2D Upper Bound Analysis of ECAE through 2θ-Dies

The manufacture of new advanced materials requires the development of new material forming deformation techniques. Equal Channel Angular Extrusion (ECAE), also known as Equal Channel Angular Pressing (ECAP), is one of the prominent Severe Plastic Deformation (SPD) techniques1-32. ECAE involves one or more extrusion passes of a lubricated billet through a die with two intersecting channels of equal cross-section. This process has been very promising for the production of bulk ultra fine-grained materials with special properties7,17. The large cumulative shear strain in ECAE leads to refinement of grains and increased boundary length. The resulting materials show a combination of a very high strength and ductility7,17. The mechanics of Equal Channel Angular Extrusion (ECAE) attracts a lot of research efforts from different fields of theoretical, experimental and industrial science1-32. Many contributions have been focused on the introduction of slip line fields11,30,31, Upper Bound Method (UBM) with continuous trial velocity fields1-6,12-19,29,32, Navier-Stokes equations22,27,28, fractal geometry techniques, discrete elements, FEM computational methods2,12,14,15,20,23-25 etc. The main computational approach in3-5 has been grounded on the introduction of cylindrical coordinate systems with the center a symmetrical plastic deformation zone, deriving a kinematically admissible velocity field, computation of the only non-zero strain rate field component ( ) r d dt θ e , generation of an equation for dissipated power balance and further minimization of the expression for dissipated power by optimization parameter differentiation. Laptev et al., 2014 have applied a one-parameter UBM (i.e. UBM for 1 DOF) with a trial DVF to the study of ECAE of an ideally plastic and strain-hardening metal workpiece through a die with channel intersection angle 2θ=90°, where the optimization parameter is the dead zone height11. Perig et al., 2009 and Perig et al., 2014 have applied a twoparameter UBM (i.e. UBM for 2 DOFs) with a trial DVF to the study of ECAE of an ideally plastic metal workpiece through a die with channel intersection angle 2θ=90°, where optimization parameters are the dead zone height and the length of the intermediate bottleneck within dead zone21,26. Perig et al., 2008 have applied a FEM simulation for a numerical description of copper workpiece flow through a 2θ-die[20]. Perig et al., 2010 have applied the Navier-Stokes equations in the curl transfer form for the numeric finitedifference description of viscous material flow through an ECAE die with channel intersection angle 2θ=90°[22]. Perig et al. have applied a QForm FEM simulation for the numerical description of copper workpiece flow through a die with channel intersection angle 2θ=90°, where external die wall is made in the shape of a hyperbola, parabola and catenary curve23. Perig et al. have applied a QForm FEM simulation for the numerical description of copper workpiece flow through a die with channel intersection angle 2θ=90°, with external and internal radii in the channel intersection zone24. Perig et al. have applied physical and numerical simulation techniques to derive the technological 2D Upper Bound Analysis of ECAE through 2θ-Dies for a Range of Channel Angles


Introduction
The manufacture of new advanced materials requires the development of new material forming deformation techniques.Equal Channel Angular Extrusion (ECAE), also known as Equal Channel Angular Pressing (ECAP), is one of the prominent Severe Plastic Deformation (SPD) techniques  . ECAEinvolves one or more extrusion passes of a lubricated billet through a die with two intersecting channels of equal cross-section.This process has been very promising for the production of bulk ultra fine-grained materials with special properties 7,17 .The large cumulative shear strain in ECAE leads to refinement of grains and increased boundary length.The resulting materials show a combination of a very high strength and ductility 7,17 .
The main computational approach in [3][4][5] has been grounded on the introduction of cylindrical coordinate systems with the center a symmetrical plastic deformation zone, deriving a kinematically admissible velocity field, computation of the only non-zero strain rate field component ( ) , generation of an equation for dissipated power balance and further minimization of the expression for dissipated power by optimization parameter differentiation.Laptev et al., 2014 have applied a one-parameter UBM (i.e.UBM for 1 DOF) with a trial DVF to the study of ECAE of an ideally plastic and strain-hardening metal workpiece through a die with channel intersection angle 2θ=90°, where the optimization parameter is the dead zone height 11   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 28 .
The rigid blocks approach to the UBM of metal flow in ECAE through a 2θ-die with a trial Discontinuous Velocity Field (DVF) has not been addressed, however, in previous known publications  .
All previous research  has not fully addressed the mechanics of ECAE through a 2θ-die with channel intersection angle 2θ>0° and 2θ≠90° with an introduction of UBM with trial DVF. Thisinsufficient analysis of energy-power parameters during ECAE through 2θ-die was the stimulus that led to the research reported in the present article, which is the first upper bound approach to workpiece 2D plastic flow through the application of the UBM with DVF.

Physical Simulation of ECAE Through 2θ-Segal Die
Plasticine based physical simulation experiments in Figures 1a, c, 9 and 10 are the new original experimental results, obtained personally by the author.The initial plasticine physical models of the workpiece in Figures 1a, c and 10 were made from navy blue plasticine in Figure 1a and light blue plasticine in Figures 1c and 10 in the shapes of rectangular parallelepipeds.These rectangular parallelepipeds were frozen, the front sides of parallelepipeds were marked, and through-holes in parallelepipeds were perforated.After that the plasticine parallelepipeds with through-holes were frozen again.Simultaneously the plasticine pieces with different colors were heated to the half-solid state and were placed into through-holes of parallelepipeds through the application of squirt without needle.In this way the initial circular plasticine gridline was formed within the plasticine model of the workpiece.Simple new original physical simulation experiments of workpiece model flow during ECAE through a 2θ-die with the channel width a =27 mm and the channel intersection angle 2θ=105° are shown in Figures 1, 9 and 10.The ECAE die model is manufactured from kapron in Figure 1a and wood in Figures 1c and 10  The applied punching force in Figure 1 is marked as P, the applied punching pressure in Figures 3, 4, 6a and (1)-( 2) is marked as p, and the direction of P and p defines the processed workpiece model flow direction sequentially from the top along the inlet channel AOE of 2θ-die (from AO to EB) to the right and down along the outlet channel OEB (from AE to OB).It has been found experimentally that in the external corner AEB of 2θ-die the dead zone AEB of plastic material flow appears.As follows from Figures 1a, c and 10 with circular gridlines shown, in the neighborhood of the outlet 2θ-die channel OEB, deformed elliptical markers indicate two independent types of Severe Plastic Deformations (SPD).
The first material deformation results in unevenness of the workpiece plastic flow in the outlet die channel OEB and is associated with the relative rotation of the major axes for every deformed elliptical marker with respect to axis EB.This relative rotation of elliptical markers as rigid solids takes place for every marker in outlet 2θ-die channel OEB and indicates processed workpiece material macroscopic rotation.The closed loops of macroscopic rotation, located in the outlet channels of the ECAE dies, are shown in Figures 1b, d.Another type of material deformation is associated with relative bending of elliptical markers and the appearance of inflection points for every marker.This type of deformation is defined as rotational inhomogeneity.

Upper Bound Analysis of ECAE Through 2θ-Segal Die
The upper bound theorem equation according to the works 8,9 has the following form: where (dE/dt) Johnson-Kudo is defined in 8,9 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 8,9 as "the shear stress"; (l ij ) Johnson-Kudo and (Δu / ij ) Johnson-Kudo are defined in 8,9 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 8,9 as "the frictional resistance, the length of contact and the rate of relative slip between triangle k and the contacting tool surface" (Johnson & Kudo, 1962) 8,9 .
According to the UBM (Figure 2a) a trial velocity field (Figure 2b) for a 2θ-die has to be introduced.
The trial velocity field can be continuous, discontinuous or mixed.In the present work a discontinuous velocity field for the 2θ-die has been used in Figure 2b.The 2D plane model of the metal workpiece in the 2θ-die has been divided into 4 rigid triangular sections AOC (number 1), OCD (number 2), ODB (number 3), and CDE (number 4), as shown in Figure 2a.The appearance of a symmetrical dead metal zone CDE in the shape of the rigid triangular block CDE numbered 4, which is adjacent to the die external angle E, lies within both inlet AOE and outlet EOB die channels (Figure 2a) and has the assumed height h = ax, where a is the channel width, and 0 ≤ x ≤ cot(θ).It is necessary to note that an assumption concerning the formation of an asymmetrical dead zone CDE (number 4) for metal ECAE through a 2θ-die is completely unacceptable because possible asymmetry of dead zone CDE (number 4) would result in a violation of metal incompressibility for the rigid blocks division used in Figure 2a.
We also will suppose that ECAE occurs with no backpressure and the constant plastic friction between the workpiece and 2θ-die walls is independent of the normal stress s n and is acting only for inlet l AC and outlet l DB lengths.The friction stress τ f we will define according to the Siebel (Tresca) friction law as τ f = mk, where 0 ≤ m ≤ 1 is the plastic friction factor in the Siebel (Tresca) friction law, the shear strength of the extruded material is the plastic constant, i.e. k is the maximum tangential stress for material with flow stress s S .
We will calculate punching pressure p at the entrance line AO.Corresponding to the partitioning scheme for Segal 2θ-die in Figure 2a, a velocity hodograph has been shown in Figure 2b.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 for ECAE of the rigid-plastic and nonhardening workpiece material in ( 1) is expressed by the following algebraic equation (Figures 3, 4 and 6a): where p is an applied ECAE punching pressure; l i-j are the lengths of common boundaries or interface joints for rigid blocks i and j within the 2θ-die rigid blocks partitioning scheme (Table 1); l AC = l DB are the lengths of friction zones; [V i-j ] are the velocities of relative sliding for blocks i and j; i, j = 1, 2, 3 (Table 1); V 1 and V 3 are material velocities in the inlet AOE and outlet OEB 2θ-die channels respectively.Moreover, due to the ECAE process symmetry requirement we have the following identities: The terms in Equation 2 have been expressed as the functions of pressing velocity V 1 and the relative length of the symmetric dead zone x = h/a in Table 1.After substitution of obtained relationships from Table 1 in Equation 2 and algebraic transformation, the following elementary formula for calculation of relative ECAE punching pressure has been derived (Figures 3, 4 and 6a): Calculations from Equation 3 are shown in Figures 3,  4 and 6a.
The correctness of derived algebraic expression (3) may be confirmed by substituting θ = 45° in (3).This case corresponds to ECAE of metal through a die with channel intersection angle 2θ = 90°, studied in the work 11 of Laptev et al.In this case the derived Formula 3 coincides with expression (3) of the work in 11 .
According to the UBM the best approximation of the real punching pressure p/2k through 2θ-die of Segal corresponds to the minimum of Expression 3, i.e. requires the solution of equation ∂(p/2k)/∂x = 0.The analytic algebraic analysis of (3) shows that the minimum of this function for ECAE punching pressure (p/2k) takes place when the relative height has the value The derived general analytic Formula 4 coincides with algebraic expression (4) of the work 11 in the case of ECAE through a die with channel intersection angle 2θ = 90°.
Using Equations 3 and 4 we may estimate the relative height x of dead zone CDE (number 4 in Figure 2a) and the value p/2k of the relative punching pressure during ECAE through a Segal 2θ-die.It has been found from Equation 4 that in some cases the global minimum of relative punching pressure p/2k during ECAE through a 2θ-die is not reached at positive values of relative length x of dead zone CDE in Figure 1a.This means that the dead zone CDE does not always appear during ECAE through 2θ-die.The development of a dead zone CDE can be seen in Figure 5.
Table 1.The lines of discontinuity and sliding velocities for a 2θ-Segal die.

Velocity discontinuity lines i-j l i-j
( ) derived with UBM according to the Formula 4 for x=h/a.So the dead zone CDE (Figure 2a) of plastic flow always appears in the case of non-zero friction factor m≠0 for ECAE through a Segal die with channel intersection angle 2θ=90°.However, in the case of ECAE through the Segal die with channel intersection angle 2θ=105° the dead zone CDE (Figure 2a) forms and appears only for the friction factor values m≥0.258 (Figure 5).During ECAE through the die with 2θ=120° the dead zone CDE (Figure 2a) appears only for the friction factor values m≥0.5 (Figure 5).ECAE is a SPD technique for grain refinement.Therefore the estimation of resulting plastic ECAE shear is also important.The total ECAE shear g S (Figure 6b) is the sum of the shears on the discontinuity inclined lines CO and DO in Figure 2a, i.e. (5) The meaning and the correction of linear superposition of finite deformations in ( 5) is shown in Figures 7, 8 and Table 2 through the good agreement of ( 5) with the slip line analysis results.The comprehensive background for linear superposition of finite deformations in ( 5) was made by Johnson & Kudo 8 , Kudo 9 , and Laptev et.al. 10,11 .
It is known that where n i j V − is a velocity component orthogonal to a discontinuity line l i-j (Table 1).
Using (6) and Table 1 we will calculate g with the following expression:  Plotted curve families in Figure 5 have been derived on the basis of (4) with the limitation for x=h/a that 0 ≤ x ≤ cot(θ).
In Figure 5 the graphical dependencies have been shown for the relative height of dead zone CDE with respect to friction factor m for different 2θ-die angles, which have been  From the hodograph in Figure 1b and Table 1, from which The derived general analytic Formula 8 for accumulated plastic shear coincides with algebraic Expression 8 of the work 11 in the case of ECAE through a die with channel intersection angle 2θ° = 90°.
Then the equivalent plastic strain during ECAE may be calculated as follows: ( The relative area of the dead zone CDE in Figure 2 may be estimated as follows: ( ) ( ) The derived general analytic formula (10) for relative area of the dead zone coincides with algebraic expression (16) of the work 11 in the case of ECAE through a die with channel intersection angle 2θ = 90°.

Discussion of Results for ECAE Through 2θ-Segal Die
The results (2)-( 10) derived from the UBM have been compared with Segal's Slip Lines (SL) solution for summary shear g Segal 30,31 for θ>0.The relative ECAE pressure p/2k corresponding to the slip line analysis Segal has calculated in 30,31 by the following analytical formula: where ( ) According to 30,31 the summary shear by Segal is as follows: ( ) The results of the comparison for total plastic shear (8), ( 4), ( 12) and ( 13) have been presented in Figures 7, 8 and Table 2.
As expected, the values of accumulated ECAE plastic shear g S (Figures 7, 8 and Table 2), obtained by the upper bound analysis (UBM), are higher than corresponding ones derived by the slip line theory (SLM), the differences ranging from negligible to 16.4%.The relative divergence (disagreement) of results was evaluated by the formula: where R SLM and R UBM are the values obtained by the slip line (SLM) and upper bound (UBM) theories respectively.
The evaluation shows (Figure 7 -Figure 8 and Table 2) that for the values of friction factor 0 ≤ m ≤ 1 and die channel intersection angle 2θ=90° the maximum divergence of results d max for total shear g S by two-way calculations is For the Segal die with 2θ=105° and friction factor 0 ≤ m ≤ 1 the maximum difference d max for total shear g S by two-way calculations is d max (g)|(2θ=105°) =3.662%.And for the Segal die with 2θ=120° and friction factor 0 ≤ m ≤ 1 the maximum difference d max for total shear g S by two-way calculations is d max (g)|(2θ=120°) =5.323%.
Further increase of friction factor m in Figures 5-8 leads to the rise of the extrusion punching pressure p/2k (Figure 6a) and to the remarkable decrease in total plastic shear γ S (Figures 6b, 7 and 8).

Dead Zone Analysis
The geometric analysis of local flow during ECAE allows estimate the dead zone relative area S/a 2 using UBM (10) and physical simulation techniques (Figures 9 and 10).
UBM Formulae 4 and 10 for the die with 2θ=105° and maximum friction factor m=1.0 yield the following values (x) UBM = 0.428 and (S/a 2 ) UBM = 0.0726 for dead zone geometry in the case of the plastic flow of non-hardening workpiece material.Averaged physical simulation results for visually observable dead zone area in Figures 9 and 10 for the viscous flow of plasticine workpiece models show that <S/a 2 > = 0.1382, i.e. physical simulation dead zone area is in 1.91 times larger than theoretical one, derived with UBM Formulae 4 and 10.
The derived divergence of results for relative dead zone area is based on the different rheological features during flow of metal materials and soft physical models of workpieces However the derived divergence of results for relative dead zone area does not reduce in any way the importance of physical simulation techniques for quality assessment of ECAE geometric features during workpiece flow through the 2θ-dies.

Conclusions
The application of the upper bound theory (UBM) in the form of one-parameter rigid blocks method with trial   numerically derived formation of a dead zone for material plastic flow in Figure 5.
Also the increase of ECAE pressure and decrease of total plastic shear resulting from an increase in friction during ECAE through a Segal 2θ-die is well predicted.
The numerical estimation shows that the relative divergence of results d(g S ) for two-way accumulated plastic shear γ S calculations in the range of friction factor values 0 ≤ m ≤ 1 is not more than 5.337% for 2θ=90˚, 3.662% for 2θ=105˚, and 5.323% for 2θ=120˚.Thus the maximum disagreement in summary shear does not exceed 5.337% for 90˚≤2θ≤120˚.
Both theories predict significant decrease in p/2k and g S when the 2θ angle increases from 60˚ to 135˚.Increasing friction factor m leads to rise of the extrusion punching pressure and to a remarkable decrease in total plastic shear g S .So the upper bound (UBM) results for accumulated plastic shear g S based on discontinuous velocity field (DVF) are in good agreement with classical Segal's slip line solution (SLM) for 0 ≤ m ≤ 1.
The proposed upper bound approach can be applied to further analysis of ECAE with angular dies of more complex geometries.
recommendation concerning rationalization of ECAE of workpieces through 2θ-dies with 2θ<90° by a change of the classical punch shape25 .Perig et al. have applied the Navier-Stokes equations in the curl transfer form for a numerical finite-difference description of viscous material ECAE flow through an S-shaped multiple angle die with movable inlet wall 27 .Perig et al. have applied the Navier-Stokes equations in the curl transfer form for the numeric finite-difference description of viscous material flow through an ECAE die

Figure 1 .
Figure 1.Physical simulation of ECAE through non-rectangular 2θ-dies with channel intersection angles 2θ=75° (a, b) and 2θ=105° (c, d) showing dead zones AEB formation as well as the formation of material macroscopic rotation in outlet channels OEB (a, c); and experimental vector fields of macroscopic rotation gradients (b, d).

Figure 2 .
Figure 2. The rigid blocks partitioning scheme of ECAE through a non-rectangular Segal 2θ-die (a) and corresponding velocity hodograph (b).

Figure 3 .
Figure 3. Spatial surface graphs (d) for relative ECAE punching pressure p/2k versus dead zone relative length x and friction factor m for Segal 2θ-dies with channels intersection angles 2θ=90° (surface of Floral White color in a, d), 2θ=105° (surface of Sky Blue color in b, d), and 2θ=120° (colored surface in c, d).

Figure 8 .
Figure 8.Comparison of Perig's upper bound (UBM, •••) and Segal's slip lines (SL, □□□) solutions for different values of friction factor m. Computational dependencies for summary plastic shear g S (a, b) with respect to the friction factor m, derived with UBM and slip line method (SLM) during ECAE of non-hardening material through Segal dies with channel intersection angles 2θ=120° (a) and 2θ=135° (b).
discontinuous velocity field to the analysis of ECAE through a Segal 2θ-die correctly describes the essential features for this SPD process like the appearance of a dead zone (DZ) and its increase as a function of external friction.Physical modeling with circular gridlines and plasticine workpiece models in Figures 1, 9 and 10 confirms the

Figure 10 .
Figure 10.The physical simulation-derived geometry of observable dead zone during ECAE of plasticine workpiece model through the die with 2θ=105°, where S/a 2 = 0.1444.
21,26g et al., 2009 and Perig et al., 2014have applied a twoparameter UBM (i.e.UBM for 2 DOFs) with a trial DVF to the study of ECAE of an ideally plastic metal workpiece through a die with channel intersection angle 2θ=90°, where optimization parameters are the dead zone height and the length of the intermediate bottleneck within dead zone21,26.
24e with channel intersection angle 2θ=90°, where external die wall is made in the shape of a hyperbola, parabola and catenary curve23.Perig et al. have applied a QForm FEM simulation for the numerical description of copper workpiece flow through a die with channel intersection angle 2θ=90°, with external and internal radii in the channel intersection zone24.Perig et al. have applied physical and numerical simulation techniques to derive the technological

Table 2 .
Comparison of total plastic shears, derived by Perig's upper bound and Segal's slip lines solutions for different values of friction factor m.
through the ECAE die with channel intersection angle 2θ=105° in Figures 9 and 10.