ECAP process improvement based on the design of rational inclined punch shapes for the acute-angled Segal 2 θ-dies : CFD 2-D description of dead zone reduction

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., 2013aPerig et al., , 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 ABab 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).
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; Re- jaeian 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 α 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 (2θ 0 = 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 • .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 2θ 0 -inclined or 2θ 0 -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 2θ 0inclined or 2θ 0 -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 2θ 0 -inclined or 2θ 0 -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 mod-els, forced by the external action of classical rectangular and novel modified 2θ 0 -inclined or 2θ 0 -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 2θ 0 -inclined or 2θ 0 -beveled punch shapes.
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 2θ 0 -inclined or 2θ 0beveled 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 ABCabc with channel intersection angle 2θ = 75 • using a standard punch (4) with rectangular shape (2θ 0 = 90 • ) in Fig. 1b and novel modified 2θ 0 = 75 • -inclined or 2θ 0 = 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  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 (2θ 0 = 90 • ) in Fig. 1b with the new 2θ 0 -inclined or 2θ 0 -beveled punch AD-ad with 2θ 0 = 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 BCbc 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.
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 2θ 0 -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  -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 2θ 0 -inclined or 2θ 0 -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.Instabilities of the numerical solutions, which appear at the outlet frontiers cC , propagate upstream.
CFD-derived computational flow lines in Fig. 3b directly show the reduction of dead zone area dDbc when we use the modified 2θ 0 -inclined or 2θ 0 -beveled punch shape, where 2θ = 2θ 0 = 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 (2θ 0 = 90 • ).CFD-derived computational di-  agrams for ECAP punching pressure in Figs.9-10 show that the application of the standard rectangular punch with 2θ 0 = 90 • requires lower punching pressures (Fig. 10).The CFD-based simulation in Figs.9-10 indicates that the use of the modified 2θ 0 -inclined or 2θ 0 -beveled punch shapes requires higher punching pressures for ECAP of viscous incompressible continuum through the acute-angled Segal 2θdies with 2θ < 90 • .
Higher values of pressure for modified 2θ 0inclined or 2θ 0 -beveled punch shapes in comparison with the standard rectangular punch with 2θ 0 = 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 2θ 0 -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 2θ 0 -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 2θ 0 -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).   . 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 2θ 0 = 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 2θ 0 -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 , taking into account the standard rectangular and improved 2θ 0 -inclined or 2θ 0 -beveled punch shapes.
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.