*3.1. Equivalent Plastic Strain*

The equivalent plastic strain during the pressing stroke was monitored through the four points located at the cross-sections of the mid-length of the workpiece as illustrated in Figure 1b. Figure 4 shows the variation in the equivalent plastic strain tracing through points 1 to 4 located on the cross-section of the mid-length of the billet as it passes through the die for one pass. As a general overview observation, the deformation behavior of the four points is similar to the deformation commencing after approximately 40% of the pressing stroke. Thereafter, the equivalent plastic strain increases rapidly when the billet is pressed through the intersection or deformation zone of the ECAP channel, corresponding to 50% of the stroke, and, finally, it remains steady during the subsequent ECAP process.

On close inspection, there is a difference in the magnitude of the equivalent plastic strain at different points after one pass of ECAP. Thus, shortly after entering the intersection of the ECAP die, corresponding to approximately between 44% and 50% of the stroke, point 1 and point 3 located near the outer corner and inner corner of the die, respectively, have higher strains than those of points 2 and 4. This is because points 1 and 3 are deformed before points 2 and 4, but the local strains change with further pressing whereby each point reaches a peak value of strain and remains approximately constant after approximately 60% of the stroke. It is found that the highest equivalent plastic strain of ~1.04 occurs at point 3 located near the inner channel angle; the lowest of ~0.58 is at point 1 located near the outer corner of the die; and the strains at points 2 and 4 are identical at ~0.75, where this value lies between points 1 and 3. This result demonstrates the deformation inhomogeneity that exists in the billet after one pass of ECAP. Similar trends were observed in earlier research [23–25], in which deformation begins to take place at points near the outer and

inner curves of the die, but the point near the outer curve has a lower deformation rate because it necessarily travels a longer distance.

**Figure 4.** Equivalent plastic strain variation tracing through points 1, 2, 3 and 4 located on the cross-section of the mid-length of the billet during ECAP processing through one pass.

The equivalent plastic strain distribution contours on the transverse cross-section (X-plane) and longitudinal section (Z-plane) through ECAP for one pass are plotted in Figure 5a,b, respectively. Figure 5a shows that the strain distribution varied along the vertical direction, in which the strain values at the bottom are lower than those at the top of the billet, while the strain distribution along the horizontal direction is reasonably uniform. The values of the equivalent plastic strain at the top, middle and bottom are consistent with the local strain at points 1–4 as observed during ECAP through one pass depicted in Figure 4. In addition, the strain distributions on the transverse cross-section are consistent with the microhardness distributions measured on the Cu-Zr alloy in an earlier study [20]. The low effective plastic strain at the bottom corresponds to an area of low microhardness, which verifies the results from the FE simulation.

**Figure 5.** Equivalent plastic strain distribution contours for (**a**) transverse cross-section and (**b**) longitudinal direction through ECAP for one pass.

In practice, the lower values of the equivalent plastic strain at the bottom of the workpiece in Figure 5a are associated with the formation of a corner gap between the die and the workpiece at the outer corner as observed in Figure 5b. As the corner gap is formed during pressing, the workpiece is no longer in contact with the die wall, and this leads to a lower degree of deformation and, consequently, to a lower imposed strain and a lower hardness.

The presence of a corner gap is usually found during the ECAP processing of strainhardening materials. Thus, the formation of a corner gap at the outer corner of the ECAP channel was reported through experimental observations [5,12,26] and numerical modelling [11,12,17,24,27–29]. In a study based on using pure aluminum as a model material [26], an examination of the formation of the corner gap between a strain-hardening material and a quasi-perfect plastic material led to the conclusion that a larger corner gap is formed in the material with the higher strain-hardening rate. It was also pointed out that the less deformation at the outer corner was due to a bending effect more than a shearing effect [28]. Another study examined the equivalent plastic strain rate in the plastic deformation zone and showed that the strain rate at the bottom surface was lower than that at the top surface, and, in addition, the strain rate at the bottom surface decreased with increasing angle Ψ [17].

The evolution of the accumulative equivalent plastic strains for these four points in consecutive ECAP processing of up to eight passes was monitored and plotted as shown in Figure 6. Since each point was rotated by 90◦ according to processing route *BC*, the position of each point has a change of 90◦ in the channel leading to a change in the strain path and strain increment in consecutive passes. It is therefore obvious that the equivalent plastic strain of each point continued to increase with the increasing number of passes, but the increment in each point is different depending on the position of the point in the die for a given pass number. For example, after two passes, the strain increments of points 1, 2, 3 and 4 were 0.63, 0.87, 0.80 and 0.55, respectively. This means that the highest increment was found in point 2 because this point was rotated to the inner curve of the die. Conversely, the lowest increment occurred in point 4 because it was moved to the outer curve of the die. Similar results were also reported in a study of 6061 Al alloy in a circular cross-sectional ECAP process with a die angle (*Φ*) of 90◦ using route *BC* and a finite volume method (FVM) simulation [30].

**Figure 6.** Equivalent plastic strain variation tracing through points 1, 2, 3 and 4 located in the cross-section of the mid-length of the billet during ECAP processing up to a total of 8 passes.

The average equivalent plastic strain (*ε p ave*) from the FE simulation was calculated by taking the average of the strain values over the cross-section at the mid-length of the billet, which were obtained by the following equation:

$$
\varepsilon\_{\text{avu}}^p = \frac{1}{n} \sum\_{i=1}^n \varepsilon\_i^p \, , \tag{2}
$$

where *n* is the number of nodes in the cross-section, and *ε p <sup>i</sup>* is the equivalent plastic strains at node *i*. The average strain per pass was given earlier as Equation (1). Therefore, the average equivalent plastic strain obtained by simulation can be plotted as a function of the number of passes and compared directly to the analytical model dictated by Equation (1) using a die geometry of *Φ* = 110◦ and *Ψ* = 20◦ as shown in Figure 7. It is readily apparent that the average equivalent plastic strains from the simulation are very close to the analytical values calculated from Equation (1). This agreement validates the results obtained by the simulation. Figure 7 also shows that the average equivalent plastic strain increases with the increase in the pass number.

**Figure 7.** Average FEM simulation and the calculated equivalent plastic strains using Equation (1) from [2].

#### *3.2. Effect of Number of Passes on the Strain Homogeneity*

In addition to the magnitude of the accumulated strain, the strain homogeneity is also important in the design of the ECAP process. In practice, the effect of ECAP passes on the overall strain homogeneity can be assessed directly by measuring the degree of inhomogeneity and noting that lower values indicate better homogeneity.

The degree of inhomogeneity can be quantified by two different methods: using an inhomogeneity index (*Ci*) or a coefficient of variance (*CV<sup>ε</sup> <sup>p</sup>* ). The inhomogeneity index (*Ci*) is based on the difference between two extreme strain values as defined by

$$\mathbf{C}\_{i} = \left(\frac{\boldsymbol{\varepsilon}\_{\max}^{p} - \boldsymbol{\varepsilon}\_{\min}^{p}}{\boldsymbol{\varepsilon}\_{\text{grav}}^{p}}\right),$$

where *ε p max*, *ε p min* and *ε p ave* denote, respectively, the maximum, minimum and average of the equivalent plastic strain. By contrast, the coefficient of variance (*CV<sup>ε</sup> <sup>p</sup>* ) uses the standard deviation, which depends on the distribution of strain considering the value of the strain at all nodes in the section as defined by

$$CV\_{\varepsilon^p} = \frac{Std\ \varepsilon^p}{\varepsilon^p\_{\text{grav}}},\tag{4}$$

where *Std ε<sup>p</sup>* is the standard deviation of the equivalent plastic strain. This latter value measures the amount of dispersion of the equivalent plastic strain at each node around the average strain as defined by

$$Std\ \varepsilon^p = \left(\frac{1}{n}\sum\_{i=1}^n \left(\varepsilon\_i^p - \varepsilon\_{ave}^p\right)^2\right)^{1/2},\tag{5}$$

The strain distribution homogeneity for different passes during ECAP was determined using these two methods, where the inhomogeneity index (*Ci*) and the coefficient of variance (*CV<sup>ε</sup> <sup>p</sup>* ) were estimated across the transverse section in the mid-length of the billet as shown in Figure 8. It is apparent that the values of the inhomogeneity index, *Ci*, are higher than the coefficient of variance, *CV<sup>ε</sup> <sup>p</sup>* , for all conditions, and this is consistent with earlier reports [15,29]. This difference arises because *Ci* is based on the difference between the maximum and the minimum values, whereas *CV<sup>ε</sup> <sup>p</sup>* is based on the distribution of strain for all nodes in the section. However, both *Ci* and *CV<sup>ε</sup> <sup>p</sup>* give the same tendency whereby their values decrease as the pass number increases, and this is especially true in the earlier stage of deformation as in passes 1–3. However, after four passes, the degree of homogeneity exhibits no significant change because the strains at all nodes on the crosssection are taken into consideration for the coefficient of variance. Therefore, this was selected to represent the strain homogeneity of ECAP.

**Figure 8.** Inhomogeneity in equivalent plastic strain indicated by the inhomogeneity index and by the coefficient of variance.

In processing by ECAP, a sample experiences intense plasticity as it is pressed through the region of the intersection of the two parts of the channel, where this may be denoted as the plastic deformation zone (PDZ). The evolution of strain homogeneity with the increment of the number of passes can be explained by the strain distribution within the PDZ at the intersection of the die channel as shown in Figure 9. The results demonstrate that the variation in the equivalent plastic strain distribution in the PDZ decreases as the

pass number increases as, for example, by 0.13 to 0.88 for one pass and 5.10 to 5.70 for eight passes, and this results in the development of a reasonable degree of deformation homogeneity in consecutive passes.

**Figure 9.** Equivalent plastic strain distribution contours of the workpiece during ECAP processing at 60% of stroke: (**a**) 1 pass, (**b**) 2 passes, (**c**) 4 passes and (**d**) 8 passes.

#### *3.3. The Steady-State Zone*

The steady-state zone is the region along the billet axis where strain is relatively uniform. To define the steady-state zone in an ECAP workpiece, the transverse crosssections were cut perpendicular to the pressing direction throughout the length of the workpiece as shown in Figure 10. Then, the average equivalent plastic strains and the coefficients of variance were calculated over the cross-sectional surface and plotted over the length of the workpiece through 1, 2, 4 and 8 passes of ECAP as shown in Figure 11a,b, respectively.

**Figure 10.** Schematic illustration showing the cross-sections in the analysis for the steady-state zone.

**Figure 11.** The plots of (**a**) average equivalent plastic strains and (**b**) coefficients of variance on different cross-sections over the length of the workpiece for various numbers of passes.

Figure 11a shows that the average equivalent plastic strain increases with the increasing number of passes for all sections along the length of the workpiece. Moreover, during ECAP through one pass and two passes, the values of the average equivalent plastic strain are relatively constant along the length and decrease slightly near the front of the workpiece due to the lack of any constraint at the exit of the ECAP die. In consecutive passes with route *B*C, the maximum value of the average equivalent plastic strain lies at the back of the workpiece close to the load application point, and then it decreases with the distance away from the application point until it reaches a certain value and remains steady in the middle portion of the workpiece over a length of about 40 mm. It then becomes lower again near the front of the workpiece as is evident in the workpiece after ECAP for four and eight passes.

Figure 11b represents the degree of strain homogeneity in each cross-section over the length of the billet measured using the coefficient of variance, *CV<sup>ε</sup> <sup>p</sup>* , where a lower value indicates a better degree of homogeneity. These results show that the coefficient of variance decreases as the number of passes increases, where the *CV<sup>ε</sup> <sup>p</sup>* values for the workpieces after ECAP for four and eight passes are lower than 0.1. It is apparent that higher values of *CV<sup>ε</sup> p* exist at the front and back of the workpiece within distances of about 15 mm from either end for all conditions.

Finally, Figure 12 shows the equivalent plastic strain distribution contours along the length of the workpiece through ECAP after different numbers of passes. An inspection shows that there is a significant variation in equivalent plastic strain from the bottom to the top of the workpiece after ECAP for one and two passes as shown in Figure 12a,b, but this variation is gradually eliminated with the increasing number of ECAP passes. From the results plotted in Figures 11 and 12, it is concluded that the workpiece may be conveniently divided into three regions corresponding to the head, the intermediate steady-state zone and the tail. The steady-state zone is the region of reasonable homogeneity located between the non-uniformly deformed head at the front of the workpiece and the tail at the back of the workpiece. This is evident in Figure 12d, where the steady-state zone is clearly marked.

In this study, the length of the steady-state zone of the Cu-0.1 wt% Zr alloy after ECAP through eight passes is ~40 mm in the middle portion of the billet between the head and the tail. Minimizing the lengths of the non-uniform regions in the head and tail is of technological importance in order to maximize the length of the useful homogeneous portion of the ECAP sample. It was reported earlier that, if the angle Ψ is maintained constant, the non-uniform part in the head is longer in a strain-hardening material than that in a perfectly plastic material [8]. Furthermore, if the ECAP facility has a capability of applying back pressure, this will contribute to the uniformity of the metal flow during the ECAP operation because the application of back pressure increases the homogeneity and leads to a filling of the outer corner of the die so that the deformation zone becomes closer to that of a localized shear band [31–33].

**Figure 12.** Equivalent plastic strain distribution contours along the length of the workpiece through ECAP for (**a**) 1 pass, (**b**) 2 passes, (**c**) 4 passes and (**d**) 8 passes.

The goal in the design of the ECAP process is to maximize the magnitude of the strain, to minimize the inhomogeneity and to minimize the force that is needed. In order to achieve these objectives, it is important to obtain an understanding of the effect of various parameters, including the die geometry, the material properties, the processing conditions and their interactions on the overall deformation behavior. The present investigation shows that the use of 3D FEM simulation provides an important contribution toward achieving a much improved understanding of the ECAP deformation process.

#### **4. Conclusions**


steady-state zone and the tail. In this study, the steady-state zone extends over approximately 40 mm in length along the longitudinal axis of the workpiece.

**Author Contributions:** Methodology, J.W.-N., N.N., C.K. and T.G.L.; validation, J.W.-N., N.N. and C.K.; investigation, J.W.-N., N.N. and C.K.; resources, J.W.-N., N.N., C.K. and T.G.L.; data curation, J.W.-N., N.N. and C.K.; writing—original draft preparation, J.W.-N., N.N. and C.K.; writing—review and editing J.W.-N., N.N., C.K. and T.G.L.; supervision, J.W.-N., N.N., C.K. and T.G.L.; funding acquisition, J.W.-N., C.K. and T.G.L. All authors have read and agreed to the published version of the manuscript.

**Funding:** The work was supported by the European Research Council under ERC Grant Agreement No. 267464-SPDMETALS, the KMITL Research Fund under Project Number KREF01590 and Thammasat Postdoctoral Fellowship under Project Number TUPD12/2564.

**Data Availability Statement:** The raw and processed data generated during this study will be made available upon reasonable request.

**Acknowledgments:** We appreciate the assistance of the staff in the Mechanical Engineering Laboratory in KMITL.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**

