**1. Introduction**

The fabrication of bulk ultrafine-grained (UFG) materials has been widely investigated over the last two decades due to the potential to produce metals with superior mechanical and physical properties. Several techniques are available for fabricating these materials, but major emphasis has been placed on the use of equal-channel angular pressing (ECAP), in which a sample is pressed through a die constrained within an internal channel that is bent through a sharp angle [1]. In this procedure, a shear strain is introduced as the sample passes through the bend in the channel, and this is a very effective severe plastic deformation (SPD) processing method for producing bulk UFG materials for use in engineering applications. In ECAP processing, the cross-section of the sample remains unchanged so that the pressing may be repeated for multiple passes in order to achieve the required strain level.

The equivalent plastic strain, *ε*, introduced in a single pass through the die is given by a relationship of the form [2]:

$$\varepsilon = \frac{1}{\sqrt{3}} \left( 2 \cot \left( \frac{\Phi}{2} + \frac{\Psi}{2} \right) + \Psi \right. \tag{1}$$

where *Φ* is the angle subtended by the two parts of the channel, and *Ψ* is the outer arc of curvature at the point of intersection of the two channels. The relationship in Equation (1) is therefore only a function of the die geometry, and it provides a very useful and simple procedure for estimating the average strain introduced during ECAP processing.

**Citation:** Wongsa-Ngam, J.; Noraphaiphipaksa, N.; Kanchanomai, C.; Langdon, T.G. Numerical Investigation of Plastic Strain Homogeneity during Equal-Channel Angular Pressing of a Cu-Zr Alloy. *Crystals* **2021**, *11*, 1505. https:// doi.org/10.3390/cryst11121505

Academic Editor: Wojciech Polkowski

Received: 7 November 2021 Accepted: 1 December 2021 Published: 3 December 2021

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Nevertheless, an understanding of the local strain distribution during ECAP processing is also important since the strain imposed during processing relates directly to the internal microstructure, and, ultimately, it characterizes the overall material properties.

The finite element method (FEM) is a powerful tool that can be used to understand the deformation behavior of a material during the ECAP process. Earlier works on FEM simulations of the ECAP process were carried out by using two-dimensional (2D) simulations [3–9]. In these 2D finite element models, plane strain conditions were generally assumed in order to calculate the effects of the die geometries, the processing conditions and the material properties on the deformation behavior and the inhomogeneity of the materials. However, 2D simulation is used for a billet with a square cross-section, and it is not generally applicable to the round cross-sectional workpieces that are used in ECAP when employing a solid die [10–12]. Recently, three-dimensional (3D) FEM simulations were effectively performed in an analysis of deformation behavior during the first pass of the ECAP process [10,11,13–15]. Nevertheless, there are only limited reports related to multi-pass ECAP processing using 3D FEM for circular cross-sectional workpieces, and all of these reports are directed at the deformation of aluminum and aluminum alloys [16–18].

Because of this deficiency, the present investigation was initiated in order to use a 3D FEM simulation to investigate the deformation behavior and homogeneity evolution of the multi-pass ECAP processing, up to eight passes, of a copper alloy of Cu-0.1 wt.% Zr with billets with circular cross-sections.

#### **2. The Principles of Finite Element Simulations**

The commercial software Abaqus/Explicit version 2016 [19] was used to simulate the multi-pass ECAP processing of the Cu-0.1 wt.% Zr alloy. The workpiece and the die geometry were modeled according to descriptions in earlier experiments conducted at University of Southern California (USC), USA [20,21]. Specifically, the workpiece was in the form of a cylindrical billet with a diameter of 10 mm and a length of 70 mm, and it was processed using a rigid solid die with a channel angle (*Φ*) of 110◦ and an outer corner angle (*Ψ*) of 20◦. These angles were therefore used in the simulation.

Figure 1a shows a schematic representation of an ECAP assembly. For convenience in discussion, a local coordinate system was set as x, y and z axes. The plane normal to the x, y and z axes are henceforth designated the X-plane, Y-plane and Z-plane, respectively. Four points (1–4) were marked on the cross-section of the mid-length of the billet, and these points were used to monitor the accumulated equivalent strain during the ECAP processing as shown in Figure 1b. Consecutive passes of the ECAP processing were modeled using an interconnected multi-channel die corresponding to the equivalent of processing route *BC*, in which the billet is rotated by 90◦ around the longitudinal axis in the same sense between each pass [22] up to a total of 8 passes as illustrated in Figure 1c.

In the simulation, the overall behavior of the billet is taken as an elastic–plastic material. This model can describe the deformation behaviour of a material under severe plastic deformation during ECAP processing. To determine the material properties, the Cu-0.1 wt.% Zr billet was annealed as in earlier experimental work [20,21], and then the tensile testing was conducted. The stress–strain curve and the material properties used for the analysis are shown in Figure 2 and Table 1, respectively. The flow stress curve of the present material was determined until the maximum strain of 0.4 and the strain-hardening exponent was 0.68. However, during the ECAP process, the cumulative strain is expected to increase continuously with the number of passes. To numerically evaluate the cumulative strain beyond the limit of the flow stress curve, the flow stress is linear extrapolated from the end of the curve using a tangent line with a strain-hardening exponent of 0.68. The billet material was modeled with C3D8R (eight-node linear brick element). The die and punch were modeled as a rigid surface, and all simulations were performed with a pressing speed of 3 mm/s. The value of the friction coefficient between the die and the billet was assumed to be 0.1; this value is recommended when processing using MoS2. An arbitrary Lagrangian–Eulerian (ALE) adaptive remeshing and mass scaling was used for

all simulations to prevent failure of the mesh due to large deformation and also to reduce the total computation time.

**Figure 1.** Schematic illustration of the ECAP process: (**a**) an ECAP assembly, (**b**) four points marked on the mid-length of the billet and (**c**) an interconnected multi-channel die.

**Figure 2.** Experimental true plastic stress–strain curve for the Cu-0.1% Zr alloy.

**Table 1.** Material properties of an annealed specimen of Cu-0.1 wt.% Zr.


Before performing the simulation, a convergence test was carried out to assess the mesh sensitivity. Seven different numbers of elements, namely 1056, 1280, 1408, 2200, 3420, 5092 and 7680, were used to calculate the total strain energy. The relationship between the total strain energy and the numbers of elements was then plotted as shown in Figure 3. Based on the result in Figure 3, a mesh consisting of 2200 elements, with element sizes of 0.2–2.0 mm, was chosen in all simulations.

**Figure 3.** Convergence test for different numbers of elements.
