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

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.


Introduction
Severe Plastic Deformation (SPD) techniques have become the prominent processing technologies  for the production of bulk ultra fine-grained materials with special physical and mechanical properties 28-32, by material pressure forming methods 25,26 . The echanics of metal flow during Equal Channel Angular Extrusion (ECAE) or Equal Channel Angular Pressing (ECAP) through the 2θ-dies of simple Segal geometry attracts a lot of research contributions from different fields of theoretical  , experimental [28][29][30][31][32] and industrial 8,25,33 science.ECAE assumes one or several extrusion passes of a lubricated billet in a 2θ-die of Segal geometry with two intersecting channels of equal cross-section in Figures 1-2.Materials' processing by ECAE results in the accumulation of large shear strains and grain refinement.The obtained materials show the combination of a very high strength and ductility.
Therefore, studies of ECAE mechanics through the 2θ-dies of simple Segal geometry form an important branch of modern applied plasticity science.
Segal 25,26 has grounded his analytic approach to ECAE metal flow with the introduction of slip line fields.Tóth et al. 28 have proposed a novel analytic approach to ECAE copper flow with the introduction of a flow line model.Milind & Date 11 have applied generalized analytical kinematic models for ECAE strain estimation.
The Upper Bound Method (UBM)-based analytical solutions of ECAE problems for metal workpiece flow through ECAE dies have been derived in works of Abrinia & Mirnia 1 , Alkorta & Sevillano 2 , Altan et al. 3 , Eivani & Karimi Taheri 4,5 , Faraji et al. 6 , Laptev et al. 7 , Luri et al. 8 , Luri & Luis 9 , Medeiros et al. 10 , Narooei & Karimi Taheri 12-14 , Paydar et al. 15 , Perig & Laptev 19 , Perig 22 , Reihanian et al. 24 , Talebanpour & Ebrahimi 27 and others.Altan et al. 3 have addressed the Upper Bound Method (UBM) with continuous trial velocity fields for description of metal ECAE motion through the Segal die.The main computational approach 3 has been based on the use of a cylindrical coordinate system with the center of a symmetrical metal deformation zone, definition of a kinematically admissible velocity field for metal flow, evaluation of the only non-zero strain rate field component d(ε rθ )/dt, formulation of an equation for dissipated power balance and further minimization of the derived expression for dissipated plastic power by optimization parameter differentiation.
The numerical finite difference simulation of polymer ECAE flow through angular dies with different geometries has been developed in the works of Perig et al. 16 , Perig & Golodenko 20,21 .Perig et al. 16 have applied the Navier-Stokes equations in the curl transfer form for the numeric finite-difference description of viscous material flow through the following dies: (I) ECAE die with channel intersection angle 2θ=90° [ 16] , (II) S-shaped multiple angle die with movable inlet wall 20 , and (III) ECAE die with channel intersection angle 2θ=90° and with parallel slants in the channel intersection zone, where the slant width is equal to the inlet and outlet channel widths 21 .
All previous research [1][2][3][4][5][6][7][8][9][10][12][13][14][15]19,22,24,27 has not fully addressed the complex geometry of the dead metal zone     CEDF (Figures 3-6) nor the influence of complex dead zone geometry CEDF (Figures 3-6) on the character of metal flow during workpiece ECAE through a Segal 2θ-die applying a two-parameter UBM in the form of a two-parameter RBM with trial DVF. This insufficientanalysis of energy-power parameters during ECAE with an introduction of the two-parameter UBM with DVF was the stimulus, which resulted in the research reported in the present article.This article is the first application of a two-parameter rigid block approach to a two-parameter UBM with DVF to workpiece 2D plastic flow through a Segal-geometry 2θ-die.
The aim of the present research is the phenomenological Upper Bound Method-based description of metal workpiece plastic ECAE flow through a 2θ angular die of Segal geometry during ECAE accounting for the complex geometry of dead zone CEDF.
The subject of the present research is the plastic flow of metal workpiece model through the ECAE process with 2θ angular die of Segal geometry.
The object of the present research is the general definition of the character of the plastic flow of a metal workpiece model through a 2θ angular die of Segal geometry with respect to the complex dead zone geometry and ECAE process parameters.
The experimental novelty of the present article is the introduction of initial circular gridlines to study workpiece flow through the ECAE process with 2θ angular dies of Segal geometry, shown in Perig's derived experimental approaches in Figures 3-6.
The prime novelty of the present research is the first time application of the two-parameter rigid block method to the estimation of punching pressure and plastic shear during metal workpiece ECAE through 2θ-die of Segal geometry, accounting for the complex geometry of dead zone CEDF in Figures 3-6.
The first experimental approach to analyze the dead metal zone shape CEDF (Figures 3-6) has been grounded on the introduction of layered models (Figure 5a) for workpiece model flow through a Segal-geometry die.However it has been found that layered models define only a general shape of dead zone CED near the corner point E without clarification of dead zone CED external contours in the inlet AOE and outlet BOE die channels in Figure 5a.So a second type of approach has been applied using a physical marker experimental method (Figures 4 and 5b), which provided us with general idea about external shape of dead zone CEDF.
It has been found that dead metal zone CEDF (Figures 3-6) in the workpiece volume has the shape near to the trial computational shape in Figure 7a.Additional physical simulation experiments required addressing to the physical models with the initial circular gridlines (Figures 3 and 5c-d).
In Figures 3-6 the physical workpiece models' flow goes within inlet channel AOC from the punch level to deformation zone entrance line AO; then towards bisector line EO and then in the outlet channel BOE from EO to BO.Using both solid markers (Figures 4, 5b), layered models (Figure 5a), and initial circular gridlines (Figures 3 and 5c-d) it has been found experimentally that in the 2θ-die external corner E the dead zone CEDF of plastic material flow appears and has a quite complex geometrical shape with additional "bottleneck" EF (Figures 3-6), i.e. with formation of additional break point F along bisecting line EO in the area of the cross point for two intersecting conditional lines EO and AB (Figures 3-6).

Two-Parameter Upper Bound Analysis for a Segal 2θ-Die
In accordance with observable results of physical simulation-based experiments in Figures 3-6 we assume the formation of a complex shape dead zone CEDF with the appearance of the additional straight arch EF, which limits the "spreading" of the metal dead zone to the CDF area in Figures 3-7.So the appearance of a symmetrical complex shape dead metal zone CEDF has been shown in Figures 3-7.
The upper bound theorem equation according to the works 34,35 has the following form: where (dE/dt) Johnson-Kudo is defined in Johnson & Kudo 34 and Kudo 35 as "the total rate of energy dissipation in the system per unit thickness in the direction normal to the plane of flow"; multiplier (k) Johnson-Kudo is defined in Johnson & Kudo 34 and Kudo 35 as "the shear stress"; (l IJ ) Johnson-Kudo and (Δu / IJ ) Johnson-Kudo are defined in Johnson & Kudo 34 and Kudo 35 as "the length of a straight boundary and the rate of relative slip between triangles "i" and "j" respectively"; (f K ) Johnson-Kudo , (l K ) Johnson-Kudo and (Δu / K ) Johnson-Kudo with subscript k are defined in Johnson & Kudo 34 and Kudo 35 as "the frictional resistance, the length of contact and the rate of relative slip between triangle k and the contacting tool surface" 34,35 ).It has been shown that the dead zone CEDF shape may be only symmetrical one with axis of symmetry EF, because possible asymmetry of dead zone CEDF leads to a violation of workpiece material incompressibility for the rigid blocks division used in Figures 3-6, 7a.The 2D plane model of the metal workpiece during ECAE in 2θ-die has been divided into 7 rigid triangular sections, as shown in Figures 3-6, 7a.
For the analysis of the ECAE metal flow the trial velocity field for UBM can be continuous, discontinuous or mixed.In the present work the Discontinuous Velocity Field (DVF) for the two-parameter UBM has been used in Figure 7b and in Tables 1-3.Derived experimental data have shown that it's better to describe the shape of the symmetrical dead zone CEDF in Figures 3-6, 7a and in Tables 1-3 with two independent parameters h=a•x and H=a•y.Here a is the width of inlet AO and outlet BO 2θ-die channels; x is the relative horizontal length of "waist" or bottleneck and y is the relative height of the dead zone in both entrance AOE and outlet BOE channels (Figures 3-6, 7a and Tables 1-3).The appearance of a symmetrical dead metal zone CEDF in the shape of the twin rigid triangular block numbered 5, which is adjacent to the 2θ-die external angle CED, locates in both inlet AOE and outlet EOB 2θ-die channels (Figures 3-6, 7a and Tables 1-3) and has the height H=a•y and length h=a•x.We also will assume that metal ECAE through a Segal 2θ-die with channel intersection angle 2θ>0° and 2θ<180° occurs with no back-pressure.Additionally we will assume that the constant plastic friction between the workpiece and 2θ-die walls AC and DB is independent of the normal stress σ N and is acting only at inlet l AC and outlet l DB lengths (Figures 3-6, 7a and Tables 1-3).
The friction stress τ F we will define according to the Siebel (Tresca) friction law as τ F =m•k, where 0.0≤m≤1.0 is the plastic friction factor in the Siebel (Tresca) friction law and the shear strength of the extruded material k=σ S /3 0.5 is the plastic constant, i.e. k is the maximum tangential stress for material with flow stress σ S .We will calculate the relative punching pressure p/2k (Figures 8a,c;9a,c) at the entrance line AO.Corresponding to the partitioning scheme in Figures 3-6, 7a and in Tables 1-3 a velocity hodograph has been shown in Figure 7b.The extruded workpiece material we will assume as rigid-plastic with no strain-hardening.The plastic friction force has been assumed as independent of sliding velocity.
The balance of external and internal power of plastic deformation has been expressed by the following algebraic equation (Figures 3-9 and Tables 1-3): where p is an applied ECAE punching pressure through the 2θ-die of Segal geometry (Figures 8a,c;9a,c); l I-J are the lengths of common boundaries or join interfaces for rigid blocks i and j (Figures 3-6, 7a and Tables 1-3); Table 1.The lines of discontinuity and sliding velocities for ECAE 2θ-die with 2θ>0° and 2θ<180° (x=h/a and y=H/a) in Figures 3-7.

Velocity discontinuity lines i -j
The lines of discontinuity and sliding velocities for ECAE 2θ-die with 2θ>0° and 2θ<180° (x=h/a and y=H/a) in Figures 3-7.
Velocity discontinuity lines i -j Table 3.The lines of discontinuity and sliding velocities for ECAE die with 2θ=90° (x=h/a and y=H/a) in Figures 4-5.

Velocity discontinuity lines
is the sine of angle β or the first algebraic equation for angle β determination in Figures 3-7; ( ) is the cosine of angle β or the second algebraic equation for angle β determination in Figures 3-7; ( ) is the tangent of angle β or the third algebraic equation for angle β determination in Figures 3-7; [V I-J ] are the velocities of relative sliding for these blocks i and j (Figures 3-6, 7a and Tables 1-3); i, j = 1, 2, 3, 4, 5; V 1 and V 3 are material velocities in the inlet AOE and outlet OEB 2θ-die channels respectively (Figures 3-7), and due to the ECAE process symmetry requirement we have The terms in Equation 2have been expressed as the functions of punching velocity V 1 and the relative dimensions of the metal dead zone CEDF in Figures 3-7: x=h/a and y=H/a.After substitution of obtained relationships in Equation 2and algebraic transformation, the following elementary formula for calculation of relative punching pressure p/2k has been derived and shown in Figures 8a,c;9a,c for two-parameter UBM (○○○): For ECAE die with 2θ=90° (Figures 4-5) Equation 3 yields According to the UBM the best approximation of the real p/2k corresponds to the minimum of expression (3), i.e. requires the solution of the following system of equations: The analysis of (5) shows that the minimum of the function (p/2k) for ECAE punching pressure during metal workpiece extrusion through the Segal 2θ-die with a channel intersection angle of 2θ>0° and 2θ<180° will occur for the real numerical solutions of the following transcendental system of algebraic equations: For ECAE die with 2θ=90° (Figures 4-5) systems (6) transforms into Using the Equations 3 and 6 we may estimate the dimensions H and h of dead zone CEDF in Figures 3-6, 7a and the value p/2k (Figures 8a,c;9a,c) of the ECAE punching pressure.In the present work the system (6) has been solved numerically in the case of fixed values for friction factor m.
ECAE is an SPD technique for grain refinement.Therefore the estimation of resulting plastic ECAE shear is also important.The total ECAE shear γ S is the sum of the shears on the discontinuity lines CO, FO and DO in Figures 3-6, 7a, i.e.

It has been known that
where V I-J N is a velocity component orthogonal to a discontinuity line l I-J (Figure 7b).
The value of the accumulated plastic shear (Figures 8b,d;9b,d) decreases with increasing friction factor.This general trend is assumed to be caused by additional metal sticking to the die walls during ECAE without lubrication.The area of dead metal zone CEDF increases with friction growth.In such cases with maximum friction takes place the destruction of   workpiece surface with formation of additional wrinkles and burrs at workpiece surface.These defects, together with the large dead zone and growing metal sticking violate the dynamics of macroscopic rotation within volume of worked material, which determines the value of accumulated plastic shear during ECAE.This causes above mentioned decrease of accumulated plastic shear (Figures 8b,d;9b,d) with the increase of the friction factor during ECAE through a Segal 2θ-die with 2θ>0° and 2θ<180°.

Comparison with Published Experimental Results
The results of the comparison of obtained two-parameter UBM values (3)-( 4), ( 6) with published experimental data 30,34 for ECAE punching pressure p/2k are shown in Figure 10, where ECAE die angle is 2θ=90º: ••••• -two-parameter UBM p/2k results, estimated accordingly to (3)-( 4), ( 6 So good agreement of proposed two-parameter UBM approach with published experimental results has been obtained for an ECAE die with channel intersection angle 2θ=90º.

Conclusions
I. Physical simulation results for ECAE material flow through a Segal 2θ-die with a channel intersection angle of 2θ>0° and 2θ<180° underline the complex geometry of dead zone for metal plastic flow.This experimentally determined fact was the stimulus resulting in the introduction of the two-parameter UBM to the metal ECAE problem for metal workpiece forcing through Segal 2θ-die with channel intersection angle of 2θ>0° and 2θ<180°.
II.The application of the two-parameter upper bound method (UBM) in the form of two-parameter rigid block method (RBM) with trial discontinuous velocity field (DVF) to the analysis of ECAE through a Segal 2θ-die with 2θ>0° and 2θ<180° correctly describes the essential geometrical features for this SPD process, such as the appearance of a dead zone (DZ) and its increase as a function of external friction m.
III. Also the increase of ECAE punching pressure p/2k and decrease of total plastic shear γ S , resulting from an increase in friction m is well predicted with two-way ECAE parameter computations.The two-parameter upper bound (UBM) results based on a discontinuous velocity field (DVF) are in good agreement with the one-parameter upper bound solution results and published experimental results of Pürçek 32 and Talebanpour & Ebrahimi 27 .
IV.The proposed two-parameter upper bound approach can be applied to further analysis of metal workpieces ECAE flow through 2θ-dies with external and internal radii in the channel intersection zone as well as for the dies for equal channel multiple angular extrusion, where angular dies have additional pairs of intersecting channels.

Nomenclature
• SPD is Severe Plastic Deformation; • ECAE is Equal Channel Angular Extrusion; • RBM is Rigid Block Method; • UBM is Upper Bound Method; • one-parameter UBM assumes dead zone shape in the form of isosceles triangle CED without additional bottleneck EF; • two-parameter UBM assumes dead zone shape CEDF in the form of two adjacent triangles CEF & DEF with formation of additional bottleneck EF; • DVF is Discontinuous Velocity Field; • DZ is Dead Zone; • a is the channel width of the ECAE die, [m]; • 2θ is the channel intersection angle of the ECAE die, [deg]; • The 2θ-die is the angular die AEO-BEO with channel intersection angle 0°<2θ<180°; • The 2θ-die of Segal geometry is the angular die AEO -BEO with channel intersection angle 0°<2θ<180° without external and internal radii in channel intersection zone; • h = ax is the horizontal length of the dead zone CEDF, i.e. the horizontal projection of bottleneck EF, [m]; • x = h/a is the relative dimensionless horizontal length of the dead zone CEDF, i.e. the first independent parameter for two-parameter UBM; • H = ya is the vertical length of the dead zone CEDF within inlet die channel AEO, i.e. the relative height of the dead zone CEDF, [m];  4), and ( 6); ■ -G.Pürçek experiment (results divergence is δ G. Pürçek (p/2k)=16.48% [32]); ♦ -B.Talebanpour experiment (results divergence is δ B. Talebanpour (p/2k)=9.76% [27]).

Figure 1 .
Figure 1.The principal scheme of ECAE billet processing through the 2θ-die.

Figure 3 .
Figure 3. Physical model of the shape of the material dead zone CEDF during ECAE through a Segal 2θ-die with channel intersection angle 2θ=75° after three ECAE passes via deformation route C.

Figure 4 .
Figure 4. Physical model of the shape of material dead zone CEDF during ECAE through a Segal 2θ-die with channel intersection angle 2θ=90°.

Figure 5 .
Figure 5. Physical models of the shape of material dead zone CEDF during ECAE through a Segal die with channel intersection angle 2θ=90° with an introduction of layered workpiece models (a), solid marker techniques (b), and initial circular gridlines (c, d).

Figure 6 .
Figure 6.Physical model of the shape of material dead zone CEDF during ECAE through a Segal 2θ-die with channel intersection angle 2θ=105°.
Two-parameter Rigid Block Approach to Upper Bound Analysis of Equal Channel Angular Extrusion Through a Segal 2θ-die parameter Rigid Block Approach to Upper Bound Analysis of Equal Channel Angular Extrusion Through a Segal 2θ-die

Table 4 .
Comparison of upper bound-estimated computational results for ECAE punching pressure p/2k and total accumulated plastic strain γ S derived with the introduction of two-way UBM techniques for Segal 2θ-dies.