe full text article is available as a PDF file from https://ms.copernicus.org/articles/6/41/2015/ms-6-41-2015.pdf
This article is focused on a 2-D fluid dynamics description of punch shape
geometry improvement for Equal Channel Angular Extrusion (ECAE) or Equal
Channel Angular Pressing (ECAP) of viscous incompressible continuum through
acute-angled Segal 2θ-dies with 2θ< 90∘. It has been shown both
experimentally with physical simulation and theoretically with computational
fluid dynamics that for the best efficiency under the stated conditions, the
geometric condition required is for the taper angle 2θ0 of the
inclined oblique punch to be equal to the 2θ angle between the inlet
and outlet channels of the Segal 2θ-die. Experimentally and
theoretically determined rational geometric condition for the ECAP punch
shape is especially prominent and significant for ECAP through the acute
angled Segal 2θ-dies. With the application of Navier-Stokes equations
in curl transfer form it has been shown that for the stated conditions, the
introduction of an oblique inclined 2θ0-punch results in dead
zone area downsizing and macroscopic rotation reduction during ECAP of a
viscous incompressible continuum. The derived results can be significant
when applied to the improvement of ECAP processing of both metal and polymer
materials through Segal 2θ-dies.
Introduction
For the last 20 years a number of research efforts in materials science
related fields have been focused on wider development, implementation,
commercialization and improvement of new material forming methods known as
Severe Plastic Deformation (SPD) schemes (Boulahia et al., 2009; Haghighi et
al., 2012; Han et al., 2008; Laptev et al., 2014; Minakowski, 2014;
Nagasekhar et al., 2006; Nejadseyfi et al., 2015; Perig et al., 2013a, b, 2015;
Perig and Laptev, 2014; Perig, 2014; Rejaeian and Aghaie-Khafri, 2014). The classical SPD
processing method is Segal's Equal Channel Angular Extrusion (ECAE) or Equal
Channel Angular Pressing (ECAP) material forming technique (Segal, 2004).
ECAE or ECAP realization is based on one or several extrusion passes of a
lubricated metal or polymer material through a die with two intersecting
channels of equal cross-section (Segal, 2004). Materials' processing by ECAP
results in the accumulation of large shear strains and material structure
refinement with physical properties enhancement (Boulahia et al., 2009;
Nagasekhar et al., 2006; Nejadseyfi et al., 2015; Segal, 2004). The standard
die geometry ABC-abc for ECAP processing is the so-called Segal 2θ-die
geometry, where the inlet AB-ab and outlet BC-bc die channels have an intersection
angle 2θ (Figs. 1–2). Moreover Segal 2θ-dies have neither
external nor internal radii at the channel intersection points B; b (Figs. 1–2).
Physical simulation with soft model-based experiments of punch
shape (1, 4) influence on ECAP flow of viscous continuum (2) through
acute-angled Segal die ABC-abc with channel intersection angle 2θ= 75∘< 90∘:
(4) classical punch of rectangular shape dD in (b); (1) modified
shape of inclined 2θ0-punch dD in (a) and (c), where
2θ0= 2θ; (3) the experimentally derived shape of the dead
zone for material flow during ECAP; the schematic diagrams of macroscopic
rotation (e) and rotational inhomogeneity (d) formation during viscous
continuum ECAP.
In recent years we have seen major research interest in the introduction of
fluid mechanics techniques (Minakowski, 2014; Perig et al., 2010; Perig and
Golodenko, 2014a, b; Rejaeian and Aghaie-Khafri, 2014) to the solution of
ECAP problems. This interest is results from growing application of ECAP SPD
techniques to processing of polymers (Boulahia et al., 2009; Perig et al.,
2010; Perig and Golodenko, 2014a, b) and powder materials
(Haghighi et al., 2012; Nagasekhar et al., 2006) where viscosity effects
become essential.
At the same time the phenomenological description of polymer materials flow
through Segal 2θ-dies with Navier-Stokes equations has not been
adequately addressed in previously known publications (Minakowski, 2014;
Perig et al., 2010; Perig and Golodenko, 2014a, b; Rejaeian and Aghaie-Khafri, 2014).
This underlines the importance of the present research, dealing
with fluid dynamics 2-D simulation of material flow through the acute-angled
Segal 2θ-dies with channel intersection angles of 2θ> 0∘ and
2θ< 90∘.
Another problem during ECAP material processing through the acute-angled
Segal 2θ-dies with 2θ< 90∘ is connected with the formation of large
dead zones (3) in the viscous material flow in Fig. 1b as well as enormous
and dangerous mixing Δα of viscous material (2) in
Fig. 1b and e during viscous continuum ECAP through acute-angled dies
with channel intersection angles of 2θ< 90∘ when standard classical
rectangular punches (4) are applied (Fig. 1b). So simple physical
simulation experiments in Fig. 1b for viscous continuum ECAP through the
die ABC-abc with 2θ= 75∘ confirm the disadvantages of using a standard
punch (4) with rectangular shape AD-ad (2θ0= 90∘) in Fig. 1b. It is very
important to note that known approaches in published articles (Boulahia et
al., 2009; Haghighi et al., 2012; Han et al., 2008; Laptev et al., 2014;
Minakowski, 2014; Nagasekhar et al., 2006; Nejadseyfi et al., 2015; Perig et
al., 2013a, b; Perig and Laptev, 2014; Perig, 2014; Rejaeian and Aghaie-Khafri, 2014;
Segal, 2004; Wu and Baker, 1997) have never addressed the
possibility of changing the standard rectangular punch shape AD-ad in Fig. 1b
for material ECAP through acute-angled Segal dies with 2θ< 90∘.
Soft physical model of the workpiece after 3 ECAP passes through
Segal 2θ-die via route C with modified shape of 2θ0-inclined
or 2θ0-beveled punch, where 2θ= 2θ0= 75∘.
This fact emphasizes the importance and underlines the prime novelty of the
present article addressing the viscous fluid dynamics description of the
influence of classical (Fig. 1b) and novel modified 2θ0-inclined
or 2θ0-beveled (Figs. 1a, c and 2)
punch shape AD-ad on viscous flow features of processed workpieces during ECAP
SPD pressure forming through acute-angled Segal 2θ-dies with channel
intersection angles of 2θ> 0∘ and 2θ< 90∘.
Aims and scopes of the article – prime novelty statement of research
The present article is focused on the experimental and theoretical
description of viscous workpiece flow through 2θ acute-angled
angular dies of Segal geometry during ECAP by a classical rectangular punch
and a novel modified 2θ0-inclined or 2θ0-beveled punch.
The aim of the present research is the phenomenological continuum mechanics
based description of viscous workpiece flow through the 2θ
acute-angled angular dies of Segal geometry during ECAE with an application
of classical rectangular and novel modified 2θ0-inclined or
2θ0-beveled punch shapes.
The subject of the present research is the process of ECAP working through
the 2θ acute-angled angular dies of Segal geometry with viscous flow
of polymeric workpiece models, forced by the external action of classical
rectangular and novel modified 2θ0-inclined or 2θ0-beveled punch shapes.
The object of the present research is to establish the characteristics of
the viscous flow of workpiece models through the 2θ acute-angled
angular dies of Segal geometry with respect to workpiece material rheology
and geometric parameters of different punch shapes on viscous ECAP process.
The experimental novelty of the present article is based on the introduction
of initial circular gridlines to study the punch shape influence on viscous
workpiece ECAP flow through the 2θ angular acute-angled dies of
Segal geometry.
The prime novelty of the present research is the numerical finite-difference
solution of Navier-Stokes equations in the curl transfer form for the
viscous workpiece flow through 2θ acute-angled angular dies of Segal
geometry during ECAP, taking into account the classical rectangular and
novel modified 2θ0-inclined or 2θ0-beveled punch shapes.
Physical simulation study of punch shape influence on viscous flow
Physical simulation techniques using plasticine workpiece models are often
used in material forming practice (Chijiwa et al., 1981; Han et al., 2008;
Laptev et al., 2014; Perig et al., 2010, 2013a, b, 2015; Perig and Laptev, 2014;
Perig and Golodenko, 2014a, b; Perig, 2014; Sofuoglu and Rasty, 2000; Wu and Baker, 1997).
In order to estimate the character of viscous flow during ECAP through a
2θ acute-angled angular die of Segal geometry ABC–abc under the
action of a classical rectangular punch and a novel modified 2θ0-inclined
or 2θ0-beveled punch shapes we have utilized
physical simulation techniques in Figs. 1–2. The plasticine workpiece models
in Figs. 1–2 have been extruded through a ECAP die ABC–abc with channel
intersection angle 2θ= 75∘ using a standard punch (4) with rectangular
shape (2θ0= 90∘) in Fig. 1b and novel modified 2θ0= 75∘-inclined
or 2θ0= 75∘-beveled punch (1) in Figs. 1a, c and 2 as
the first experimental approach to polymeric materials flow (Figs. 1–2).
The aim of the physical simulation is an experimental study of dead zone
abc formation and deformation zone abc location during viscous ECAP flow of
workpiece plasticine models under the external action of rectangular and
inclined punches. The physical simulation in Figs. 1–2 is also focused on the
experimental visualization of rotary modes of SPD during ECAP of viscous
polymer models for the different punch geometries. The experimental results
in Figs. 1–2 are original experimental research results, obtained by the authors.
The plastic die model of ECAP die ABC-abc with channel intersection angle
<ABC =<abc = 2θ= 75∘ and the width of inlet aA
and outlet cC die channels 35 mm is shown in Figs. 1–2. Potato flour was used
as the lubricator in Figs. 1–2.
The main experimental visualization technique in Figs. 1–2 is based on the
manufacture of the initial plasticine physical models of the workpieces in
the shapes of rectangular parallelepipeds, freezing of these rectangular
parallelepipeds, marking the initial circular gridlines on the front sides
of the frozen parallelepipeds, perforation of through-holes in the
parallelepipeds at the centers of the initial circular gridlines, repeated
freezing of the plasticine (Fig. 1) parallelepipeds, heating of the
plasticine (Fig. 1) pieces with different colors to the half-solid state,
and placing the half-solid multicolor plasticine (Fig. 1) into the
through-holes of the frozen parallelepipeds using a squirt without needle technique.
In this way the initial plasticine-based (Fig. 1) circular gridlines were
marked throughout the initial plasticine (Fig. 1) workpieces. The initial
circular gridlines transform into deformed elliptical ones as workpieces
flow from inlet to outlet die channels during ECAP (Figs. 1a, c
and 2). The gridline-free dead zones (p. b) were visualized through the
physical simulation techniques introduction in Figs. 1–2. It was found that
dead zone (p. b) formation takes place in the vicinity of the external angle
abc of channel intersection zone Bb. It was experimentally shown that the best
reduction of dead zone size (3) for an ECAE die with 2θ= 75∘
could be achieved through the replacement of the standard rectangular
punch AD-ad with (2θ0= 90∘) in Fig. 1b with the new 2θ0-inclined
or 2θ0-beveled punch AD-ad with 2θ0= 75∘.
It was experimentally found in Figs. 1–2 that the deformation zone BCDc
during ECAP of the viscous models is not located in the channel intersection
zone Bb but is located in the beginning of the outlet die channel BC-bc. The
relative location of the elliptical markers in outlet die channel BC-bc show
the formation of two rotary modes of SPD during ECAP (Fig. 1).
Checking the successive locations of one color elliptical markers in Fig. 1,
we see that the major axis of every elliptical marker rotates with respect
to the axis of the outlet die channel bc. We define the term of macroscopic
rotation as the relative rotation of the major axis of an elliptical marker
with respect to the flow direction axis bc. The macroscopic rotation is the
first visually observable rotary mode during ECAP forming of the viscous
workpiece model.
Computational flow lines for the Segal die with 2θ= 75∘
for the coordinate steps ξ‾= 1.10 mm;
η‾= 1.44 mm for the rectangular punch dD (2θ0= 90∘) (a)
and for the inclined punch dD (2θ0= 75∘) (b),
where time iteration step is t‾it= 610 µs,
transition time is t‾tr= 11.3 s.
Visual comparison of Fig. 1b with Figs. 1a, c and 2 shows
that the macroscopic rotation is an unknown function of ECAP die channel
intersection angle 2θ and 2θ0-punch shape geometry.
However under SPD ECAP treatment some deformed elliptical markers within the
viscous material have additional bending points and have the form of
“commas” or “tadpoles” in Figs. 1–2. If the elliptical marker has an
additional bending point during ECAP, then we will call the vicinity of the
marker with this “waist” as a zone of rotational inhomogeneity within the
workpiece material, which is usually located at the beginning of the outlet
die channel BC-bc in Figs. 1–2. The rotational inhomogeneity is the second
visually observable rotary mode during ECAP forming of the viscous workpiece
model, which strongly depends on the ECAP die channel intersection angle
2θ and 2θ0-punch shape geometry.
The experimental results in Figs. 1–2 have indicated the formation of the
following zones within worked materials' volumes: (I) the dead zone
(p. b); (II) the deformation zone BCDc; (III) the macroscopic rotation zone
(BC-bc), and (IV) the zone of rotational inhomogeneity (BC-bc). The complex
of physical simulation techniques in Figs. 1–2 introduces the initial
circular gridlines technique with the application of plasticine workpieces
with the initial circular colorful gridlines in the shape of initial
colorful cylindrical plasticine inclusions (Fig. 1). The application of the
initial circular gridlines experimental technique and the introduction of a
novel modified 2θ0-inclined or 2θ0-beveled punch
shapes has not been addressed in previous known ECAP research (Boulahia et
al., 2009; Haghighi et al., 2012; Han et al., 2008; Laptev et al., 2014;
Minakowski, 2014; Nagasekhar et al., 2006; Nejadseyfi et al., 2015; Perig et
al., 2013a, b; Perig and Laptev, 2014; Perig, 2014; Rejaeian and Aghaie-Khafri, 2014;
Segal, 2004; Wu and Baker, 1997).
The proposed complex of experimental techniques for physical simulation of
SPD during ECAP in Figs. 1–2 will find the further applications in the study
of viscous ECAP through the dies with more complex Iwahashi, Luis-Perez,
Utyashev, Conform and equal radii geometries for the different punch shape
geometries and different routes of multi-pass ECAP working.
Computational dimensionless flow function ψ for the Segal
die with 2θ= 75∘ for the rectangular punch dD
(2θ0= 90∘) (a) and for the inclined punch dD
(2θ0= 75∘) (b).
Computational dimensionless curl function ζ for the Segal
die with 2θ= 75∘ for the rectangular punch dD
(2θ0= 90∘) (a) and for the inclined punch dD
(2θ0= 75∘) (b).
Computational u components of flow velocities for the Segal die
with 2θ= 75∘ for the rectangular punch dD
(2θ0= 90∘) (a) and for the inclined punch dD
(2θ0= 75∘) (b).
Computational v components of flow velocities for the Segal die
with 2θ= 75∘ for the rectangular punch dD
(2θ0= 90∘) (a) and for the inclined punch dD
(2θ0= 75∘) (b).
Computational dimensionless full flow velocities w for the Segal
die with 2θ= 75∘ for the rectangular punch dD
(2θ0= 90∘) (a) and for the inclined punch dD
(2θ0= 75∘) (b).
Computational dimension punching pressure for the Segal die with
2θ= 75∘ for the rectangular punch dD (2θ0= 90∘) (a, c)
and for the inclined punch dD (2θ0= 75∘) (b, d).
Computational dimensionless punching pressures for plasticine
viscous liquid flow through Segal dies with 60∘≤ 2θ≤ 110∘
for the rectangular punch dD (2θ0= 90∘) (○) and
for the modified 2θ0-inclined or 2θ0-beveled punch dD
(2θ0= 2θ) (□).
Numerical simulation study of punch shape influence on flow lines, and punching pressure during viscous ecap flow through Segal 2θ-dies
In order to derive the mathematical model of the viscous material flow
during ECAP through the acute-angled Segal 2θ-die taking into account
the punch shape AD-ad effect on viscous flow dynamics we will apply the
Navier-Stokes equations (Appendices A–D). The results of the numerical
simulation study are shown in computational diagrams in Figs. 3–10.
Computational results in Figs. 3–10 illustrate the punch shape
influence on geometry (Fig. 3), kinematics (Figs. 4–8) and dynamics
(Figs. 9–10) of the viscous flow during ECAP. Computational plots in
Figs. 3–10 are based on a finite-difference solution of the
Navier–Stokes equations in curl transfer form Eqs. (A1)–(A2) with initial Eq. (B1)
and boundary Eqs. (C1)–(C7) conditions.
Instabilities of the numerical solutions, which appear at the outlet
frontiers cC (Figs. 3–10), propagate upstream.
CFD-derived computational flow lines in Fig. 3b directly show the reduction
of dead zone area dDbc when we use the modified 2θ0-inclined or
2θ0-beveled punch shape, where 2θ= 2θ0= 75∘ (Fig. 3b). CFD-derived computational flow lines in Fig. 3a
also outline the largest dead zone area dDbc when we use the standard punch
(Fig. 3a) with rectangular shape (2θ0= 90∘). CFD-derived
computational diagrams for ECAP punching pressure in Figs. 9–10 show
that the application of the standard rectangular punch with 2θ0= 90∘
requires lower punching pressures (Fig. 10). The CFD-based
simulation in Figs. 9–10 indicates that the use of the modified
2θ0-inclined or 2θ0-beveled punch shapes requires
higher punching pressures for ECAP of viscous incompressible continuum
through the acute-angled Segal 2θ-dies with 2θ< 90∘.
Higher values of punching pressure for modified 2θ0-inclined or
2θ0-beveled punch shapes in comparison with the standard
rectangular punch with 2θ0= 90∘ in Figs. 9–10 result from
the fact that the compressive strains in such schemes are higher than shear strains.
So in order to force the plasticine model through the 2θ-die by the
modified 2θ0-inclined punch we have to apply higher punching
force in order to reach the necessary shear stresses. This fact is shown in Figs. 9–10.
The increased punching pressure required for the modified 2θ0-inclined
punches and for the acute angled 2θ-dies with
2θ< 90∘ results in decreased dead zone in angle b and a decreased shear
stress component (Figs. 6b, 7b, 8b, 9b, d, and 10).
For the modified 2θ0-inclined punches and the obtuse angled
2θ-dies with 2θ> 90∘ the decreased punching pressure results from
increased effective punch area dD and increased shear stress component (Fig. 10).
Discussion of derived results
The technological issue addressed in this article has direct industrial
importance in material forming applications. The introduction of the fluid
dynamics numerical simulation (Figs. 3–10) provides us with a better
understanding of physical simulation results in Figs. 1–2.
Addressing Eqs. (A1)–(A2) in Appendix A again, the partial derivatives of
dimensionless flow function ψ define the flow velocity components:
∂ψ/∂y=u; ∂ψ/∂x= (-v).
In the 3-D spatial diagrams for flow function ψ in Fig. 4 near
the die corner b with rectangular Cartesian coordinates (0, -40) we have
the following effect of punch shape ad-AD on dead zone dDb size. With the
application of a rectangular punch with 2θ0= 90∘ in Figs. 1b, 3a, 4a, 5a, 6a,
7a, 8a, 9a, c, 10 we see a large dead zone
dDb with zero flow function ψ= 0 (Fig. 4) and zero flow velocities
u= 0 (Fig. 6); v= 0 (Fig. 7). But with the introduction of an inclined
2θ-punch with 2θ0= 75∘ (inclined punch in
Figs. 1a, c, 2, 3b, 4b, 5b, 6b, 7b, 8b, 9b, d, 10) we see a smaller dead
zone size dDb. Computational flow lines (Fig. 3) are the lines near which flow
function ψ (Fig. 4) is constant ψ= const. The computed effect in
Fig. 4, which shows the absence of the “sawtooth” shape of the
ψ-function over the die area dDb confirms that the dDb area is just the dead zone
and not a vortex or eddy zone with circulating flow. Figure 5 show us that the
curl function ζ= 0 is also zero in the dead zone dDb.
Polycrystalline material is a natural composite, which contains ultra fine
single crystals and amorphous viscous fluid between single crystals for
fastening and connecting these single crystals among themselves.
Laminar-flow layers of such amorphous fluid move with different velocities
as well as single crystal sides, adjacent to laminar-flow layers. Curl
ζ (Eq. A2) characterizes single crystal relative rotation during its
linear displacement along the flow lines in Fig. 5. As a result of internal
friction the contacting facets of single crystals become smooth like
smoothing of river or sea pebbles under action of viscous flow.
This is the hydrodynamic explanation of the increase of the material
plasticity during ECAP, which follows from the computational diagrams in
Figs. 3–10. Under the action of mechanical loads at the boundaries of
the contacting facets of single crystals, there appear no micro-cracks
because of their flatness. The curl is zero in dead zone dDb. So in the
material dead zone dDb no smoothing of single crystals facets takes place. As a
result, material plasticity cannot be improved in the material dead zone dDb.
Such hydrodynamic illustrations (Figs. 3–10) directly confirm
experimentally derived results (Figs. 1–2) with physical simulation
of punch shape effect on material flow kinematics during ECAE through the
acute-angled Segal 2θ-die.
Conclusions
In the present work we addressed the 2θ0-punch shape effect on
material flow dynamics during ECAP through the numerical solution of the
boundary value problem Eqs. (A1)–(A2), (B1), (C1)–(C7) for Navier–Stokes
equations in curl transfer form (Figs. 3–10), taking into account the
standard rectangular and improved 2θ0-inclined or 2θ0-beveled punch shapes.
Both physical (Fig. 1b) and fluid dynamics (Figs. 3a, 4a, 5a, 6a, 7a, 8a, 9a, c, 10)
simulations show that the application of a standard rectangular punch with
2θ0= 90∘ for workpiece ECAP through acute-angled Segal 2θ-dies
with 2θ< 90∘ is highly undesirable because of the resulting large
material dead zone areas dDb in the neighborhood of the external die angle
2θ=<(abc).
Both physical (Figs. 1a, c and 2) and fluid dynamics
(Figs. 3b, 4b, 5b, 6b, 7b, 8b, 9b, d, 10) simulations reveal that the introduction of
2θ0-inclined or 2θ0-beveled punch shapes with
dDbc for material ECAP processing through the acute-angled Segal 2θ-dies
with 2θ< 90∘ and 2θ0= 2θ is a very promising technique
because of minimal material dead zone areas dDb and the resulting minimal
material waste in the neighborhood of external die angle 2θ=<(abc),
e.g. for 2θ= 75∘.
Navier–Stokes equations in curl transfer form
The curl transfer equation in dimensionless variables will have the
following form (Roache, 1976):
∂ζ∂t=-Re∂(uζ)∂x+∂(vζ)∂y+∂2ζ∂x2+∂2ζ∂y2,
where the dimensionless curl function will be defined as (Fig. 5):
ζ=∂u∂y-∂v∂x.
Initial conditions for curl transfer equation
We now study the steady-state regime of viscous flow for a physical model of
polymer material (Figs. 3–10). So the initial conditions we will
assume in the form of a rough approximation to the stationary solution
(Figs. 3–10):
ui,j0=0;vi,j0=0;ζi,j0=0;ψi,j0=0.
Boundary conditions for curl transfer equation
The boundary conditions for the die walls we will define as the viscous
material “sticking” to the walls of the die (Figs. 3–10).
At the inner upper boundary DBC (Figs. 3–10) we have
ψi,j=1;ζi,j=1.
At the external lower boundary dbc (Figs. 3–10) we have
ψi,j=0;ζi,j=0.
For the punch frontal edge dD (Figs. 3–10) we have
ψ10,-10=1;ψ9,-11=1-2/N;ψi,j=ψi+2,j+2-2/N,
where N is the quantity of ordinate steps along the channel width.
For the angular points, which are located in the vertices of the concave
angles b and B (Figs. 3–10) we have
ζi,j=0.
For the angular point D (Figs. 3–10) of the convex angle in the
finite-difference equation, written for the mesh point (10, -11) we have the
following curl
ζ10,10=2ψ10,-11.
For the angular point D (Figs. 3–10) of the convex angle in the
finite-difference equation, written for the mesh point (11, 10) we have the curl
ζ10,10=2ψ11,10.
At the outlet line cC we have
ψN+1,j=ψN-3,j-2ψN-2,j+2ψN,j;ζN+1,j=ζN-3,j-2ζN-2,j+2ζN,j.
Numerical values of physical parameters for the problem
The numerical results of integration of curl transfer Eqs. (A1)–(A2)
with initial Eq. (B1) and boundary Eqs. (C1)–(C7) conditions are outlined in
Figs. 3–10 for the following numerical values:
the dimensional width of inlet and outlet die channels is
a‾= 35 mm;
the dimensional length of die channel is L‾= 16 ⋅a‾= 16 ⋅ 35 × 10-3 m = 0.56 m;
the dimensional average ECAP punching velocity is
U‾0= 0.1 × 10-3 m s-1;
the dimensional time of processed workpiece material motion in die channel
is t‾*=L‾/U‾0= 0.56/(0.1 × 10-3) = 5600 s;
the maximum value of dimensionless curl is ζ= 1;
the dimensional curl is
ζ‾=ζ⋅U‾0/a‾= (1 × 0.1 × 10-3 m s-12)/(35 × 10-3 m) = 2.86 × 10-3 s-1;
the dimensional average angular velocity of rotation for viscous material
layers is ω‾=|rotw|/2=ζ‾/2= 1.43 × 10-3 s-1;
the number of turns for viscous material layers during the time of workpiece
material motion in die channel is
N*=ω‾t‾*/2π= (1.43 × 10-3× 5600)/2 × 3.14) = 1.27;
the dimensional density of the viscous plasticine physical model of extruded
polymer material is ρ‾= 1850 kg m-3;
the dimensional plasticine yield strength is σ‾s= 217 kPa (Sofuoglu and Rasty, 2000);
the dimensional specific heat capacity of plasticine material is
c‾= 1.004 kJ/(kg K-1);
the dimensional thermal conductivity is λ‾= 0.7 J/ (m s-1 K-1) (Chijiwa et al., 1981);
the dimensional punching temperature is t‾temp= 20 ∘C;
the dimensional dynamic viscosity for viscous Newtonian fluid model of
plasticine workpiece during ECAE is η‾vis= 1200 kPa s-1;
the dimensional kinematic viscosity for viscous Newtonian fluid model of
plasticine workpiece during ECAE is
ν‾vis=η‾vis/ρ‾= 1.2 × 106/1850 = 648.648 m2 s-1;
Reynolds number is
Re=U‾0a‾ρ‾/η‾vis=U‾0a‾/ν‾vis= 5.396 × 10-9;
the half number of coordinate steps along the x and y axes is N= 40;
the number of coordinate steps along the x and y axes is
2 ×N= 80;
the relative error of iterations is e= 1/1000;
the dimensional time moment for the first isochrone building is
t1= 100 s;
die channel intersection angle of Segal die is 2θ= 75∘;
punch shape inclination angles adD are 2θ0= 90∘ (rectangular punch in
Figs. 1b, 3a, 4a, 5a, 6a, 7a, 8a, 9a, c, 10) and 2θ0= 75∘ (inclined punch in
Figs. 1a, c, 3b, 4b, 5b, 6b, 7b, 8b, 9b, d, 10);
the dimensional horizontal and vertical coordinate steps along the x– and
y axes are ξ‾= 1.10 mm and η‾= 1.44 mm
for angular die with 2θ= 75∘;
the dimensional time iteration step is τ‾=t‾it= 610 µs
for ECAP die with 2θ= 75∘;
the dimensional transition time is t‾tr=11.3 s for ECAP die
with 2θ= 75∘.
Acknowledgements
Authors thank three “anonymous” referees for their valuable notes and suggestions. Authors
are thankful to Mechanical Sciences Editors and Copernicus GmbH Team for this great opportunity
to publish our original research at your respectful periodical Mechanical Sciences under a Creative
Commons License.
Edited by: A. Barari
Reviewed by: three anonymous referees
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