Sensitivity analysis of the ECAE process via 2 k experiments design

In this work the theoretical solutions based upon the upper-bound theorem recently proposed by Pérez and Luri [Mech. Mater. 40 (2008) 617] for the equal channel angular extrusion process (ECAE) are analyzed by performing a 25 central composite factorial analysis. The uniaxial mechanical properties of commercial pure aluminium are considered by assuming isotropic nonlinear work-hardening combined to von Mises and Drucker isotropic yield criteria to predict the ECAE load and the effective plastic strain. From the proposed 25 factorial analysis, the main parameters affecting the ECAE pressure may be ranked as: (1) Friction factor, (2) die channels intersection angle, (3) outer and (4) inner die corners fillet radii and lastly, (5) plunger velocity. Alternatively, the effective plastic strain is mainly controlled by the die channels intersection angle and, in a less extent, by the outer and inner die corners fillet radii. Keyword: upper-bound theorem, factorial analysis, equal channel angular extrusion.


INTRODUCTION
The equal channel angular extrusion process (ECAE) is a severe plastic deformation process employed to produce bulk ultra-fine grained materials with improved mechanical properties [1,2].In the ECAE process, a well lubricated billet is forced to pass through a two-channel die with constant crosssectional area.The workpiece undergoes a large amount of plastic strain by simple shear within the deformation zone located at the channels die intersection [3].Thus, the knowledge of the kinematics of deformation is essential to understand the basic mechanisms controlling the grain refinement in the ECAE process.The consideration of material nonlinear work-hardening to predict the ECAE pressure, assuming a frictionless condition with an outer die corner radius, was firstly proposed by Alkorta and Sevillano [4].Their analytical solution is based upon the upper-bound theorem and provides a good agreement with numerical predictions determined from a plane-strain finite element model.Later on, Pérez [5,6] evaluated the effects of equal fillet die radii located at the die channels intersection with the help of the upper-bound theorem and finite element simulations, respectively.Although the analytical model proposed by Pérez neglected the effects of material work-hardening, the benefits of adopting a non-zero inner die corner radius were revealed up to a maximum value from which the predominant deformation mode is bending.
Eivani and Taheri [7] presented the first upper-bound solution in which both the friction conditions and nonlinear work-hardening behavior were considered for a die geometry containing only the outer die corner radius.By varying the die channels angle between 90º and 135º and for a given friction factor, they reported that both the effective von Mises plastic strain and normalized extrusion pressure decreases as the outer die corner radius increases.Besides, it is verified that the effect of the die channels intersection angle prevails over both tribological conditions and other geometrical or rheological parameters.Also, Eivani and Taheri [8] also analyzed the effects of the formation of a dead metal zone in sharp-corner dies and established an explicit dependence of the resulting strain per ECAE pass with the friction factor from the minimization of the extrusion force.A better agreement with the measured load was achieved by this recent work by comparison with the earlier results [7].
Pérez [5] and Luri et al. [9] developed theoretical expressions for the shear strain calculations considering all the possible die configurations.The authors showed a gain close to 11% of the effective plastic strain per pass when the inner radius is 2.67 times larger than the outer die fillet radius.Recently, Pérez and Luri [10] developed upper-bound solutions for the extrusion pressure, considering all die geometry possibilities and including the friction effects for perfectly plastic materials.The authors pointed out that the increasing of the inner fillet radius leads to an elevation of effective plastic strain combined with higher extrusion pressure levels.
Based upon the review presented here above, it is clear the need for more general modelling techniques to describe the effects of the relevant parameters on the strains and mechanical properties resulting from the ECAE, namely, tooling geometry, billet material, friction conditions and processing velocity.Into this context, the present work firstly aims at providing a sensitivity analysis with the help of the 2K central composite factorial design to evaluate the influence of these parameters on the effective plastic strain and the extrusion pressure for commercial pure aluminum and some typical die configurations by means of the variance analysis.

Extrusion pressure
The upper-bound solutions developed by Pérez and Luri [10] for the extrusion pressure, p, for all possible die geometries and including frictional effects are recalled.The tooling configurations are shown in Figure 1 where the inner and outer fillet radii are defined by Rinner and Router, respectively, and have local origin along line O.At the same time, Φ is the die channels intersection angle whereas β denotes the angle associated to nonzero fillet radii values.Also, r and x define the radial and horizontal directions.Considering the material point q and its position vector Oq and assuming the constant velocity V 0 hypothesis for both the plunger and the point q, the extrusion pressure of rectangular samples can be calculated by, where κ is the material pure shear yield stress and f is the Tresca's friction factor.Also, H, L and W denote the billet total height, width and thickness, respectively.
According to Pérez and Luri [10] the angle β is given by,

Plastic Material Behaviour
The billet material plastic behaviour is assumed as isotropic and temperature independent including nonlinear work-hardening with strain-rate effects.Moreover, the yield surface shape influence on the pure shear yield stress κ and, therefore, on the extrusion pressure is evaluated by considering both von Mises and Drucker [11] isotropic yield criteria.Thus, the plastic loading condition is defined as, where f denotes the yield function, F (σ ij ) is a first degree homogeneous function of the Cauchy stress tensor, σ ij = σ kk δ ij + S ij , defining the yield surface shape whereas σ y is the uniaxial yield stress identified as a function of the equivalent plastic strain and strain-rate scalar measures.
The von Mises and Drucker yield criteria are defined for the second and third invariants of the deviatoric stress components of the Cauchy stress tensor, S ij , that is, And assuming in-plane pure shear (S 12 = S 21 = κ other S ij = 0) combined with Equation 3, where c is a material constant satisfying the condition -27/8 ≤ c ≤ 2.25 [12] for the yield locus convexity.Drucker's yield criterion is suited to describe the crystallographic yield loci of both isotropic f.c.c. and b.c.c.metals.In the present work, the parameter c is assumed to be equal to 2.0.This value has been adopted by Ferron et al. [13] to fit the isotropic f.c.c.yield loci determined by Barlat and Lian [14] with the Bishop and Hill [15] model.
The uniaxial tension yield stress σ y is calculated by means of the average stress obtained from the material Swift hardening law with multiplicative strain-rate sensitivity as, wherein p ε is the effective plastic strain, and the strain-rate effect is accounted for by introducing the dwelling time in the ECAE deformation zone, tD, defined in section 2.3.Also, B, ε 0 , n and m denote strength coefficient, pre-strain, work-hardening exponent and strain-rate sensitivity exponent, respectively.Equation.6 is numerically solved through the trapezoidal rule in order to accurately determine the mean stress.It should be noted that Equation 6is restricted to a constant strain-rate deformation process.This assumption is adopted hereafter based upon the idea of a total time resulting from the ECAE deformation zone geometry.In the following, we assume that the elastic strains are small in comparison to the resulting ess and can, thus be neglected.
The plastic strain-rate components are determined assuming isotropic work-hardening from the associated flow rule applied to the yield function, see Equation 3, as, plastic strains from the ECAE proc ε that is,

Effective plastic strain
where the Eq 2 ε ε = ε uation 8 is valid to von Mises and Drucker criteria, once for a pure shear stress state the third invariant of the tensor Sij vanishes.Therefore, the total effective plastic strain is obtained by integrating Equation 8, namely, and assuming a constant shear strain-rate as one obtains: where, according to Figure 1, the plastic shear strain-rate components are defined as V 0 / x by assuming x = L into the regions AEB and DFC.Also, for the portion ABCD the associated shear strain contribution is equal to V 0 / r [10].The solutions for the shear plastic strain associated to die geometries presented in Figure 1 were proposed by Pérez [5] and Luri et al. [9], that is,

Deformation time
In the present work, the during which the billet undergoes severe plastic deformation along the die channels intersection was assumed as the contributions from the regions AEB, ABCD and DFC depicted on Figure 1 by considering that the inlet and outlet surfaces AE and DF have the same length.Thus, for the continuous kinematically admissible velocity field defined by a constant velocity V 0 , the time between the inlet and outlet surfaces is given by,

The 2 K Factorial Central Composite Factorial Design
The methodology proposed by Montgomery [16] for the single-replicate 2 K central composite design of experiments is adopted for theoretical simulations to classify the interesting parameters influence on both extrusion pressure and effective plastic strain by means of a variance analysis.The parameters considered are die geometry (Rinner, Router and Φ), friction conditions (Tresca friction factor f) and plunger velocity (V 0 ).These parameters and their corresponding values are listed in Table 1.
To perform the variance analysis related to central composite factorial design, the calculations of parameters effect (A, B,..., K), sum of the squares (SSA,B,…, K) for each individual effect, total sum of squares (SSTA,B,…, K), pure quadratic curvature SSPQ, error (E) and mean error are needed.The Equations 14 to 19 define each one of these variables, that is, where nr, nF, nCP, F y and CP y denote number of replicates, numbers of factorial and central points, averages between factorial and central points, respectively.Finally, the variance, F o , is defined by, where the index l takes into account from I to N-th value in the summation over the most important effects on either p or p ε denoted by SSl.Also, the combinations between the considered parameters for each case simulated are listed on Table 2.

RESULTS AND DISCUSSION
The mechanical properties considered in the present work are related to the commercial pure aluminium tested in uniaxial tension by Bressan et al. [17], according to Equation.(6), defined by B = 235 MPa, ε 0 = 0.045, n = 0.21 and m = 0.027.Firstly, an evaluation of the adopted yield criteria and friction effects on the extrusion pressure, p, is realized assuming a die with Φ = 90º wherein the die fillet radii are taken equal to zero together with a plunger velocity (V 0 ) value of 2.5 mm / s.Then, a 2K central composite factorial design is employed to classify, in order of relevance, the die geometrical, frictional conditions and process parameters upon the predictions of either p and assuming the billet dimensions as H = 75 mm and L = W = 15 mm.The levels assumed for each parameter are listed in the Tab. 1.  Figure 2 presents the effects of the plasticity criterion defined in terms of the ratio κ/σ y equal to 0.54 for Drucker and 0.58 for von Mises isotropic descriptions and the friction factor f on the ECAE pressure assuming Φ = 90º, Rinner = Router = 0 mm along with V 0 = 2.5 mm / s.As expected, one can observe the existence of a direct effect from the frictional conditions, namely, the ECAE pressure increases significantly with the friction factor f or in a less extent with the yield stress ratio κ/σ y .The Drucker yield surface presents a flattening between plane tension/compression and pure shear stress states which is responsible for the decreasing of the ratio κ/σ y in comparison to the von Mises yield criterion.Hereafter, the Drucker isotropic yield criterion is adopted for all the analysis related to factorial analysis on the pressure predictions.
Table 3 presents the variance (F o ) analysis associated to the 25 central composite design employed in the present work.In relation to the pressure, p, the influence of the parameters considered for the factorial design can be classified in order of importance as: (1) friction factor m, (2) intersection die angle Φ, (3) outer fillet radius Router, (4) inner fillet radius Rinner and (5) the plunger velocity V 0 , respectively.As expected, in the case of the effective plastic strain the variance results confirmed the large dependence only with the die geometrical parameters ordered as: (1) intersection die angle Φ, (2) inner fillet radius Rinner and (3) outer fillet radius Router.

CONCLUSIONS
Analytical investigations based upon the upper-bound method, including the material strain-rate effects and two isotropic plasticity yield criteria are proposed in the present work in order to evaluate the extrusion pressure and the effective plastic strain associated to the processing of a commercial pure aluminium.The effects of the plasticity criteria on the extrusion pressure are evaluated to point out the formulation responsible to processing load decreasing.Finally, a variance analysis based on the 2 5 central composite factorial design was performed to quantify the relevance of these parameters on the ECAE pressure and the effective plastic strain.From these analyses, the following conclusions can be outlined: 1) The analysis of the influence of yield surface shape and friction conditions on the extrusion pressure proved to be a useful tool to better understand the frictional conditions effects arising from a single pass of ECAE at room temperature.In particular, the isotropic Drucker yield criterion is more appropriate to reproduce the pure shear and plane tension /compression stress states than the von Mises criterion and, thus, should be adopted in the analytical predictions of fcc materials deformed via ECAE; 2) From the performed variance analysis, the ECAE parameters most affecting the extrusion pressure can be classified in the following order of importance: (1) friction factor, (2) intersection die channels angle, (3) outer fillet radius, (4) inner fillet radius and (5) the plunger velocity, respectively.Also, for the effective plastic strain the significance order for the affecting parameters is: (1) intersection die channels angle, (2) inner fillet radius and (3) outer fillet radius.

λε
denotes the plastic multiplier.It is well known that both von Mises and Drucker plasticity criteria are first-degree hom ous stress functions.Thus, by applying e Euler identity combined to the equivalent plastic work-rate on the Equation7, one can verify that the plastic multiplier is equal to the e plastic conjugated of the effective stress measure σ .In this way, the effective plastic strai i terms of th on Mises yield criterion is obtained by multiplying both sides of Equation7

Figure 2 :
Figure 2: Influence of yield criterion and friction conditions on the pressure.

Table 1 :
Parameters for the ECAE 2K factorial design.Also, the rotatability parameter α is considered when the central and axial points are added on the factorial analysis.Thus, for the 25 central composite design we have,

Table 2 :
Combinations for the 25 factorial design treatment*.

Table 3 :
Combinations for the 25 factorial design treatment.