Numerical and experimental study of the behavior of a polyamide during the ECAE process using a ${105}^{\circ }$${105}^{\circ }$105^(@)105^{\circ}${105}^{\circ }$ die

The equal channel angular extrusion (ECAE) is a processing technique used to induce substantial plastic deformation in materials while preserving the original cross-sectional dimensions of the specimen. The basic principle of the process is illustrated in Fig. 1. Initially developed by Segal et al. [1], this technique has attracted considerable interest as an effective method for enhancing material properties through severe plastic deformation. The process is based on simple shear deformation, achieved by pressing the workpiece through a die with two intersecting channels of equal cross-section. ECAE has proven highly effective in producing ultrafine grain structures in metallic materials [2,3] and in inducing significant molecular orientation in polymeric materials, $[4,5,6]$$[4,5,6]$[4,5,6][4,5,6]$[4,5,6]$ both of which result in notable improvements in mechanical properties.
Nowadays, polyamides and other polymers are increasingly replacing metallic materials in various industrial domains. They are widely employed in applications such as mechanical components and electronic devices (e.g., covers, couplers, gears) owing to their relatively low manufacturing cost and versatility [7]. In this context, the equal channel angular extrusion (ECAE) process has been extensively used to enhance the mechanical properties of polymers, and numerous studies have addressed its effects on different polymeric systems. For instance, investigations on linear low-density polyethylene (LLDPE) revealed notable morphological changes induced by ECAE [4]. Sue et al. [8] reported that, in the case of polycarbonate (PC), extrusion must be performed at temperatures slightly below the glass transition to ensure effective processing. Similarly, Li et al. [9] demonstrated that the mechanical properties of PC can be tailored by varying the processing routes and the number of passes. Xia et al. [10] examined the effect of molecular anisotropy on the impact strength of polycarbonate and showed that the improvement in impact resistance is directly related to the changes in molecular orientation induced by the process. Furthermore, Xia et al. [11] identified crystallinity and molecular orientation as key factors influencing the dynamic mechanical properties of ECAE-oriented semi-crystalline polyethylene terephthalate (PET), with improvements observed in both bending and torsional storage moduli.

The influence of different ECAE routes has also been investigated for other polymers. Weon et al. [12] studied polymethyl methacrylate (PMMA) and reported significant variations in tensile, fracture toughness, flexural, and ballistic impact properties depending on the route applied. Recent studies on polypropylene further highlighted the strong dependence of microstructural evolution and mechanical performance on the processing route (e.g., route A versus route C ) under back pressure, confirming notable differences in deformation behavior [13].
Beyond experimental work, several reviews have addressed the structural transformations induced by severe plastic deformation in polymers, including semi-crystalline and amorphous polymers, polymer blends and composites, and polymer powders. These works also consider advanced variants of the process, such as equal channel multi-angular extrusion (ECMAE), and their impact on material properties. [14] In parallel, numerical studies have contributed to a better understanding of polymer deformation under ECAE. One notable contribution developed efficient finite element analysis (FEA) models and demonstrated the crucial influence of the constitutive law on predictive accuracy. Comparisons between J2-plasticity, Bergstrom-Boyce, and three-network models revealed significant differences in extrusion load, deformed geometry, and shear strain distribution. The study also emphasized the strong effect of friction on punch force, highlighted the limitations of the plane strain assumption in the presence of billet-die interactions, and showed that sharp die corners can generate a "dead zone" that substantially alters extrusion force and strain distribution [15].
The ECAE process has also been employed to modify the aspect ratio and orientation of clay nanoparticles in nylon-6/clay nanocomposites, as demonstrated by Weon and Sue [16]. Wang et al. [17] provided valuable insights into lamellar formation and relaxation in shear-deformed polyethylene terephthalate (PET) using an in situ time-resolved synchrotron small-angle X-ray scattering (SAXS) technique. More recently, combined numerical and experimental investigations have been carried out to assess the influence of key geometrical and processing parameters on the viscoplastic behavior of polymers during the ECAE process [15,18].
Although Equal Channel Angular Extrusion (ECAE) is a well-established severe plastic deformation technique widely used for metallic materials, its application to polymers remains extremely limited. Existing experimental studies on polymer ECAE are scarce and have been almost exclusively restricted to a small number of channel angles, namely ${90}^{\circ },{120}^{\circ }$${90}^{\circ },{120}^{\circ }$90^(@),120^(@)90^{\circ}, 120^{\circ}${90}^{\circ },{120}^{\circ }$, and ${135}^{\circ }$${135}^{\circ }$135^(@)135^{\circ}${135}^{\circ }$. Moreover, these studies mainly focus on process feasibility and macroscopic deformation, while practical limitations such as extrusion instability at low angles and excessive specimen curvature at higher angles have received little attention. To date, no experimental study has reported the application of ECAE to polyamide materials, despite their extensive industrial use and strong sensitivity to processing-induced deformation. In addition, the evolution of mechanical properties, particularly hardness, under ECAE processing has not yet been investigated for polyamides. In this context, the present work aims to address these gaps by experimentally applying the ECAE process to polyamide for the first time, investigating an intermediate channel angle of ${105}^{\circ }$${105}^{\circ }$105^(@)105^{\circ}${105}^{\circ }$, and analyzing the resulting deformation behavior and hardness evolution. This approach provides new insights into the suitability of ECAE for polyamides and contributes to extending severe plastic deformation concepts to polymeric materials.
In this study, the deformation behavior of polyamide (PA) is investigated using a ${105}^{\circ }$${105}^{\circ }$105^(@)105^{\circ}${105}^{\circ }$ ECAE die, with the aim of enhancing both the magnitude and homogeneity of plastic strain while minimizing sample warping. To achieve this objective, the paper is structured as follows: Section 2 describes the finite element modeling, while Section 3 compares analytical predictions with FEM results. Section 4 presents the numerical findings on the effects of channel angle, intermediate channel length, and friction on strain distribution, pressing force, and the variation factor V, considering both single-turn and double-turn dies. Section 5 reports the experimental results obtained with 1-ECAE and 2-ECAE configurations, focusing on curvature evolution and hardness. Finally, Section 6 summarizes the main conclusions of the work.

The general principle of the ECAE process is illustrated in Fig. 1. The tooling consists of a block incorporating either two (Fig. 1a) or three (Fig. 1b) intersecting channels of identical cross-section. During processing, a specimen inserted into the entrance channel is extruded through the exit channel by means of a punch. When using a 2-ECAE die, the material undergoes two successive shear deformations within a single pass. Compared with the conventional 1 ECAE configuration, this design allows the accumulation of higher plastic strains in fewer passes, thereby improving strain homogeneity and enhancing the efficiency of molecular orientation.

Theoretically, the $\mathrm{\Phi }$$\mathrm{\Phi }$Phi\Phi$\mathrm{\Phi }$ angle is arbitrary; the increment of equivalent plastic strain caused by shear as a result of one pass through the channel can be calculated as follows [19]:

where $\phi$$\phi$varphi\varphi$\phi$ is the corner angle.
The total equivalent plastic strain for N cycles is:

In the case of the modified ECAE process, known as equal-channel multiple angular extrusion (ECMAE) [20], the billet is extruded through a die consisting of several channels of identical cross-sections intersecting at fixed angles. Under these conditions, the equivalent plastic strain can be expressed as:

where n is the number of intersection angles of the channels, it should be noted that. ECAE equipment realizes simple shear deformation with varied orientation of shear planes and extrusion with varied deformation intensity.

The mechanical response of polyamide under compression and various strain rates exhibits pronounced timedependent and rate-sensitive characteristics that cannot be accurately captured by conventional elastic-plastic formulations. To represent this behavior more realistically, the present study employs the Perzyna viscoplastic model [21], combined with a Voce-type nonlinear isotropic hardening law [22]. This constitutive framework provides a physically consistent description of the rate-dependent plastic flow and the saturation of hardening typically observed in polymeric materials. The model is formulated under small strain assumptions and assumes an additive decomposition of the total strain rate into its elastic and plastic components.

where ${\dot{\mathit{\epsilon }}}^e$${\dot{\mathit{\epsilon }}}^e$epsi^(˙)^(e)\dot{\boldsymbol{\varepsilon}}^{e}${\dot{\mathit{\epsilon }}}^e$ and ${\dot{\mathit{\epsilon }}}^p$${\dot{\mathit{\epsilon }}}^p$epsi^(˙)^(p)\dot{\boldsymbol{\varepsilon}}^{p}${\dot{\mathit{\epsilon }}}^p$ denote the elastic and plastic strain rates, respectively.
This decomposition assumes that the material response can be separated into a recoverable (elastic) part and a timedependent irreversible (plastic or viscoplastic) part, as commonly adopted in rate-dependent plasticity formulations. [21,22,23]
The elastic behavior of the polyamide is considered linear and isotropic, following Hooke's law:

where ${\mathbf{C}}^e$${\mathbf{C}}^e$C^(e)\mathbf{C}^{e}${\mathbf{C}}^e$ is the isotropic elasticity tensor defined by Young's modulus E and Poisson's ratio $\nu$$\nu$nu\nu$\nu$. The tensor components can be expressed as:

where ${\delta }_{ij}$${\delta }_{ij}$delta_(ij)\delta_{i j}${\delta }_{ij}$ is Kronecker delta ( ${\delta }_{ij}=1$${\delta }_{ij}=1$delta_(ij)=1\delta_{i j}=1${\delta }_{ij}=1$ if $\mathrm{i}=\mathrm{j}$$\mathrm{i}=\mathrm{j}$i=j\mathrm{i}=\mathrm{j}$\mathrm{i}=\mathrm{j}$, otherwise 0 ), $\lambda$$\lambda$lambda\lambda$\lambda$ and $\mu$$\mu$mu\mu$\mu$ are the Lamé parameters.
This formulation ensures that the stress-strain relationship remains consistent with the principles of small deformations and isotropic elasticity.
The elastic parameters $E$$E$EE$E$ and $v$$v$vv$v$ are determined from the initial slope of the experimental stress-strain curves, prior to the onset of plastic deformation. The elastic response provides the foundation upon which the viscoplastic flow rule is subsequently built to describe the rate-dependent inelastic behavior of the polyamide under various strain rates.
The time-dependent viscoplastic behavior of polyamide is modeled using the Perzyna overstress formulation [22,23,24], which assumes that plastic flow is activated once the equivalent stress exceeds a rate-independent yield surface. This approach has been widely used for metallic and polymeric materials exhibiting rate-dependent plasticity [21,25,26]. The total inelastic strain rate is therefore governed by a viscous flow function that depends on the overstress and the strain-rate sensitivity of the material. The onset of plastic flow is governed by a von Mises yield criterion expressed as:

where ${\sigma }_{eq}$${\sigma }_{eq}$sigma_(eq)\sigma_{e q}${\sigma }_{eq}$ is the equivalent von Mises stress, and $\mathbf{s}$$\mathbf{s}$s\mathbf{s}$\mathbf{s}$ denotes the deviatoric part of the Cauchy stress tensor $\mathit{\sigma }$$\mathit{\sigma }$sigma\boldsymbol{\sigma}$\mathit{\sigma }$ defined as:

The isotropic hardening is described by a Voce-type law, which captures the saturation of hardening at large strains [27,28]:

where ${\sigma }_0$${\sigma }_0$sigma_(0)\sigma_{0}${\sigma }_0$ is the initial yield stress, Q the saturation stress (maximum hardening amplitude), and b a material parameter controlling the rate of saturation.
This formulation has been widely validated for polymers exhibiting nonlinear hardening [27,29,30]. To account for rate effects, the viscoplastic flow is described using Perzyna's overstress theory [23]. The plastic strain rate is expressed as:

where $\mathbf{n}=\frac{\partial f}{\partial \mathit{\sigma }}=\frac{3}{2}\frac{\mathbf{s}}{{\sigma }_{eq}}$$\mathbf{n}=\frac{\partial f}{\partial \mathit{\sigma }}=\frac{3}{2}\frac{\mathbf{s}}{{\sigma }_{eq}}$n=(del f)/(del sigma)=(3)/(2)(s)/(sigma_(eq))\mathbf{n}=\frac{\partial f}{\partial \boldsymbol{\sigma}}=\frac{3}{2} \frac{\mathbf{s}}{\sigma_{e q}}$\mathbf{n}=\frac{\partial f}{\partial \mathit{\sigma }}=\frac{3}{2}\frac{\mathbf{s}}{{\sigma }_{eq}}$ is the flow direction (normal to the yield surface), and $\dot{\gamma }$$\dot{\gamma }$gamma^(˙)\dot{\gamma}$\dot{\gamma }$ is the plastic consistency parameter (or viscoplastic multiplier) defined by the flow function:

where $\eta$$\eta$eta\eta$\eta$ is a viscosity coefficient controlling the rate dependence, m is the strain-rate sensitivity exponent, and $⟨\cdot ⟩$$⟨\cdot ⟩$(:*:)\langle\cdot\rangle$⟨\cdot ⟩$ denotes the Macaulay brackets, ensuring plastic flow occurs only when $\mathrm{f}>0$$\mathrm{f}>0$f > 0\mathrm{f}>0$\mathrm{f}>0$.
This formulation ensures that viscoplastic flow occurs only when the stress state exceeds the instantaneous yield surface, thus maintaining thermodynamic consistency. The equivalent plastic strain rate ${\dot{\overline{\epsilon }}}^p$${\dot{\overline{\epsilon }}}^p$bar(epsi)^(˙)^(p)\dot{\bar{\varepsilon}}^{p}${\dot{\overline{\epsilon }}}^p$ is computed as:

This set of equations fully defines the elasto-viscoplastic response of the polyamide under multiaxial loading. The model provides a realistic description of strain-rate sensitivity and strain hardening, which are characteristic of semi-crystalline polymers. [29, 30, 31]
It is worth noting that the constitutive model employed in this study is valid under isothermal conditions and assumes a homogeneous, isotropic material response subjected to monotonic severe plastic deformation without damage initiation or evolution. The model is formulated at a macroscopic scale and captures the nonlinear elasto-viscoplastic behavior of the polymer; however, it does not explicitly account for microstructural mechanisms, such as the evolution of crystalline and amorphous phases during deformation. Consequently, the model is expected to provide reliable predictions for the ECAE process under the investigated processing conditions, while its applicability may be limited for non-isothermal loading, cyclic deformation, or deformation regimes involving significant microstructural transformations or damage mechanisms.

To identify the six parameters $\{E,{\sigma }_0,Q,b,\eta ,m\}$$\left\{E,{\sigma }_0,Q,b,\eta ,m\right\}${E,sigma_(0),Q,b,eta,m}\left\{E, \sigma_{0}, Q, b, \eta, m\right\}$\{E,{\sigma }_0,Q,b,\eta ,m\}$ of the proposed elasto-viscoplastic model, polyamide specimens were tested under three different true strain rates of ${10}^{-3},{10}^{-2}$${10}^{-3},{10}^{-2}$10^(-3),10^(-2)10^{-3}, 10^{-2}${10}^{-3},{10}^{-2}$, and ${10}^{-1}{\mathrm{s}}^{-1}$${10}^{-1}{\mathrm{s}}^{-1}$10^(-1)s^(-1)10^{-1} \mathrm{~s}^{-1}${10}^{-1}{\mathrm{s}}^{-1}$ at room temperature ( ${25}^{\circ }\mathrm{C}$${25}^{\circ }\mathrm{C}$25^(@)C25^{\circ} \mathrm{C}${25}^{\circ }\mathrm{C}$ ). The uniaxial compression tests were conducted using a Shimadzu universal testing machine equipped with a 50 kN load cell to accurately measure the applied force. The experimental stress-strain data were continuously recorded through the Trapezium acquisition software. The displacement of the upper platen was controlled by adjusting the crosshead speed, which was determined according to the imposed strain rate. Each specimen was compressed up to $55\mathrm{\%}$$55\mathrm{\%}$55%55 \%$55\mathrm{\%}$ true strain between two parallel compression plates to ensure uniform deformation.
From the recorded crosshead displacement and the measured applied force F during the test, the nominal stress-strain curves were obtained using the Shimadzu software integrated into the testing system. In order to convert the nominal quantities into true stress and true strain, the material was assumed to be isotropic and incompressible, and the deformation
was considered homogeneous. The axial displacement was obtained from the crosshead movement of the testing machine. Knowing the initial specimen height ${\mathrm{L}}_0$${\mathrm{L}}_0$L_(0)\mathrm{L}_{0}${\mathrm{L}}_0$, the true axial strain was calculated from the measured displacement $\mathrm{\Delta }\mathrm{L}$$\mathrm{\Delta }\mathrm{L}$DeltaL\Delta \mathrm{L}$\mathrm{\Delta }\mathrm{L}$ using the following relation:

The true stress becomes:

where ${\sigma }_N=F/S_0$${\sigma }_N=F/S_0$sigma_(N)=F//S_(0)\sigma_{N}=F / S_{0}${\sigma }_N=F/S_0$, is the nominal stress and ${\mathrm{S}}_0$${\mathrm{S}}_0$S_(0)\mathrm{S}_{0}${\mathrm{S}}_0$ is the initial section.
Three cylindrical specimens were tested for each deformation rate of ${10}^{-3},{10}^{-2}$${10}^{-3},{10}^{-2}$10^(-3),10^(-2)10^{-3}, 10^{-2}${10}^{-3},{10}^{-2}$, and ${10}^{-1}{\mathrm{s}}^{-1}$${10}^{-1}{\mathrm{s}}^{-1}$10^(-1)s^(-1)10^{-1} \mathrm{~s}^{-1}${10}^{-1}{\mathrm{s}}^{-1}$ to ensure statistical reliability. The specimens had a length of 20 mm and a diameter of $10\mathrm{mm}(\mathrm{L}=2\mathrm{D})$$10\mathrm{mm}(\mathrm{L}=2\mathrm{D})$10mm(L=2D)10 \mathrm{~mm}(\mathrm{~L}=2 \mathrm{D})$10\mathrm{mm}(\mathrm{L}=2\mathrm{D})$. Fig. 2 shows the true stress-strain curves obtained from compression tests on polyamide at three different strain rates: $\dot{\epsilon }={10}^{-1}{\mathrm{s}}^{-1},{10}^{-2}{\mathrm{s}}^{-1}$$\dot{\epsilon }={10}^{-1}{\mathrm{s}}^{-1},{10}^{-2}{\mathrm{s}}^{-1}$epsi^(˙)=10^(-1)s^(-1),10^(-2)s^(-1)\dot{\varepsilon}=10^{-1} \mathrm{~s}^{-1}, 10^{-2} \mathrm{~s}^{-1}$\dot{\epsilon }={10}^{-1}{\mathrm{s}}^{-1},{10}^{-2}{\mathrm{s}}^{-1}$ and ${10}^{-3}{\mathrm{s}}^{-1}$${10}^{-3}{\mathrm{s}}^{-1}$10^(-3)s^(-1)10^{-3} \mathrm{~s}^{-1}${10}^{-3}{\mathrm{s}}^{-1}$. A clear strain-rate sensitivity can be observed. At the lowest strain rate $\left({10}^{-3}{\mathrm{s}}^{-1}\right)$$\left({10}^{-3}{\mathrm{s}}^{-1}\right)$(10^(-3)s^(-1))\left(10^{-3} \mathrm{~s}^{-1}\right)$\left({10}^{-3}{\mathrm{s}}^{-1}\right)$, the material exhibits a more gradual increase in stress with strain, reflecting greater ductility and enhanced molecular mobility. As the strain rate increases, both the apparent yield stress and flow stress rise significantly, illustrating the typical viscoplastic hardening behavior of semi-crystalline polymers. This rate-dependent strengthening is attributed to the restricted motion of crystalline lamellae and the limited molecular rearrangements within the amorphous regions at higher loading rates. Consequently, the polyamide displays a stiffer mechanical response and higher resistance under rapid deformation. At larger strains, the curves tend to converge, indicating the onset of hardening saturation, which is consistent with nonlinear isotropic hardening laws such as the Voce model or rate-dependent formulations like the Perzyna viscoplastic model.

The constitutive model is defined by a set of material parameters identified from experimental stress-strain curves obtained at different strain rates, namely $\{E,{\sigma }_0,\mathcal{Q},b,\eta ,m\}$$\left\{E,{\sigma }_0,\mathcal{Q},b,\eta ,m\right\}${E,sigma_(0),Q,b,eta,m}\left\{E, \sigma_{0}, \mathcal{Q}, b, \eta, m\right\}$\{E,{\sigma }_0,\mathcal{Q},b,\eta ,m\}$. Parameter identification is performed using uniaxial compression tests conducted at strain rates of ${10}^{-3},{10}^{-2}$${10}^{-3},{10}^{-2}$10^(-3),10^(-2)10^{-3}, 10^{-2}${10}^{-3},{10}^{-2}$, and ${10}^{-1}{\mathrm{s}}^{-1}$${10}^{-1}{\mathrm{s}}^{-1}$10^(-1)s^(-1)10^{-1} \mathrm{~s}^{-1}${10}^{-1}{\mathrm{s}}^{-1}$. The optimization procedure aims to minimize the root mean square (RMS) error between the experimental and simulated stress-strain curves. This identification strategy yields a consistent set of parameters capable of accurately reproducing both the rate-dependent yielding and the saturation of strain hardening, in agreement with recent studies [29, 30].

The identified parameters are summarized in Tab. 1, and Fig. 3 shows a good agreement between the experimental data and the numerical simulations. The number of significant digits used in the reported parameters has been adjusted to reflect the experimental uncertainties and the accuracy of the parameter identification procedure. A simple sensitivity analysis was carried out by varying the main material parameters by $\pm 5\mathrm{\%}$$\pm 5\mathrm{\%}$+-5%\pm 5 \%$\pm 5\mathrm{\%}$. The results indicate that the numerical predictions remain stable within this range, confirming the robustness of the model.

At room temperature, the strain-rate dependence observed in Fig. 2 remains relatively limited. However, a consistent trend can still be identified over the investigated range of deformation rates, which is sufficient for the calibration of the constitutive model parameters. The calibrated model accurately reproduces the experimental stress-strain responses within the considered deformation domain, demonstrating its robustness and predictive capability under the studied conditions. It should be noted that the present calibration is focused on deformation rates relevant to the ECAE process investigated in this work. A more pronounced strain-rate sensitivity is expected at higher deformation rates and/or different temperature levels, and a more extensive experimental-numerical validation under such conditions is planned as part of future work.

The simulations were carried out using the finite element code MSC.Marc under plane-strain conditions. The sample dimensions were set to 10 mm (width) $\times 10\mathrm{mm}$$\times 10\mathrm{mm}$xx10mm\times 10 \mathrm{~mm}$\times 10\mathrm{mm}$ (thickness) $\times 100\mathrm{mm}$$\times 100\mathrm{mm}$xx100mm\times 100 \mathrm{~mm}$\times 100\mathrm{mm}$ (length). For the die geometry, a channel angle of ${\mathrm{\Phi }}_1={\mathrm{\Phi }}_2=\mathrm{\Phi }={105}^{\circ }$${\mathrm{\Phi }}_1={\mathrm{\Phi }}_2=\mathrm{\Phi }={105}^{\circ }$Phi_(1)=Phi_(2)=Phi=105^(@)\Phi_{1}=\Phi_{2}=\Phi=105^{\circ}${\mathrm{\Phi }}_1={\mathrm{\Phi }}_2=\mathrm{\Phi }={105}^{\circ }$ was considered (corresponding to a ${105}^{\circ }$${105}^{\circ }$105^(@)105^{\circ}${105}^{\circ }$ die), while the two outer corner angles were varied. The inner corners were modeled with a radius of $\mathrm{r}=1\mathrm{mm}$$\mathrm{r}=1\mathrm{mm}$r=1mm\mathrm{r}=1 \mathrm{~mm}$\mathrm{r}=1\mathrm{mm}$ [32].
The selection of the channel angle is a critical parameter in the ECAE process, as it directly governs the imposed strain and the material flow behavior. Previous experimental studies on polymer ECAE have mainly focused on channel angles of ${90}^{\circ }$${90}^{\circ }$90^(@)90^{\circ}${90}^{\circ }$, ${120}^{\circ }$${120}^{\circ }$120^(@)120^{\circ}${120}^{\circ }$, and ${135}^{\circ }$${135}^{\circ }$135^(@)135^{\circ}${135}^{\circ }$. For polymers, and particularly for polyamide, severe experimental limitations have been reported for ${90}^{\circ }$${90}^{\circ }$90^(@)90^{\circ}${90}^{\circ }$ dies, including material blockage at the channel corner and unstable extrusion conditions, which were also observed during preliminary trials in the present work. On the other hand, for channel angles greater than or equal to ${120}^{\circ }$${120}^{\circ }$120^(@)120^{\circ}${120}^{\circ }$, extrusion is feasible; however, the curvature of the extruded specimens remains relatively high, limiting deformation efficiency and dimensional stability.
Based on these observations, an intermediate channel angle of ${105}^{\circ }$${105}^{\circ }$105^(@)105^{\circ}${105}^{\circ }$ was selected in this study as a compromise between process feasibility and effective deformation. This angle facilitates material flow while simultaneously contributing to curvature reduction of the extruded specimens. The present work, therefore, focuses on the experimental and numerical investigation of polyamide processed by ECAE at this intermediate channel angle.
In the modeling, both the die and the ram were assumed to behave as rigid bodies. A displacement of 100 mm was applied to the ram in the extrusion direction. Initially, the friction coefficient between the sample and the die channels was assumed to be zero, corresponding to frictionless contact conditions for the simulations. The finite element calculations were performed under the plane strain assumption. The sample was discretized using four-node isoparametric plane strain elements with reduced integration. A total of 4000 elements was used to mesh the sample, providing a satisfactory compromise between computational efficiency and solution accuracy. The selected mesh density was sufficiently refined to accurately capture the localized deformation occurring during the ECAE process. Based on theoretical [3] and experimental [33] analyses, the material behavior of the sample was modeled as elasto-viscoplastic. To evaluate the plastic strain distribution, the selected cross-section was taken 20 mm below the first and the second inner corners (see Fig. 4).

To compare the analytical solution given by Eqn. (1) with the FEM results for different corner angles ( $\phi$$\phi$varphi\varphi$\phi$ ), a perfectly plastic material behavior was assumed at this stage. This assumption neglects hardening and viscous effects, thereby isolating the influence of geometrical factors. The equivalent plastic strain values at the midsection of the workpiece were plotted as a function of the channel angle ( $\mathrm{\Phi }={105}^{\circ }$$\mathrm{\Phi }={105}^{\circ }$Phi=105^(@)\Phi=105^{\circ}$\mathrm{\Phi }={105}^{\circ }$ ). The results, presented in Fig. 5(a), show that the solid line corresponds to Eqn. (1), while the data points represent the FEM predictions. It can be seen that a good agreement has been found between the two approaches, as the maximum relative error does not exceed $2.6\mathrm{\%}$$2.6\mathrm{\%}$2.6%2.6 \%$2.6\mathrm{\%}$, which occurs for a corner angle of $\phi ={60}^{\circ }$$\phi ={60}^{\circ }$varphi=60^(@)\varphi=60^{\circ}$\phi ={60}^{\circ }$ (see Fig. 5(b)).
The slight discrepancies between the analytical and numerical results can be attributed to the simplifying assumptions of the analytical formulation, whereas FEM accounts for more realistic stress-strain distributions and boundary conditions. Overall, the close correlation between both approaches demonstrates the reliability of the analytical model for estimating the strain generated during the equal channel angular extrusion (ECAE) process.
Moreover, the results show that increasing the corner angle leads to a progressive decrease in equivalent plastic strain, indicating that sharper die angles induce greater shear deformation in the material.

In the ECAE process, ensuring homogeneous plastic strain distribution within the bulk material during successive passes is crucial. This section aims to identify the optimal conditions that enhance strain homogeneity by examining the influence of the corner angle $(\phi )$$(\phi )$(varphi)(\varphi)$(\phi )$ and the length $(\mathrm{L})$$(\mathrm{L})$(L)(\mathrm{L})$(\mathrm{L})$ of the second channel in the two-elbow die.
The simulations were performed for the case of the ${105}^{\circ }$${105}^{\circ }$105^(@)105^{\circ}${105}^{\circ }$ die (1-ECAE) following the conditions stated in section 3, and the results are illustrated in Fig. 6. The equivalent plastic strain distribution appears relatively uniform and homogeneous. At a low shear angle $(\phi ={15}^{\circ })$$\left(\phi ={15}^{\circ }\right)$(varphi=15^(@))\left(\varphi=15^{\circ}\right)$(\phi ={15}^{\circ })$, the deformation remains fairly uniform and constrained, indicating weak strain localization. When the angle increases to $\phi ={30}^{\circ }$$\phi ={30}^{\circ }$varphi=30^(@)\varphi=30^{\circ}$\phi ={30}^{\circ }$, distinct deformation bands begin to emerge, reflecting stronger shear effects. At $\phi ={45}^{\circ }$$\phi ={45}^{\circ }$varphi=45^(@)\varphi =45^{\circ}$\phi ={45}^{\circ }$, these bands become more pronounced, revealing intensified plastic flow and higher strain concentration within the material. Finally, at $\phi ={60}^{\circ }$$\phi ={60}^{\circ }$varphi=60^(@)\varphi=60^{\circ}$\phi ={60}^{\circ }$, the deformation is highly localized, with sharp shear zones dominating the microstructure, highlighting the strong influence of larger shear angles in promoting localized plastic instabilities.

To provide further detail, Fig. 7 presents the evolution of the equivalent plastic strain along the selected cross-section of the sample for different corner angles $\phi$$\phi$varphi\varphi$\phi$. The results show that the lowest equivalent plastic strain occurs at $\phi ={60}^{\circ }$$\phi ={60}^{\circ }$varphi=60^(@)\varphi=60^{\circ}$\phi ={60}^{\circ }$, whereas the highest strain level is reached at $\phi ={15}^{\circ }$$\phi ={15}^{\circ }$varphi=15^(@)\varphi=15^{\circ}$\phi ={15}^{\circ }$. In all cases, strain is concentrated near the die corner, where intense shear deformation develops. The strain magnitude increases as the corner angle decreases, indicating that sharper die angles promote more pronounced plastic deformation. Nevertheless, the ${105}^{\circ }$${105}^{\circ }$105^(@)105^{\circ}${105}^{\circ }$ die offers a balance between strain concentration and uniformity, as the deformation is distributed over a wider region compared with sharper angles. This confirms that die geometry plays a critical role in strain distribution during ECAE, thereby directly affecting the efficiency of microstructural refinement.

It is interesting to note that the degree of homogeneity of the strain distribution can be assessed through a variation factor V , which is a key parameter for quantifying strain uniformity. Indeed, a higher value of V corresponds to greater heterogeneity of deformation. This factor is given by the following expression[34,35].

where ${\dot{\epsilon }}_{\text{ave}}$${\dot{\epsilon }}_{\text{ave }}$epsi^(˙)_("ave ")\dot{\varepsilon}_{\text {ave }}${\dot{\epsilon }}_{\text{ave }}$ is the average of the equivalent plastic strain values computed on $n$$n$nn$n$ integration points, and ${\dot{\epsilon }}_i$${\dot{\epsilon }}_i$epsi^(˙)_(i)\dot{\varepsilon}_{i}${\dot{\epsilon }}_i$ is the equivalent plastic strain value of a given integration point along the sample width.
Tab. 2 summarizes the numerical results of the maximum and average values of the equivalent plastic strain, together with the corresponding variation factor, for different corner angles $(\phi )$$(\phi )$(varphi)(\varphi)$(\phi )$ in the case of the ${105}^{\circ }$${105}^{\circ }$105^(@)105^{\circ}${105}^{\circ }$ die. These indicators provide complementary insights: while the maximum and average strains reflect the overall deformation intensity, the variation factor highlights the degree of strain heterogeneity within the sample. Such a comparison is essential to assess the influence of die geometry on both the magnitude and uniformity of plastic deformation, thereby allowing a more comprehensive evaluation of the process efficiency. It can be observed that the variation factor decreases as the corner angle $\phi$$\phi$varphi\varphi$\phi$ decreases. Moreover, the average equivalent plastic strain ranges from 0.609 (at $\phi ={60}^{\circ }$$\phi ={60}^{\circ }$varphi=60^(@)\varphi=60^{\circ}$\phi ={60}^{\circ }$ with $\mathrm{V}=28.9\mathrm{\%}$$\mathrm{V}=28.9\mathrm{\%}$V=28.9%\mathrm{V}=28.9 \%$\mathrm{V}=28.9\mathrm{\%}$ ) to 0.723 (at $\phi ={15}^{\circ }$$\phi ={15}^{\circ }$varphi=15^(@)\varphi=15^{\circ}$\phi ={15}^{\circ }$ with $\mathrm{V}=19.8\mathrm{\%}$$\mathrm{V}=19.8\mathrm{\%}$V=19.8%\mathrm{V}=19.8 \%$\mathrm{V}=19.8\mathrm{\%}$ ) in the investigated case. Based on the variation factor criterion, it can be concluded that the best homogeneity of the plastic strain distribution is achieved when the corner angle is $\phi ={15}^{\circ }$$\phi ={15}^{\circ }$varphi=15^(@)\varphi=15^{\circ}$\phi ={15}^{\circ }$.

Fig. 8 presents the numerical results of the pressing force required for extrusion of polyamide (PA) using a ${105}^{\circ }1$${105}^{\circ }1$105^(@)1105^{\circ} 1${105}^{\circ }1$-ECAE die. It can be observed that, at the onset of extrusion, the pressing force increases progressively until reaching its maximum value of 6.3998 kN for $\phi ={15}^{\circ }$$\phi ={15}^{\circ }$varphi=15^(@)\varphi=15^{\circ}$\phi ={15}^{\circ }$, and 5.8626 kN for $\phi ={30}^{\circ };5.5388\mathrm{kN}$$\phi ={30}^{\circ };5.5388\mathrm{kN}$varphi=30^(@);5.5388kN\varphi=30^{\circ} ; 5.5388 \mathrm{kN}$\phi ={30}^{\circ };5.5388\mathrm{kN}$ for $\phi ={45}^{\circ }$$\phi ={45}^{\circ }$varphi=45^(@)\varphi=45^{\circ}$\phi ={45}^{\circ }$, and 5.3353 kN for $\phi ={60}^{\circ }$$\phi ={60}^{\circ }$varphi=60^(@)\varphi=60^{\circ}$\phi ={60}^{\circ }$. It should be noted that after the sample exits the first channel junction, the pressing force stabilizes and remains nearly constant until the specimen is fully extruded.
Furthermore, a dimpled trend is observed in the force-plateau region of Fig. 8. Since friction was neglected in the present numerical model, this behavior cannot be associated with a stick-slip mechanism. Instead, it is attributed to the intrinsic non-linear response and intermittent viscoplastic flow of the material under severe shear deformation. Similar oscillatory trends have been reported in the literature for equal-channel angular extrusion processes, notably in the work of Zaïri et al. [5].

This section investigates the effect of friction on the pressing force and the equivalent plastic strain distribution in the 1 ECAE die with a corner angle of $\phi ={15}^{\circ }$$\phi ={15}^{\circ }$varphi=15^(@)\varphi=15^{\circ}$\phi ={15}^{\circ }$. The interaction between the billet and the die wall is modeled according to Coulomb's law, considering four friction coefficients: $\mathrm{f}=\{0,0.1,0.2,0.3\}$$\mathrm{f}=\{0,0.1,0.2,0.3\}$f={0,0.1,0.2,0.3}\mathrm{f}=\{0,0.1,0.2,0.3\}$\mathrm{f}=\{0,0.1,0.2,0.3\}$. Fig. 9 illustrates the evolution of the plastic strain distribution along the sample thickness.

The results demonstrate that friction exerts a significant influence on both the intensity and the uniformity of plastic deformation. In the absence of friction ( $f=0$$f=0$f=0f=0$f=0$, red curve), the strain distribution is relatively low and homogeneous, indicating nearly ideal sliding conditions with limited resistance from the die walls. As the friction coefficient increases to $0.1,0.2$$0.1,0.2$0.1,0.20.1,0.2$0.1,0.2$, and 0.3, a progressive rise in plastic strain is observed, especially near the inlet zone (normalized distance $<0.2$$<0.2$< 0.2<0.2$<0.2$ ), where shear localization becomes more pronounced. This increase is accompanied by a steeper slope at the entrance of the channel, reflecting accelerated deformation, followed by stabilization further along the cross-section. For instance, at $\mathrm{f}=0.3$$\mathrm{f}=0.3$f=0.3\mathrm{f}=0.3$\mathrm{f}=0.3$ (black curve), the equivalent plastic strain exceeds 0.85 , significantly higher than in the other cases, suggesting more constrained flow and improved strain homogenization across the section. Mechanistically, the additional resistance generated by friction forces the material to undergo more intense shear deformation to pass through the die channels. While moderate friction appears beneficial by enhancing strain development and homogenization, excessive friction may introduce drawbacks such as flow instabilities, defect formation, and accelerated die wear. Overall, these findings emphasize the crucial role of interfacial friction in the ECAE process, showing that its proper control is essential to optimize both the intensity and the uniformity of plastic strain distribution.

Fig. 10 presents the equivalent plastic strain distribution for four different second-channel lengths, $\mathrm{L}=20,30,40$$\mathrm{L}=20,30,40$L=20,30,40\mathrm{L}=20,30,40$\mathrm{L}=20,30,40$, and 50 mm , considering a section of $10\times 10{\mathrm{mm}}^2$$10\times 10{\mathrm{mm}}^2$10 xx10mm^(2)10 \times 10 \mathrm{~mm}^{2}$10\times 10{\mathrm{mm}}^2$, and a corner angle of $\phi ={15}^{\circ }$$\phi ={15}^{\circ }$varphi=15^(@)\varphi=15^{\circ}$\phi ={15}^{\circ }$. The results reveal that the plastic strain distribution is generally non-uniform at the sample extremities. In contrast, a more homogeneous distribution is observed in the central region. Furthermore, the folding phenomenon appears markedly more pronounced when the second-channel length is short ( $\mathrm{L}=20\mathrm{mm}$$\mathrm{L}=20\mathrm{mm}$L=20mm\mathrm{L}=20 \mathrm{~mm}$\mathrm{L}=20\mathrm{mm}$ ), indicating that a reduced channel length amplifies strain localization and flow instabilities.

For more clarity, Fig. 11 illustrates the distribution of the equivalent plastic strain as a function of the distance from the bottom along the selected cross-section of the sample (see Fig. 4(b)), for each corner angle and for different second-channel lengths. The results clearly show that the highest plastic strain levels are obtained at a corner angle of $\phi ={15}^{\circ }$$\phi ={15}^{\circ }$varphi=15^(@)\varphi=15^{\circ}$\phi ={15}^{\circ }$, consistently across all tested channel lengths ( $\mathrm{L}=20,30,40$$\mathrm{L}=20,30,40$L=20,30,40\mathrm{L}=20,30,40$\mathrm{L}=20,30,40$, and 50 mm ). In contrast, the lowest plastic strain values are observed for $\phi ={60}^{\circ }$$\phi ={60}^{\circ }$varphi=60^(@)\varphi=60^{\circ}$\phi ={60}^{\circ }$. This suggests that the corner angle $\phi$$\phi$varphi\varphi$\phi$ has a predominant influence on the intensity of plastic deformation, while the second-channel length mainly affects the homogeneity of strain distribution.
In order to highlight the effect of the second channel length, the equivalent plastic strain along the selected section has been plotted for different lengths and for $\phi ={15}^{\circ }$$\phi ={15}^{\circ }$varphi=15^(@)\varphi=15^{\circ}$\phi ={15}^{\circ }$ as shown in Fig. 12. It can be seen that the highest level of plastic strain is obtained for $\mathrm{L}=20$$\mathrm{L}=20$L=20\mathrm{L}=20$\mathrm{L}=20$ and 30 mm , however, for $\mathrm{L}=40$$\mathrm{L}=40$L=40\mathrm{L}=40$\mathrm{L}=40$ and 50 mm a slight difference has been observed.
The average equivalent plastic strain values obtained from the numerical simulations, along with the corresponding variation factors for the different geometries, are summarized in Tab. 3. It can be observed that the average plastic strain ranges between 1.24 and 1.52 across all cases. However, the results indicate that the most homogeneous strain distribution is achieved when the inner angle is $\phi ={15}^{\circ }$$\phi ={15}^{\circ }$varphi=15^(@)\varphi=15^{\circ}$\phi ={15}^{\circ }$ and for a channel length $\mathrm{L}=20\mathrm{mm}$$\mathrm{L}=20\mathrm{mm}$L=20mm\mathrm{L}=20 \mathrm{~mm}$\mathrm{L}=20\mathrm{mm}$. This suggests that shorter channel lengths facilitate a more balanced redistribution of strain, thereby enhancing microstructural uniformity. Consequently, the
configuration with $\mathrm{L}=20\mathrm{mm}$$\mathrm{L}=20\mathrm{mm}$L=20mm\mathrm{L}=20 \mathrm{~mm}$\mathrm{L}=20\mathrm{mm}$ proves to be the most favorable, as it minimizes strain localization while promoting overall homogeneity in the material.

The magnitude of the pressing force required to extrude the sample is a key factor in defining the practical operating limits of the ECAE device. In this study, the pressing force for polyamide (PA) extrusion was numerically determined by evaluating the nodal reaction forces applied at the top surface of the sample during the process. Fig. 13 illustrates the temporal evolution of this force for a cross-section of $10\times 10{\mathrm{mm}}^2$$10\times 10{\mathrm{mm}}^2$10 xx10mm^(2)10 \times 10 \mathrm{~mm}^{2}$10\times 10{\mathrm{mm}}^2$, considering various second-channel lengths and different corner angles $(\phi )$$(\phi )$(varphi)(\varphi)$(\phi )$. It can be observed that the pressing force increases sharply at the initial stage of extrusion, corresponding to the onset of shear deformation at the die entrance. Once the sample passes through the elbows and establishes a steady-state flow, the force tends to stabilize. The magnitude of the force is strongly influenced by both the channel length and the corner angle, with sharper angles generally requiring higher pressing forces due to the more pronounced shear localization. These results highlight the trade-off between achieving higher plastic strain levels and maintaining manageable pressing forces, which is crucial for the practical design and optimization of ECAE dies.

The maximum punch force obtained from the numerical simulations for the different die geometries is summarized in Tab. 4. It can be observed that the maximum punch force increases as the second-channel corner angle decreases. Overall, the required force ranges between 10.7295 kN and 13.2554 kN across all configurations. Notably, the most homogeneous plastic strain distribution was achieved at a maximum pressing force of 13.2554 kN , corresponding to a channel length of $\mathrm{L}=20\mathrm{mm}$$\mathrm{L}=20\mathrm{mm}$L=20mm\mathrm{L}=20 \mathrm{~mm}$\mathrm{L}=20\mathrm{mm}$ and a corner angle of $\phi ={15}^{\circ }$$\phi ={15}^{\circ }$varphi=15^(@)\varphi=15^{\circ}$\phi ={15}^{\circ }$. However, the reduced maximum pressing force (approximately 10.73 kN ) was obtained for $\phi ={60}^{\circ }$$\phi ={60}^{\circ }$varphi=60^(@)\varphi=60^{\circ}$\phi ={60}^{\circ }$ and $\mathrm{L}=40\mathrm{mm}$$\mathrm{L}=40\mathrm{mm}$L=40mm\mathrm{L}=40 \mathrm{~mm}$\mathrm{L}=40\mathrm{mm}$ (see Tab. 4 and Fig. 14).

At the onset of extrusion, the pressing force begins to rise as the material comes into contact with the first outer corner of the second channel. At this stage, the initiation and subsequent growth of a shear band occur until the sample head fully passes through the first elbow, corresponding to the first plastic deformation zone. The curve then reaches a first plateau, which reflects the steady-state plastic flow as the material traverses this elbow. Upon crossing the second plastic deformation zone, a second plateau with slight undulations appears, attributable to the interaction between the plastic deformations generated by both elbows of the die.

In this section, the effects of friction on the equivalent plastic strain distribution and the pressing force are analyzed for the optimized geometries of a ${105}^{\circ }$${105}^{\circ }$105^(@)105^{\circ}${105}^{\circ }$ 2-ECAE die with $\phi ={15}^{\circ }$$\phi ={15}^{\circ }$varphi=15^(@)\varphi=15^{\circ}$\phi ={15}^{\circ }$ and $\mathrm{L}=\{20,30,40,50\mathrm{mm}\}$$\mathrm{L}=\{20,30,40,50\mathrm{mm}\}$L={20,30,40,50mm}\mathrm{L}=\{20,30,40,50 \mathrm{~mm}\}$\mathrm{L}=\{20,30,40,50\mathrm{mm}\}$. The interaction between the tool and the billet is modeled using Coulomb's law of friction, and numerical simulations were conducted with four friction coefficients: $\mathrm{f}=\{0,0.1,0.2,0.3\}$$\mathrm{f}=\{0,0.1,0.2,0.3\}$f={0,0.1,0.2,0.3}\mathrm{f}=\{0,0.1,0.2,0.3\}$\mathrm{f}=\{0,0.1,0.2,0.3\}$.

Fig. 15 presents the distribution of equivalent plastic strain across the sample for different lengths of the second channel. The results indicate that, under this configuration, friction exerts only a moderate influence on the homogeneity of strain distribution. However, as the friction coefficient increases, localized strain intensification tends to occur near the die walls, which may adversely affect deformation uniformity. From a practical standpoint, extrusion can be carried out without lubrication, particularly at low friction levels, but the use of a suitable lubricant is strongly recommended. Lubrication not only promotes a more homogeneous strain distribution but also reduces tool wear, minimizes the risk of surface defects, and ensures greater stability of the material flow during extrusion. Consequently, appropriate friction management becomes a critical parameter in achieving both microstructural refinement and long-term tool durability in the ECAE process.
It is observed that friction has a significant impact on both the homogeneity and the magnitude of the equivalent plastic strain in the extruded polyamide (PA). Specifically, as the friction coefficient increases, the plastic strain also rises. To illustrate this effect, Fig. 16 presents the equivalent plastic strain contours for extrusion lengths $\mathrm{L}=30\mathrm{mm}$$\mathrm{L}=30\mathrm{mm}$L=30mm\mathrm{L}=30 \mathrm{~mm}$\mathrm{L}=30\mathrm{mm}$ and $\mathrm{L}=50\mathrm{mm}$$\mathrm{L}=50\mathrm{mm}$L=50mm\mathrm{L}=50 \mathrm{~mm}$\mathrm{L}=50\mathrm{mm}$ under different friction coefficients, where the color variations correspond to different strain levels. These results reveal the classical features of the ECAE process, emphasizing the role of friction in amplifying strain localization. Therefore, it is recommended to minimize friction as much as possible in order to prevent the initiation of micro-cracks at the billet-die interface and to ensure a more uniform strain distribution.

Fig. 17 presents the evolution of the variation factor for different friction coefficients in the case of the ${105}^{\circ }2$${105}^{\circ }2$105^(@)2105^{\circ} 2${105}^{\circ }2$-ECAE die. The results clearly demonstrate that friction has a significant impact on the homogeneity of the equivalent plastic strain distribution. The numerical values of the variation factor and the average equivalent plastic strain are summarized in Tab. 5. It can be observed that the average plastic strain varies between 1.45 and 1.64 across all studied cases, indicating a relatively high level of deformation. However, the variation factor, which reflects the degree of strain heterogeneity, exhibits its maximum values when friction is absent $(f=0)$$(f=0)$(f=0)(f=0)$(f=0)$ for all channel lengths. This outcome suggests that, in the absence of interfacial resistance, the material flow is less constrained, leading to uneven strain localization. Conversely, a moderate friction coefficient, particularly $\mathrm{f}=0.2$$\mathrm{f}=0.2$f=0.2\mathrm{f}=0.2$\mathrm{f}=0.2$, appears to promote better strain homogenization, as evidenced by the lowest value of the variation factor obtained for $\mathrm{L}=20\mathrm{mm}$$\mathrm{L}=20\mathrm{mm}$L=20mm\mathrm{L}=20 \mathrm{~mm}$\mathrm{L}=20\mathrm{mm}$. This behavior can be explained by the fact that a controlled amount of friction enhances shear transfer between the die and the billet, thereby redistributing deformation more uniformly across the sample cross-section. Nevertheless, excessive friction ( $f=0.3$$f=0.3$f=0.3f=0.3$f=0.3$ ) tends to elevate strain levels but at the expense of introducing localized heterogeneities and potential risks of surface defects. These findings highlight the dual role of friction in the ECAE process: while a complete absence of friction induces non-uniform strain fields, a moderate coefficient can improve homogeneity, whereas excessive values may degrade material quality.

The evolution of the pressing force as a function of time for different channel lengths and friction conditions is presented in Fig. 18. It is evident that the maximum punch force increases with the rise in the friction coefficient, reflecting the additional resistance introduced at the billet-die interface. Quantitatively, the maximum pressing force varies between 13.1520 kN and 86.6768 kN for all simulated cases (see Tab. 6).

This trend is consistent with the mechanics of the ECAE process, where higher friction enhances energy dissipation and impedes smooth sliding of the material along the die walls, thereby requiring greater extrusion loads. However, excessive friction may also promote flow localization, leading to non-uniform strain fields and potential surface damage. Interestingly, the analysis shows that the most homogeneous plastic strain distribution was obtained when the pressing force reached 50.4739 kN , corresponding to a channel length of $\mathrm{L}=20\mathrm{mm}$$\mathrm{L}=20\mathrm{mm}$L=20mm\mathrm{L}=20 \mathrm{~mm}$\mathrm{L}=20\mathrm{mm}$ and a friction coefficient of $\mathrm{f}=0.2$$\mathrm{f}=0.2$f=0.2\mathrm{f}=0.2$\mathrm{f}=0.2$. This result highlights the critical balance between applied load and strain uniformity: moderate friction levels contribute to improved strain homogenization without excessively increasing the extrusion force. Consequently, this case can be considered as an optimal compromise between mechanical efficiency and microstructural refinement.