An Investigation of the Performance of Equal Channel Angular Pressed Copper Electrodes in Electric Discharge Machining

Abstract

This study examines the mechanical, thermal, and electrical properties of copper tool electrodes processed via Equal Channel Angular Pressing (ECAP), with a specific focus on their performance in Electrical Discharge Machining (EDM) applications. A novel Crystal Plasticity Finite Element Method (CPFEM) framework is employed to model anisotropic slip behavior and microscale deformation mechanisms. The primary objective is to elucidate how initial crystallographic orientation influences hardness, thermal conductivity, and electrical conductivity. Simulations are performed on single-crystal copper for three representative Face Centered Cubic (FCC) orientations. Using an explicit CPFEM model, the study examines texture evolution and deformation heterogeneity during the ECAP process of single-crystal copper. The results indicate that the <100> single-crystal orientation exhibits the highest Taylor factor and the most homogeneous distribution of plastic equivalent strain (PEEQ), suggesting enhanced resistance to plastic flow. In contrast, the <111> single-crystal orientation displays localized deformation and reduced hardening. A decreasing Taylor factor correlates with more uniform slip, which improves both electrical and thermal conductivity, as well as machinability, by minimizing dislocation-related resistance. These findings make a novel contribution to the field by highlighting the critical role of crystallographic orientation in governing slip activity and deformation pathways, which directly impact thermal wear resistance and the fabrication efficiency of ECAP-processed copper electrodes in EDM.

1. Introduction

Electrical discharge machining (EDM) is classified among unconventional fabrication techniques [1,2]. During EDM, material removal occurs through electric discharges (sparks) generated in the gap between the tool electrode (shortly referred to as the electrode) and the workpiece. The energy released during each spark pulse causes localized melting and vaporization of the electrode and workpiece material surfaces. The total cost of a component fabricated via EDM is primarily influenced by the electrode, including material and fabrication costs, as well as the duration of the operation [3].

The electrode wear, one of the process’s limitations, is affected by several factors, including the thermal and electrical conductivity of the electrode material. These properties affect how efficiently electrical energy is transferred during discharge and how spark heat dissipates. Although EDM does not involve mechanical loading, the high-energy discharges generate microscopic stresses in the material. Therefore, the electrode’s durability, structural stability, and homogeneity are critical factors that directly influence EDM performance and electrode service life [4,5]. Furthermore, the machinability of electrode material is an important consideration, as it substantially affects electrode production costs. If the chosen material is difficult to shape, it becomes impractical for EDM applications and severely limits its usability [6].

Pure copper and its alloys are the most widely used material for EDM electrodes, primarily due to their excellent electrical and thermal conductivity and easy fabrication [7,8,9]. They exhibit desirable mechanical strength, high thermal conductivity, good oxidation resistance, and dimensional stability, in addition to being easy to machine. Copper electrodes, with higher thermal conductivity, dissipated heat more effectively, reducing localized melting and wear.

Equal-Channel Angular Pressing (ECAP) is a widely used severe-plastic-deformation (SPD) technique that improves hardness, extends wear life [10,11] and enhances corrosion resistance [12,13,14] in a range of alloys, particularly aluminum- and zinc-based systems. Magnesium and nickel alloys subjected to three ECAP passes exhibited a nearly 50% reduction in surface roughness and an over 50% decrease in abrasive wear, resulting in enhanced machining precision [15,16]. In the case of pure copper, optimized control of grain size and dislocation density has been shown to yield high electrical conductivity alongside increased tensile strength and elongation [17]. For Zn-Ag-Mg alloy, suppressing twin boundaries and reducing dislocation density further contribute to improved ductility with successive ECAP passes [18]. Texture evolution during deformation also plays a significant role; initial [111]<110> textures transform into stronger [110]<112> Brass textures, particularly after four passes, at which point electrical conductivity has been observed to reach up to 96.1% [19]. Similarly, for copper rods, Ciemiorek et al. [20] reported an initial drop in electrical conductivity after the first ECAP pass, followed by a recovery and eventual increase through subsequent passes up to eight. Shaban et al. [21] also confirmed that ECAP strengthens pure copper billets without significantly compromising their electrical performance. SPD has been demonstrated to significantly refine the microstructure of copper, thereby enhancing its mechanical performance. Yield and ultimate tensile strengths were reported to increase by approximately 85% and 80%, respectively. Despite these improvements, electrical conductivity was only marginally reduced (about 3% IACS), and hydrostatically extruded electrodes showed over 60% lower wear rates in EDM tests compared to their commercial counterparts [22]. The other study on EDM using ECAP-processed copper electrodes reported approximately 6% lower electrode wear compared to as-received samples, while maintaining a consistent material removal rate. This resulted in the lowest wear ratio observed after a single pass [23]. Additionally, Gopal et al. [24] observed a general trend of increasing hardness and decreasing electrical conductivity in copper electrodes subjected to EDM with each successive ECAP pass. Nevertheless, the magnitude of these changes was found to diminish progressively after the initial passes.

The literature review reveals a scarcity of studies investigating the use of ECAP-processed electrodes to improve the performance of the EDM process, particularly in terms of reducing electrode wear and facilitating electrode fabrication. Moreover, only a few studies [25,26] have systematically evaluated ECAP performance using numerical methods. Therefore, the current study aims to determine ECAP processing conditions that promote microstructural states conducive to improved mechanical strength, thermal transport, and electrical conductivity in pure copper electrodes, using a Crystal Plasticity Finite Element Method (CPFEM) approach. In this study, a novel rate-dependent crystal plasticity algorithm is developed and implemented as a user-defined material subroutine (VUMAT) in Abaqus/Explicit. The algorithm is utilized to investigate the influence of crystallographic deformation orientations, specifically <100>, <110>, and <111>, on the electrical and thermal conductivity, as well as the hardness of copper electrodes, which are critical factors affecting their wear behavior and machinability in EDM. The proposed model is employed to perform CPFEM simulations on single-crystal copper under ECAP-induced boundary conditions, enabling a detailed assessment of orientation-dependent responses. The effect of the ECAP process on the hardness, thermal conductivity, and electrical conductivity of copper electrodes is analyzed and compared with data from the literature. To the best of our knowledge, this is the first study to integrate CPFEM-based crystallographic analysis with EDM performance metrics for ECAP-processed copper electrodes.

2. Finite Element Model

The CPFEM simulation for a single-pass ECAP uses the commercial FE software Abaqus 2023. Figure 1a provides a 3D CAD model representation of the ECAP mold and the FE mesh configuration utilized for the analysis.The mesh properties employed in the CPFEM model (Figure 1b) are detailed in

Table 1. CPFEM mesh model parameters of simulated ECAP process.

Model	Element Type	Element Number	Model Type
Mold	Linear tet-C3D4	381,163	Rigid
Punch	Explicit-Hex-C3D8R	735	Rigid
Polycrystal Dummy	Explicit-Hex-C3D8R	758,462	Deformable (VM)
Electrode	Explicit-Hex-C3D8R	2560	Deformable (CP)

. The single-crystal copper electrode underwent meshing using 2560 linear hexahedral C3D8R elements, selected based on dilatational wave speed and the deformation speed ratio to prevent distortion issues. Meanwhile, the rigid punch is meshed using 504 linear hexahedral C3D8R elements. The ECAP mold is meshed utilizing 381,163 linear tetrahedral elements. The technical drawing of ECAP mold is given in Figure 1c. Increasing the $\varphi$ angle (Figure 1d) leads to a decrease in both strain homogenization and hardness. To avoid these adverse effects and to ensure a more uniform strain distribution [27], a $\varphi$ angle of 90° is selected in the present study. The ECAP mold and punch are treated as rigid bodies, representing non-deformable structures.

Encastre boundary conditions ($u_1=u_2=u_3=R_1=R_2=R_3=0$) are applied to the mold via a rigid body reference point definition. The square electrode boasts a cross-sectional area of 16 ${\mathrm{mm}}^2$ ($4\mathrm{mm}\times 4\mathrm{mm}$) and spans a length of 50 mm. In the contact algorithm, explicit general contact is employed with a Coulomb friction coefficient ($\mu$) of 0.05. CPFEM analyses of copper single crystals during ECAP report that $\mu$ = 0.05 yields better agreement with experimentally observed texture than $\mu$ = 0.1 [28,29]. In all cases, the simulations are conducted at a pressing speed of 3 mm/s [30], employing a mass scaling factor of sixteen. A dummy polycrystalline specimen with the same cross-sectional geometry as the electrode sample is modeled using von Mises plasticity to isolate and eliminate the influence of the punch force on the actual specimen during simulation.

Energy balance equations are commonly used to verify whether a simulation produces an appropriate quasi-static response. A widely accepted criterion in the literature involves comparing the kinetic energy history with the internal energy. In metal forming simulations, the majority of internal energy originates from plastic deformation. For a solution to be considered quasi-static, the kinetic energy should constitute only a small fraction of the internal energy, typically less than 1–5% [31]. As illustrated in Figure 2, the kinetic energy remains below 1% of the internal energy throughout the analysis, apart from the initial stage. This confirms that the criterion is satisfied, thereby indicating a valid quasi-static response. The figure also highlights the variations in strain energy (SE) and plastic dissipation energy (PD). Moreover, the total energy (Etotal) of the model remains nearly constant during the entire simulation.

3. Methods

3.1. Crystal Plasticity Framework

Metal forming simulations using the CPFEM can be performed with both implicit and explicit algorithms. Implicit CPFEM may encounter convergence issues when applied to highly non-linear and dynamic processes [32]. On the other hand, explicit algorithms are often preferred due to their stability under contact nonlinearity and frictional discontinuities. However, CPFEM simulations remain computationally demanding due to the stress update algorithms. Numerous studies have explored the pros and cons of implicit versus explicit CPFEM [33,34].

The crystallographic slip mechanism can effectively model the material’s constitutive response in single crystals [35,36]. The overall deformation gradient $\mathit{F}={\mathit{R}}^{\mathbf{*}}\mathit{U}$, where ${\mathit{R}}^{\mathbf{*}}$ represents rigid body rotation and $\mathit{U}$ signifies the right stretch tensor, can be decomposed multiplicatively into elastic (${\mathit{F}}^{\mathit{e}}$) and plastic (${\mathit{F}}^{\mathit{P}}$) components [37] (Equation (1)). Furthermore, the elastic portion of deformation (${\mathit{F}}^{\mathit{e}}$) can be expressed as the product of the left stretch tensor (${\mathit{V}}^{\mathit{e}}$) and the rotation tensor (${\mathit{R}}^{\mathit{e}}$).

$$\mathit{F}={\mathit{F}}^{\mathit{e}}{\mathit{F}}^{\mathit{p}}$$

(1)

In the explicit formulation within Abaqus, the total rotation tensor ${\mathit{R}}^{\mathbf{*}}$ represents rigid body rotation, and it is formulated utilizing the Green-McInnis-Naghdi rate [38]. In the context of plastic deformation, the plastic component of the deformation gradient has no impact on the rigid body rotation of a single crystal. Consequently, the elastic portion of rotation ${\mathit{R}}^{\mathit{e}}$ can be equated to the total rotation ${\mathit{R}}^{\mathbf{*}}$ of the deformation, as expressed in Equation (2).

$$\mathit{F}={\mathit{R}}^{\mathbf{*}}\mathit{U}, {\mathit{R}}^{\mathbf{*}}\mathit{U}={\mathit{V}}^{\mathit{e}}{\mathit{R}}^{\mathbf{*}}{\mathit{F}}^{\mathit{p}}$$

(2)

Resolved shear stress ${\tau }^{\alpha }$ acting on the $\alpha$th slip system in the undeformed lattice can be defined as;

$${\tau }_n^{\alpha }=\sum _{\mathit{k}=\mathit{1}}^{\mathsf{\alpha }}{\mathit{C}}_{\mathit{n}}^{{\mathit{e}}_{\mathbf{0}}}{\mathit{S}}_{\mathit{n}}^{{\mathit{e}}_{\mathbf{0}}}{\tilde{\mathit{s}}}_{\mathbf{0}}^{\mathsf{\alpha }}\otimes {\tilde{\mathit{n}}}_{\mathbf{0}}^{\mathsf{\alpha }}$$

(3)

where ${\mathit{C}}_{\mathit{n}}^{{\mathit{e}}_{\mathbf{0}}}$ in Equation (3) is the elastic right Cauchy-Green strain tensor. The ${\mathit{S}}_{\mathit{n}}^{\mathit{e}\mathbf{0}}$ term represents the Piola-Second Kirchhoff stress (PK2) on the intermediate state $U_0$ as illustrated in Figure 3. ${\tilde{\mathit{s}}}_{\mathbf{0}}^{\mathsf{\alpha }}\otimes {\tilde{\mathit{n}}}_{\mathbf{0}}^{\mathsf{\alpha }}$ refers to the tensor product of the slip direction and nominal direction of the initial slip system directions. This product can be obtained by multiplying the slip system (${\mathit{s}}^{\mathsf{\alpha }},{\mathit{n}}^{\mathsf{\alpha }}$) with the Bunge-formulated Euler angles ($\alpha ,\beta ,\gamma$) from ${\tilde{\mathit{s}}}_{\mathbf{0}}^{\mathsf{\alpha }}$ = ${\mathit{Q}}_{\mathit{0}}{\mathit{s}}^{\mathsf{\alpha }}$ and ${\tilde{\mathit{n}}}_{\mathbf{0}}^{\mathsf{\alpha }}={\mathit{Q}}_{\mathbf{0}}{\mathit{n}}^{\mathsf{\alpha }}$.

The power law type rate-dependent formulation, which establishes the relationship between the slip rate on each slip system, the current yield stress ${\tau }^{\alpha }$, and the current strength $g^{\alpha }$, is presented in Equation (4) [39].

$${\dot{\gamma }}_n^{\alpha }={\dot{\gamma }}_0|{\displaystyle \frac{{\tau }_n^{\alpha }}{g_n^{\alpha }}}{|}^{m}sign\left({\tau }_n^{\alpha }\right)$$

(4)

Here, ${\dot{\gamma }}_0$ represents the reference shearing rate, and the exponent m denotes the strain rate sensitivity coefficient.

The expression for current strength $g^{\alpha }$ is defined by Equation (5). Initially, the critical resolved shear stress ${\tau }_0^{\alpha }$ is equal to $g_0$. The hardening moduli $h^{\alpha \beta }$ describe the rate of strain hardening on slip system $\alpha$ due to slip $\beta$. This phenomenon of self and latent hardening is phenomenologically characterized by $h_{\alpha \beta }=qh\left(\gamma \right)$ [40].

$${\dot{g}}^{\alpha }=\sum _{k=1}^{12}{h}^{\alpha \beta }|{\dot{\gamma }}^{\beta }|$$

(5)

The secant-type model is introduced in Equation (6) [41].

$$h_{\alpha \alpha }=h\left(\gamma \right)=h_{0}sec{h}^2|{\displaystyle \frac{h_0\gamma }{{\tau }_s-{\tau }_0}}|$$

(6)

Here, $h_0$ represents the initial hardening modulus, ${\tau }_0$ stands for the yield stress, which is equal to the initial value of the current strength $g_0^{\alpha }$, ${\tau }_s$ denotes the stage I stress, and $\gamma$ signifies the Taylor cumulative shear strain on all slip systems as presented in Equation (7).

$$\gamma =\sum _{\alpha }{\int }_0^t|{\dot{\gamma }}^{\alpha }|dt$$

(7)

Plastic deformation is induced by shear or slip ${\gamma }^{\alpha }$ occurring on slip systems. These slip planes are characterized by ${\tilde{\mathit{s}}}_{\mathbf{0}}$ and ${\tilde{\mathit{n}}}_{\mathbf{0}}$ vectors, representing the unit vector in the slip direction and the unit normal vector to the slip plane in the intermediate configuration $U_0$, as illustrated in Figure 3. The velocity gradient is defined by Equation (8);

$$\mathit{L}=\dot{\mathit{F}}{\mathit{F}}^{-\mathbf{1}}$$

(8)

Substituting Equation (1) into the equation above and then applying straightforward differentiation through the product rule results in the subsequent additive decomposition of the velocity gradient: $\mathit{L}={\mathit{L}}^{\mathit{e}}+{\mathit{F}}^{\mathit{e}}{\mathit{L}}^{\mathit{p}}{\left({\mathit{F}}^{\mathit{e}}\right)}^{-1}$. Here, ${\mathit{L}}^{\mathit{e}}$ and ${\mathit{L}}^{\mathit{p}}$ represent the elastic and plastic components of velocity gradients, respectively. As defined in the previous expression, the plastic part of the velocity gradient pertains to the intermediate configuration. Consequently, the second term represents a push forward to the current configuration. As plastic deformation is attributed to dislocation slip along multiple slip systems, the plastic part of velocity gradient ${\mathit{L}}^{\mathit{p}}$ can be determined as the sum of shear rates associated with each slip system, denoted as ${\dot{\gamma }}^{\alpha }$, according to Equation (9).

$${\mathit{L}}_{\mathit{p}}=\sum _{k=1}^{\alpha }{\dot{\gamma }}_n^{\alpha }{\tilde{\mathit{s}}}_{\mathbf{0}}^{\mathsf{\alpha }}\otimes {\tilde{\mathit{n}}}_{\mathbf{0}}^{\mathsf{\alpha }}$$

(9)

The plastic component of the deformation gradient can be expressed using Equation (10).

$${\mathit{F}}_{\mathit{n}+\mathbf{1}}^{\mathit{p}}=(\mathit{I}+\Delta t\sum _{k=1}^{\alpha }{\dot{\gamma }}_n^{\alpha }{\tilde{\mathit{s}}}_{\mathbf{0}}^{\mathsf{\alpha }}\otimes {\tilde{\mathit{n}}}_{\mathbf{0}}^{\mathsf{\alpha }}){\mathit{F}}_{\mathit{n}}^{\mathit{p}}$$

(10)

Thus, the elastic component of the deformation gradient ${\mathit{F}}_{\mathit{n}+\mathbf{1}}^{\mathit{e}}$ can be written as;

$${\mathit{F}}_{\mathit{n}+\mathbf{1}}^{\mathit{e}}={\mathit{V}}_{\mathit{n}+\mathbf{1}}^{\mathit{e}}{\mathit{R}}^{\mathbf{*}}$$

(11)

The rate of Cauchy stress is linked to the second Piola-Kirchhoff (PK2) stress, and the rate of PK2 stress can be determined by the multiplication of the rotated elastic modulus by transformed initial orientations [42]. This is represented as $\tilde{\mathit{C}}$ = ${\mathit{Q}}_{\mathbf{0}}^{\mathit{c}}\mathit{C}{{\mathit{Q}}_{\mathbf{0}}^{\mathit{c}}}^{\mathit{T}}$, where $\mathit{C}$ is the material’s elastic stiffness tensor. The PK2 stress rate is calculated by combining the material’s strain rate tensor and the Green-Lagrange strain rate (${\dot{\mathit{E}}}_{\mathit{n}+\mathbf{1}}$), as outlined in Equations (12).

$${\dot{\mathit{S}}}_{\mathit{n}+\mathbf{1}}^{{\mathit{e}}_{\mathbf{0}}}=\tilde{\mathit{C}}\mathbf{:}{\dot{\mathit{E}}}_{\mathit{n}+\mathbf{1}}$$

(12)

The corotational stress rate can be expressed using Equation (13) and the term ${\mathit{R}}^{\mathbf{*}\mathit{T}}\dot{\mathsf{\sigma }}{\mathit{R}}^{\mathbf{*}}$ can be calculated from ${\mathit{R}}^{\mathbf{*}\mathit{T}}\dot{\mathsf{\sigma }}{\mathit{R}}^{\mathbf{*}}={\displaystyle \frac{1}{det{F}_{n+1}^e}}{\mathit{U}}_{\mathit{n}+\mathbf{1}}^{\mathit{e}}{\dot{\mathit{S}}}_{\mathit{n}+\mathbf{1}}^{{\mathit{e}}_{\mathbf{0}}}{\mathit{U}}_{\mathit{n}+\mathbf{1}}^{\mathit{e}}$

$${\dot{\mathsf{\sigma }}}^{\mathbf{\circ }}={\dot{\mathit{R}}}^{\mathbf{*}\mathit{T}}\mathsf{\sigma }{\mathit{R}}^{\mathbf{*}}+{\mathit{R}}^{\mathbf{*}\mathit{T}}\dot{\mathsf{\sigma }}{\mathit{R}}^{\mathbf{*}}+{\mathit{R}}^{\mathbf{*}\mathit{T}}\mathsf{\sigma }{\dot{\mathit{R}}}^{\mathbf{*}}$$

(13)

Stress rate can be defined as $\dot{\mathsf{\sigma }}={\mathsf{\sigma }}^{\mathbf{▽}}+\mathbf{\Omega }{\mathsf{\sigma }}_{\mathit{n}}-{\mathsf{\sigma }}_{\mathit{n}}\mathbf{\Omega }$ so the increment of Cauchy stress can be calculated as $\dot{\mathsf{\sigma }}\Delta t=\mathbf{\Delta }{\mathsf{\sigma }}^{\mathbf{▽}}+\mathit{R}{\mathsf{\sigma }}_{\mathit{n}}{\mathit{R}}^{\mathit{T}}$ in which ${\mathsf{\sigma }}^{\mathbf{▽}}$ is the rate of objective, corotational stress, or the rate associated with the constitutive response [43]. Skew symmetric angular velocity tensor $\mathsf{\Omega }$ can be found from $\mathsf{\Omega }={\dot{\mathit{R}}}^{\mathbf{*}}{\mathit{R}}^{\mathbf{*}\mathit{T}}$. Hence, the update of Cauchy stress for the spatial configuration $U_s$ can be derived from ${\mathsf{\sigma }}_{\mathit{n}+\mathbf{1}}={\mathsf{\sigma }}_{\mathit{n}}+\dot{\mathsf{\sigma }}\Delta t$.

Tensile test, simple shear test, and pure shear test simulations are conducted to verify the consistency of the explicit algorithm with experimental results [44] and the UMAT code developed by Huang [45]. The boundary conditions are treated as shown in Figure 4 [46,47]. Material parameters of single crystal copper, taken from the literature [48], are compared with [100] and [111] directed single copper crystal test data. However, under simple shear boundary conditions, an inconsistency is observed with the UMAT code, as shown in Figure 5b. One of the main reasons for this situation is as explained in detail in reference [49]. The material parameters of single copper for the VUMAT subroutine are illustrated in

Table 2. Material parameters for the VUMAT single crystal copper model [50].

Model	${\mathit{C}}_{11}$	${\mathit{C}}_{12}$	${\mathit{C}}_{44}$	${\mathit{\tau }}_0$	${\mathit{\tau }}_{\mathit{s}}$	${\mathit{h}}_0$	$\dot{{\mathit{\gamma }}_0}$	n	q
	(MPa)	(MPa)	(MPa)	(MPa)	(MPa)	(MPa)	(${\mathbf{s}}^{\mathbf{-}\mathbf{1}}$)
VUMAT CP	168,000	121,400	75,400	25	115	120	0.001	17	1.4

. The algorithm of the explicit crystal plasticity user subroutine (VUMAT) is illustrated in Figure 6.

3.2. Von Mises Plasticity Framework

The von Mises plasticity model is the constitutive model used to simulate severe plastic deformations of the dummy workpiece. The true stress-strain curve of the copper specimen is illustrated in Figure 7.

4. Results and Discussion

4.1. Plastic Strain and Taylor Factor Evolution in ECAP-Processed Copper Electrodes

Figure 8 shows that the plastic equivalent strain (PEEQ) values near the top of the single-crystal copper electrode exhibit homogeneity, with PEEQ maintaining an approximately constant value across all its elements. Conversely, the strain distribution in the lower part of the electrode appears heterogeneous, with each region showing a consistent value of effective strain. The strain distribution along the axial direction remains nearly uniform except for the initial billet segment, where steady-state deformation hasn’t yet been achieved. The distribution of effective strain along the height of the billet is the characteristic of ECAP processes employing dies with external curvature. The visible gap between the electrode and the die represents an external curvature during the process, resulting in reduced shear strain levels in the outer region of the deformation zone. The electrode near the inner corner experiences more significant strain and strain gradients than the material near the outer corner. The intense deformation near the inner corner imparts a higher compressive force to the material neighboring the right-hand side interface of the entry channel. As a result, this leads to increased frictional stress at the interface on the right-hand side and a notably significant plastic strain rate.

Experimental measurement of deformation modes is not directly feasible due to the complex, multiscale nature of plastic deformation, which involves heterogeneous slip activity, internal lattice rotations, and evolving microstructural features that are difficult to observe in situ and at sufficient resolution. The CPFEM approach offers the advantage of capturing such local heterogeneities in strain and stress distribution, enabling a more detailed and mechanistic understanding of orientation-dependent plastic behavior. The extent of strain inhomogeneity can be evaluated utilizing the equation proposed as $S{H}_i={\displaystyle \frac{Acc{u}_{max}-Acc{u}_{min}}{Acc{u}_{Avg}}}$ [52]. In this equation, $S{H}_i$ represents the strain inhomogeneity index, $Acc{u}_{min}$ stands for the minimum accumulated plastic strain, $Acc{u}_{max}$ denotes the maximum accumulated plastic strain, and $Acc{u}_{Avg}$ signifies the average accumulated plastic strain. The strain inhomogeneity index and the coefficient of variation of accumulated plastic strain for different initial orientations are shown in Figure 9. The black color corresponds to the initially oriented <100> direction. The orange and blue colors represent initially oriented <110> and <111> directions, respectively. Lower values of the index indicate better homogeneity. Although the <111> initially oriented single crystal electrode exhibits more inhomogeneous deformation compared to the <100> and <110> orientations after one pass of ECAP, the resulting inhomogeneity values remain relatively close to each other. The results suggest that the heterogeneity of through-thickness accumulated plastic strain in the Z-direction is also orientation-dependent.

The analysis of deformation behavior and texture evolution in the single copper crystal involves examining specific regions and elements of the specimen, as shown in Figure 10. Initially <100> oriented single crystal copper demonstrates the highest Taylor factor ($Taylorfactor\left(M\right)={\sum }_{\alpha }{\displaystyle \frac{|{\dot{\gamma }}^{\alpha }|▵t}{{\dot{\mathsf{ϵ}}}_{eff}^p▵t}}$) when it is deformed in A deformation path within the plastic deformation zone. Conversely, initially, a <111>-oriented single-crystal copper exhibits the lowest values throughout the process. Toward the end of the process, initially <100> oriented single-crystal copper again displays higher Taylor factors than initially <111> oriented single-crystal copper. It’s important to clarify that reducing the Taylor or Schmid factors does not necessarily imply material softening. Instead, grains with higher Taylor factors experience more significant stress reduction during deformation. A decrease in the Taylor factor suggests that the grain’s orientation has shifted towards more favorable slip conditions. Smaller Taylor factor values indicate orientations more conducive to deformation, as they require lower stress levels for plastic deformation to occur. The crystals with smaller Taylor factors tend to deform more quickly than those with larger values [53]. Conversely, an increase in the Taylor factor is associated with higher energy accumulation due to deformation [54]. The presence of low Taylor factor values with a narrow distribution in microstructural orientations facilitates more uniform deformation among the grains. This homogeneous deformation can limit local increases in dislocation density, reduce electron scattering, and thereby improve electrical conductivity [55]. High electrical conductivity ensures efficient spark generation and minimizes the electrode’s energy loss, contributing to lower wear rates [56]. Moreover, metals with high electrical conductivity also tend to exhibit high thermal conductivity [57]. The thermal conductivity of the electrode material influences the transfer of spark energy within the electrode body during EDM, thereby affecting electrode wear rates. Electrodes with higher thermal conductivity dissipate heat more effectively, which reduces localized melting and subsequent wear [58]. A decrease in the Taylor factor generally enhances the machinability of the material, as it offers lower resistance to deformation during cutting, resulting in reduced tool loading and facilitated plastic flow. However, secondary factors such as chip morphology and surface integrity must be carefully considered due to their sensitivity to local deformation behavior [59]. Initially, <111>-oriented single-crystal copper electrodes exhibit a reduction in Taylor factor values, particularly under deformation routes A and B, resulting in a concurrent decrease in dislocation density. This microstructural refinement, achieved through SPD methods such as ECAP, results in lower machining forces due to enhanced plastic deformability [60]. Furthermore, the elevated electrical conductivity observed following ECAP processing along routes A and B has also been reported in the literature, correlating with the lower Taylor factor and dislocation density in the <111> orientation [61,62].

4.2. Texture Evolution of ECAP-Processed Copper Electrode

The crystallographic texture of polycrystalline materials significantly influences their thermal conductivity due to the anisotropic nature of phonon and electron transport. Specific texture components, such as cube ([100]<001>), Goss ([110]<001>), and Brass ([011]<211>), are commonly observed in FCC metals and can be identified through characteristic intensity peaks in pole figures [63]. The cube component, for example, aligns heat-conducting directions <100> with the measurement axis, often resulting in higher thermal conductivity in textured materials. Conversely, orientations such as Brass or Goss may lead to increased phonon scattering due to higher dislocation density or less favorable transport pathways. Studies have shown that a firm Cube texture can enhance thermal conductivity in rolled or annealed metals by minimizing grain boundary resistance and maximizing aligned conduction pathways [64]. Therefore, pole figure analysis not only aids in texture quantification but also provides insights into the anisotropic thermal performance of engineered materials.

The pole figures after a single ECAP pass of single-crystal copper are shown in Figure 11. According to these pole figures, Goss and Brass orientations emerge after one ECAP pass for initially <100> and <110> oriented single crystals under deformation paths A, B, and C. Similarly, Goss and Brass components are also observed in initially <111> oriented crystals under paths A and B. Therefore, the electrical conductivity of initially <100> oriented copper electrodes is expected to decrease due to the loss of the favorable <100> alignment. In contrast, for initially <110> oriented electrodes, the conductivity is expected to remain relatively stable due to the retention of suitable transport orientations. Moreover, for initially <111> oriented copper, the transformation to Goss and Brass textures under deformation paths A and B is expected to enhance electrical conductivity. Besides that, under deformation path C, shear texture components become dominant after a single pass, which may negatively impact conductivity. The obtained pole figures are in good agreement with those reported in previous studies [65,66]. Shear deformation paths, such as those imposed by SPD techniques like ECAP, significantly influence the microstructure and, consequently, the electrical conductivity of metallic materials. The applied shear strain leads to grain refinement, dislocation multiplication, and texture evolution, all of which affect electron scattering mechanisms. In particular, high-angle grain boundaries and increased dislocation density associated with shear deformation typically reduce electrical conductivity [67,68]. However, under controlled deformation paths such as route A or B, it is possible to promote favorable crystallographic textures that align low-resistivity directions with the measurement axis. This alignment can partially offset conductivity losses by facilitating electron transport along preferred orientations [69]. Thus, while SPD generally introduces microstructural defects that hinder conductivity, carefully selected deformation paths can optimize texture to mitigate these effects and even enhance electrical conductivity. The Taylor factor and pole figure results indicate that both electrical and thermal conductivities can be improved through the ECAP process, particularly under deformation paths A and B for initially <111> oriented copper electrodes.

4.3. Thermal Effects on Texture Analysis of ECAP-Processed Copper Electrodes

During the ECAP process, the imposed deformation path has a significant influence on the temperature evolution within the material, which in turn affects its microstructural development. Different ECAP routes (such as A, B, and C) result in varying shear strain distributions and heat generation patterns due to differences in strain path repetition and crystal lattice reorientation. These temperature variations, particularly under adiabatic or near-adiabatic conditions, can accelerate recovery or dynamic recrystallization in localized regions, thereby modifying grain size, dislocation density, and texture evolution [70]. The recrystallization temperature for pure copper is typically expected to be in the range of 30–40% of its melting point (1085 °C), corresponding to approximately 350–450 °C. However, severe plastic deformation techniques such as ECAP can significantly reduce this temperature to below 200 °C [71]. Therefore, to accurately evaluate the temperature rise during the ECAP process, it is necessary to perform a detailed thermal analysis under appropriately defined boundary conditions. The present study, the temperature analysis is initially carried out under adiabatic assumptions, considering only the contribution from shear deformation. However, it should be noted that frictional dissipation contributes directly to additional heat generation at the die–specimen interface, which can further accelerate recrystallization processes in localized regions. In order to illustrate the temperature rise independently of die parameters, only the contribution from plastic strain has been presented in this work. In this context, the temperature rise due to both plastic deformation work and interfacial frictional heating should be explicitly taken into account. A general equation for the temperature rise in ECAP is derived, and, as also demonstrated in previous studies in the literature, the temperature increase scales with the material strength, ram speed, and channel angle, whereas it decreases with higher density, heat capacity, and die corner angle [72,73].

Assuming adiabatic conditions, the temperature change is calculated using $\rho c\dot{\theta }=\xi {\sum }_{k=1}^N{\left({\sum }_{\alpha }{\tau }^{\alpha }{\dot{\gamma }}^{\alpha }\right)}^{\left(k\right)}$ [74]. Where $\rho$ denotes the material density, and c is the specific heat capacity. The left-hand side represents the rate of internal energy accumulation per unit volume, attributed to the temperature change under adiabatic conditions. On the right-hand side, $\xi$ is the Taylor–Quinney coefficient, which quantifies the fraction of plastic work converted into heat (typically ranging between 0.85 and 1). The double summation accounts for the plastic dissipation within all crystallographic slip systems $\alpha$, evaluated at each material integration point k. Here, ${\tau }^{\alpha }$ is the resolved shear stress, and ${\dot{\gamma }}^{\alpha }$ is the shear strain rate associated with the $\alpha$-th slip system. The expression thus captures the total rate of plastic dissipation transformed into heat and subsequently manifested as a temperature increment in the material during the plastic deformation. Figure 12 indicates that deformation route A exhibits the highest temperature increase compared to routes B and C. This observation is consistent with the results of accumulated plastic strain. A route can also be reached through the recrystallization temperature during the ECAP process. Upon reaching the recrystallization temperature, the material undergoes significant microstructural transformations driven by the stored deformation energy, primarily in the form of dislocations. The high dislocation density accumulated during severe plastic deformation is progressively reduced through recovery and recrystallization mechanisms. During this process, dislocation rearrangement and annihilation occur, followed by the nucleation and growth of new, defect-free grains. As a result, the heterogeneity in dislocation distribution is eliminated, and internal residual stresses are substantially relieved. Simultaneously, the crystallographic texture of the material is also altered: deformation-induced texture components, such as Brass, Copper, and S-type orientations (common in FCC metals), are weakened or replaced by recrystallization textures. In the case of copper, a strong Cube texture ([001]<100>) often develops due to the preferential growth of favorably oriented grains. The extent and nature of these changes are strongly influenced by factors such as prior strain, annealing temperature, and the mobility of high-angle grain boundaries [75,76,77].

5. Conclusions

This study applies an explicit CPFEM framework to assess the effect of crystallographic orientation on the deformation and performance of ECAP-processed copper electrodes for EDM. Simulations are performed for three FCC single-crystal orientations, <111>, <100>, and <110>, under one pass ECAP conditions. The <100> orientation displays the most homogeneous deformation and the highest Taylor factor, indicating strong resistance to plastic flow. In contrast, the <111> orientation shows more localized plasticity and lower hardening capacity, although with relatively uniform fields. Temperature variations are linked to accumulated plastic strain, and the recrystallization temperature is found to significantly influence texture evolution. Texture analysis and Taylor factor evaluations indicate that ECAP, particularly along paths A and B in <111> oriented electrodes, can improve electrical and thermal conductivity through favorable texture development. The reduction in Taylor factor, dependent on both orientation and path, promotes plastic softening and enhances machinability. <111> crystals also exhibit potential for slip band formation and structural homogeneity. While deformation twinning is likely in high-shear regions, particularly along <111> directions, it is not explicitly included in the present framework. The present findings underscore the pivotal role of crystallographic orientation and deformation path in enhancing the manufacturability, durability, and thermoelectric efficiency of copper electrodes employed in EDM. The computational framework developed herein establishes a direct link between orientation-dependent deformation mechanisms and the resulting functional performance, offering a solid basis for process optimization. Future investigations will focus on experimental validation to ensure the practical applicability and reliability of the predicted trends.