Tailoring Deformation Homogeneity of WE43 Alloy through Strain Rate Sensitivity and Back Pressure during Equal Channel Angular Pressing

Equal channel angular pressing (ECAP) of the magnesium alloys at room temperature owing to their limited workability is challenging. Successful ECAP processing of WE43 magnesium faces two main difﬁculties, heterogeneous distribution of the strain rate and also tensile strain accommodation on the top surface of the workpiece, leading to catastrophic segmentation of the alloy. In this paper, strain rate sensitivity (SRS) was studiedtoadoptaproperpreprocessingofthematerialbeforeECAPprocessing.TheSRSexponents,obtainedfromcompressiontests,revealedthatsolutiontreatmentreducedtheSRSofthealloy.Tomitigatestrain accommodation,anECAPcore-sheathconﬁgurationwasusedtoinducebackpressureforthesakeof deformation homogeneity improvement. A combination of experimental processing and 3D ﬁnite element method simulations was applied to the solution-treated WE43 alloy with different core sizes and sheath materials. By ﬁnding the optimized core sizes and sheath materials with higher strengths, the differences in microhardness and equivalent plastic strain were reduced. Besides, the adequate magnitude of back pressure and the imposed fully compressive stress prevent fragmentation of the WE43 core during ECAP. After stepwise modiﬁcations, the plastic strain inhomogeneity index decreased from 1.340 to 0.671.


Introduction
There has been a significant surge in the development of innovative methods for creating ultra-fine grain (UFG) materials due to their superior mechanical properties.Among numerous severe plastic deformation (SPD) techniques, equal channel angular pressing (ECAP) developed by Segal (Ref 1), is widely used as the most promising method for the enhancement of mechanical properties.However, for hard-todeform materials such as magnesium alloy with limited independent slip systems, ECAP at room temperature faces many challenges (Ref 2, 3).Appropriate preprocessing of the material and utilizing optimized ECAP parameters and techniques are troubleshooting keys to overcome this limitation.
The strain rate of plastic deformation of material during ECAP is affected by die design and processing parameters.Based on the volume flow rate rule in the main deformation zone (MDZ), the distance of material passing through the region near the inner angle is shorter than the region near the outer curvature of the die per unit time (Ref 4).This leads to a heterogeneous distribution of strain rates along the workpiece undergoing ECAP processing.Consequently, material response under mechanical force at various strain rates is an essential criterion concerning severe plastic deformation.
Strain rate sensitivity (SRS) is described as the variation of flow stress with respect to changes in strain rate, and measured by the SRS parameter, m (m= dlnr/dln _ e), where r and _ e are flow stress and strain rate, respectively (Ref 5).SRS affects the mechanical response and consequently plays a crucial role in the deformability of the material.Furthermore, the material responses to variations in strain rate are important in comprehending dynamic deformation.Based on the literature (Ref 6-9), thermo-mechanical history, grain boundaries, precipitates, and alloying elements influence the strain rate dependency of mechanical properties in magnesium alloys.Recently, some studies reported that WE43, as an age-hardenable magnesiumbased alloy containing rare earth elements such as yttrium (Y) and neodymium (Nd), exhibited strain rate sensitive deformation.According to these studies, SRS in the WE43 alloy was attributed to the competition between contributor deformation mechanisms including, basal and non-basal slip and twining, as the strain rate was changed ( Ref 10,11).In terms of the mechanical properties of the WE43 alloy, it has been reported that strain to fracture increased from 0.17 to 0.22 as the strain rate decreased from 10 À2 to 10 À4 s À1 in compression tests at room temperature (Ref 12).Nonetheless, the effects of microstructure including various precipitates of the alloy on strain rate sensitivity need to be clarified within the quasi-static range.Aiming to gain a material with desirable condition for ECAP processing, the effects of precipitates on the SRS of the WE43 alloy were investigated as a preliminary step.
Following, the deformation homogeneity by inducing back pressure during ECAP processing was studied.It has already been shown that non-uniform deformation leads to stress-strain concentration on the workpiece surface during ECAP (Ref [13][14][15].In fact, the initiated crack at the upper surface of the workpiece resulted in the occurrence of catastrophic cracking and full segmentation of the billet ( Ref 13,[16][17][18][19][20][21][22].On the other hand, the formation of a corner gap at the outer surface of the die is another source of heterogeneity during ECAP of materials with high strain hardening rates (Ref 23).The application of sufficient backpressure has been introduced as a high functional factor to prevent deformation heterogeneities (Ref 24,25).Over the past decade, a procedure, named the core-sheath method (Ref 26), has gained much attention for its simple procedure to introduce back pressure.Figure 1(a) schematically shows a core-sheath billet used for ECAP processing.
The magnitude of back pressure in core-sheath ECAP was analytically established by Derakhshande (Ref 22), as follows: where m and YTS sheath are shear factor and yield strength of the sheath after ECAP, respectively, and r is the radius of the billet and l is the length of the billet between the end of the billet and the calculation point.According to Eq (1), a higher magnitude of back pressure could be achieved by increasing YTS sheath .In addition, a higher value of l increases the back pressure.Since the length of the ECAP channel is limited, as the core length specimen decreases the back pressure increases.
Recently, successful application of the core-sheath ECAP method was reported on the processing of commercially pure magnesium (Ref 27), commercially pure titanium (Ref 28), and NiTi alloy (Ref 26) at room temperature.In our earlier works (Ref 29,30), an optimized core-sheath method, was exploited to avoid damage accumulation of the WE43 alloy during ECAP at room temperature.However, comprehensive details on the back pressure calculations and core-sheath configuration optimization were not provided and were deferred to the next paper.This paper expounds on the material preprocessing and modification route of the ECAP technique to attain the optimized core-sheath configuration used in our previous investigations.
As mentioned above, deformation homogeneity, affected by back pressure, is an important processing character that affects the damage accumulation and fracture of the material during ECAP (Ref 31).The distribution of hardness and plastic strain over the workpiece determines the deformation homogeneity of processing.Direct measurement of strain and its distribution in the sample during the ECAP procedure is challenging.However, finite element method (FEM) simulation can estimate the strain distribution (Ref 22,26,32) and stress state (Ref 33) during ECAP.
In the present work, to improve deformation homogeneity, the effects of SRS and back pressure during ECAP processing were studied.Firstly, the effects of microstructure on the SRS of WE43 alloy were studied utilizing compression tests within the quasistatic strain range at room temperature.After desirable preprocessing of the material, the deformation homogeneity of the WE43 alloy was investigated using experimental processing and 3D FEM simulation of core-sheath ECAP at room temperature.The effect of back pressure on hardness and microstructural changes was also studied.In addition, deformation inhomogeneity indices of plastic strain and microhardness were obtained along the diameter of the core workpieces.

Materials and Methods
The WE43 alloy with a chemical composition of Mg-4.47 wt.% Y-2.42 wt.% Nd-0.29 wt.% other RE was supplied in asextruded condition.The solution treatment was conducted at 525 °C for 8 h, with subsequent quenching in a solution containing 80 °C water and 10 wt.% NaCl.
Cylindrical samples of as-extruded and solution-treated alloys with a diameter of 3 mm and length of 4.5 mm (l/d = 3/ 2) for the compression tests were cut using wire electrodischarge machining along the extrusion direction of the material.The compression tests were performed under quasi-static loading over the range of strain rates from 2:4 Â 10 À4 to 1:3 Â 10 À2 s À1 at room temperature.
A dynamic explicit 3D FEM model shown in Fig. 1(b) was utilized to optimize the core-sheath configuration of ECAP.In order to investigate the deformation homogeneity of the core, longitudinal and radial paths were considered, as shown in Fig. 1(c).According to Eq (1), aiming to reach a full compressive stress state, systematic stepwise simulations were performed by changing the core size and sheath material.Since core diameter was not included in Eq (1), Step 2 of the optimization was designed and carried out to investigate the effect of core diameter on stress and strain distributions.Figure 1(d) shows the three steps of optimization of core-sheath configuration.It should be noted that these parameters changed in such a way that could be practicable under experimental ECAP processing.All simulations were conducted using a commercial software named ABAQUS.Materials behavior, including density, elastic and plastic properties of core and sheath materials, was imported to the software.In these simulations, both core and sheath with cylindrical shapes were assumed deformable parts, while the die and punch were modeled as discrete rigid shells to decrease computational cost.Notably, each deformable part was assigned to a solid homogeneous section.The linear bulk viscosity parameter and quadratic bulk viscosity parameter were set at 0.06 and 1.2, respectively.A tangential interaction with a coefficient of 0.8 was adjusted between the surfaces of the core and sheath to prevent relative displacement.The friction coefficient of 0.01 was considered between the billet and the inner surface of channels, and the constant value of 0.1 mm/s was assigned to the ram speed.Due to the possible distortion of mesh elements during simulations, the arbitrary Lagrangian-Eulerian (ALE) adaptive re-meshing method was employed to increase the accuracy of the results.Besides, mesh sensitivity diagrams were used to determine the optimum mesh size.It is worth mentioning that verification of FEM results was carried out and reported in the previous work (Ref 29).
To save time and money, ECAP was performed employing initial (Billet 1) and optimized (Billet 7) core-sheath configurations at room temperature.For the preparation of core-sheath billets, round bars of solution-treated WE43 with a diameter of 5 and 10 mm and a length of 40 and 60 mm were embedded in 1.5920 and AISI 1015 commercial steels with 20 mm in diameter and 125 mm in length, respectively.Two types of core-sheath billet were schematically represented in Fig. 2. The ECAP was conducted using hydraulic press machine and a die with internal and external channel angles of u = 90°and W = 20°, respectively.In terms of ram speed, the rate was adjusted to the constant value of 0.1 mm/s, which was accordant to the simulation condition.Aiming to lubricate the internal surface of die channels, a molybdenum disulfide spray was utilized.Images of billets after ECAP processing are shown in Fig. 2(b) and (d).Details of microhardness measurements, tensile tests, the x-ray diffraction method, and microscopic inspections are explained in the previous paper [29].

Effects of Solution Treatment on Microstructure
The effects of solution treatment on the microstructure and phases of the WE43 alloy are demonstrated in Fig. 3.The SEM images were captured in the transverse cross section perpen-dicular to the extrusion direction.The XRD pattern and SEM image demonstrated that three kinds of precipitates were presented in the as-extruded alloy.Micro-constitution of different particles obtained using energy dispersive spectroscopy (EDS) is listed in Table 1.The plate-shaped Mg 14 YNd 2 particles shown in Fig. 3(d), with corresponding peak at 2h = 40.9°,were observed along the grain boundaries of the as-extruded alloy.The Mg 14 YNd 2 is a metastable phase that forms during the hot deformation of the WE43 alloy (Ref 34).Since the alloy was supplied in extruded condition, this Mg 14 YNd 2 might have been formed on the grain boundaries during hot extrusion.However, after solution treatment, these plates were completely dissolved into the a-Mg matrix.On the other hand, globular Mg 41 Nd 5 and cuboid Mg 24 Y 5 , which are  2 summarizes the shape, average size, and volume fraction of each precipitate in as-extruded and solution-treated alloys.

Compressive Yield Strength and Strain to Fracture
Figure 4 shows the mechanical properties of the WE43 alloy under different strain rates of compression tests at room temperature.As shown in Fig. 4(a) and (b), the as-extruded alloy exhibited a drastic drop in yield stress (YS) and ultimate compressive strength (UCS) as the strain rate increased from 2:4 Â 10 À4 to 1:5 Â 10 À3 s À1 .However, with a further increase in strain rate to 1:3 Â 10 À2 s À1 , YS and UCS increased to 208 and 368 MPa, respectively.Meanwhile, after solution treatment, the mechanical response to strain rate was slighter than in the as-extruded condition.In other words, the mechanical strength of the solution-treated alloy was marginally changed with strain rate.As shown in Fig. 4(c) and (d), the YS and UCS of the solution-treated alloy against strain rate approximately remained unchanged.Due to the increase in strain rate from 2:4 Â 10 À4 to 2.6 Â 10 À3 s À1 , strain to fracture of as-extruded alloy increased by 19%, whereas solution-treated alloy exhibited only a 7% increase.
The fracture surfaces of the compression samples that were captured in normal direction to the fractures surfaces, with the strain rate of 2.6 Â 10 À3 s À1 , are shown in Fig. 4(e) and (f).Regarding different mechanical properties, fracture mechanisms were changed in the as-extruded and solution-treated alloys.In the as-extruded alloy, trans-granular fracture led to the formation of large smooth cleavage planes.After solid solution at 525 °C for 8 h, the fracture mode of the WE43 alloy was quasi-cleavage.Although compared with as-extruded alloy, more tearing ridges and some dimples were observed on the rough fracture surface of the solution-treated alloy.The strain rate sensitivity exponents, m, obtained from compression tests are shown in Fig. 5.The results show that the mechanical response to strain rate was changed after solution treatment.In strain rate from 2.4 Â 10 À4 to 1.5 Â 10 À3 s À1 , the negative m 1 ¼ À 0:279 showed that the as-extruded alloy was softened due to an increase in strain rate.The positive m 2 implies that YS increased with further increase in strain rate to 1.3 Â 10 À2 s À1 .Conversely, the solution-treated alloy exhibited a negative SRS at first.Notably, the magnitudes of m 0 1 ¼ 0:092 and m 0 2 ¼ À 0:075 were much lower than those of the asextruded alloy.
As mentioned before, the strain rate is heterogeneously distributed on the workpiece during ECAP.For ECAP processing of the WE43 alloy as a strain rate sensitive material, intense SRS of the alloy may lead to localized deformation and subsequent failure.Regarding microstructural changes after solution treatment shown in Fig. 3, precipitates remarkably affected the SRS.Considering the SRS exponents shown in Fig. 5, the as-extruded alloy with larger precipitates, a higher volume fraction, and grain boundaries with respect to the solution-treated alloy, exhibited more intense SRS.Indeed, solution treatment moderated the strain rate sensitivity of the alloys.Besides, the fracture surfaces shown in Fig. 4, demonstrated that the as-extruded alloy was more brittle than the solution-treated alloy.Altogether, aiming to minimize the probable localization of plastic strain during deformation, the solution-treated alloy with low SRS was adopted as the core workpiece for ECAP processing at room temperature.

Optimization of Core-Sheath Configuration
Figure 6 depicts the distribution of maximum principal stress on the WE43 core workpieces during ECAP on the route of three-step optimizing simulations.Tensile and compressive stresses were represented with negative and positive numbers, respectively.As expected and based on Eq (1), by increasing the distance to the tip of the core, the magnitude of the compressive stress was increased.Even though, the tensile stress of 175 MPa was imposed on the upper surface of Core 1.With constant core diameter and sheath material, shortening of core length resulted in a lower value of tensile stress.So, tensile stress decreased to 142 and 104 MPa in Core 2 and Core 3, respectively.The distribution of principal stress on Core 4 and Core 5 with respect to Core 3 demonstrated that various core diameters did not significantly affect the magnitude of tensile stress.Nonetheless, a shorter core diameter led to a more homogeneous distribution of principal stress and a shrinking stress accommodation area in Core 5. Worth noting that the higher strength of stainless steel than AISI 1015 steel (Ref 35) using the former steel as sheath material for Core 6 decreased imposed tensile stress to 45 MPa in the upper surface of the core workpieces.Changing the sheath material to 1.5920 steel with higher strength resulted in complete suppression of the tensile mode stress in Core 7. Consequently, the optimized core-sheath configuration of Billet 7 provided a fully compressive stress state during ECAP at room temperature.

Back Pressure Calculations
Calculating the imposed back pressure on the core workpieces requires the yield tensile strength of AISI 1015 steel and 1.5920 steel after ECAP.Tensile tests were carried out to obtain YTS of the sheath material in the exit channel of the die for use in FEM analysis.Figure 7 depicts the engineering stress-strain curves of the sheath materials before and after the ECAP at room temperature.After ECAP, strength significantly increased and strain to fracture decreased.YTS of the AISI 1015 steel and 1.5920 steel reached 670 and 940 MPa, respectively.Values of imposed back pressure along the length of the core were calculated according to Eq (1) and mentioned in Table 3. Imposed back pressure on the Core 1 varies between 251 and 484 MPa along the length of the core, while the magnitude of back pressure imposed on Core 7 is remarkably higher and increases from 461 to 678 MPa with increasing distance to the tip of the core.It should be noted that back pressure had a linear relationship with distance to the tip of the core.

Crack Closure During ECAP
FESEM micrograph in longitudinal cross section of Billet 1 after ECAP is shown in Fig. 8(a).A catastrophic crack in the upper surface of the Core1 workpiece led to full fragmentation of the alloy parallel to the shear direction.Figure 8(b) and (c) shows the optical microstructure of the alloy in zone I and zone II, respectively.Further elongation of the grains in zone I concerning zone II demonstrates the deformation inhomogeneity along the TD in the core workpiece.
Figure 8(d) shows the FESEM micrograph of Billet 7 in the longitudinal cross section after ECAP.A microcrack with an approximate length of 60 lm propagated and then closed on the upper surface of the Core 7 workpiece.In fact, the occurrence of crack closure prevented the full fragmentation of the alloy.Furthermore, optical micrographs of the alloy in zones III and IV, which are depicted in Fig. 8(e) and (f), respectively, showed that deformation inhomogeneity reduced in Core 7 compared to Core 1.The TEM bright-field images Core 1 and Core 7 after ECAP at room temperature are shown in Fig. 9(a) and (b).The arrows exhibit the accumulation of dislocations, which generated during the plastic deformation, were aligned with a preferred direction.Difference between the densities of dislocations of Core 1 and Core 7 clearly signifies that deformation condition was quite different.The low density of dislocations in Core 1 can be attributed to the mechanical energy consumption for crack propagation and full fragmentation of the workpiece.In comparison, owing to crack closure in upper parts of the Core 7, imposed energy stored in the material and subsequently resulted in generation of numerous dislocations.Figure 9(c) and (d) depicts micro-shear bands and the corresponding SAED pattern.The accumulation of high-density dislocations along the shear direction depicts considerable micro-shear bands generated after ECAP.Curving features of shear bands, as indicated by the arrows in Fig. 8(c), imply that the band boundaries effectively impeded the dislocation movements that harden the material.The distorted HCP lattice in the SAED pattern indicates a heavily strained structure of the WE43 core alloy after ECAP.

Microhardness Distribution on the Core
The distribution of microhardness along the length and diameter of the cores is shown in Fig. 10 (a) and (b), respectively.In the deformed zone, microhardness distribution is approximately homogeneous along the length of the core.The average value of microhardness was 15 HV higher in Core 7 compared to Core 1.Interestingly, more than 60 % of the core length experienced severe plastic straining during ECAP in both billets.By increasing the distance to the tip of the billet, microhardness value drastically decreased.Average microhardness in the non-deformed zone was predictably close to solution-treated alloy.The microhardness distribution was non-homogeneous along the core diameter, where microhardness increases from bottom to top surface of the core.Besides, the smaller microhardness differential along diameter demonstrates more homogenous deformation in Core 7 concerning Core 1.

Distribution of Equivalent Plastic Strain Along the Core
The distribution of equivalent plastic strain (PEEQ) on the core workpieces is shown in Fig. 11.PEEQ distribution was acquired throughout the center plane of workpieces, which provides a better presentation of strain distribution.Along the ED, there are three distinct deformation zones: non-deformed zone, MDZ, and deformed zone.The PEEQ imposed on the non-deformed zone was approximately zero and the drastically increased when the workpiece entered the MDZ.In the deformed zone, the PEEQ reached a steady state.However, PEEQ was increased from the bottom to the top surface of the core with an average of 1.06 in the deformed zone for both cores.Additionally, Core 7 had a smaller PEEQ differential between the top and bottom surfaces of the workpiece than Core 1.

SRS Considerations
The origin of the SRS arises from the interaction of dislocations with microstructural obstacles (Ref 36) and slip mode dependency (Ref 37) during plastic deformation.The asextruded alloy with 4.06 v.% of precipitates including Mg 14 YNd 2 , Mg 41 Nd 5 , and Mg 24 Y 5 exhibited a sharp change in SRS exponent at the strain rate of 1:5 Â 10 À3 s À1 , as shown in Fig. 5.In the as-extruded alloy with sophisticated microstructure, the increase in strain rate from 1:5 Â 10 À3 to 2:6 Â 10 À3 s À1 led to a drastic increase in the m value.Although, the solution-treated alloy with only 0.76 v.% of fine Mg 41 Nd 5 and Mg 24 Y 5 precipitates showed a slight difference in SRS at the same test condition.Admittedly, the presence of various precipitates with different shapes and sizes significantly affects the materialÕs sensitivity to strain rate.
Strain rate sensitivity influences the plastic deformation of the material during ECAP.According to the theoretical and FEM analyses (Ref 38,39), the major shear deformation of the workpiece occurs in MDZ during ECAP.Notably, the distance of the material passing through the inner corner is shorter than the outer corner in MDZ.Considering the volume flow rate rule in ECAP (Ref 4), the deformation time gradually decreases from the bottom to the upper surface of the workpiece.Moreover, as shown in Fig. 11, the magnitude of the PEEQ varies between 0.2 and 1.1 in the MDZ.Consequently, the strain rate of the plastic deformation heterogeneously distributes on the core workpiece in ECAP.Nevertheless, ECAP of a material with high SRS may lead to the localization of plastic deformation and subsequent failure.As shown in Fig. 4 and 5, the solution-treated alloy exhibited lower SRS with respect to the as-extruded alloy.Aiming to prevent the localization of plastic deformation, the solution-treated alloy was adopted for ECAP.

Back Pressure Effects
Severe straining of the material during ECAP at room temperature generates a high density of dislocations and grain boundaries (Ref 1).Furthermore, strain accumulation may lead to crack propagation and fragmentation of the workpiece.Accordingly, deformation homogeneity plays a key role in plastic deformation during ECAP.It was reported in the literature that the presence of a dead zone, called corner gap, is a source of deformation inhomogeneity during conventional ECAP.Formation of a corner gap may occur owing to lost contact between the billet and die wall at the outer curvature in MDZ ( Ref 23,[40][41][42].In this paper, FEM results indicated that using core-sheath configuration inhibited the formation of a corner gap in the WE43 core workpiece.Indeed, the core was not in contact with the die wall, since it was embedded in the sheath to serve the core-sheath billet.Moreover, the sheath acts During ECAP, shearing occurs within the workpiece when it passes through the intersection of the two die channels (Ref 33).When the workpiece enters the MDZ, its bottom surface is in compression and the upper surface is in tension.Consequently, the main middle area of the workpiece is subjected to pure shear.Under this condition, severe strong deformation is imposed on the material and the crystal lattice may be distorted severely (Ref 43).The distorted HCP crystal lattice of the WE43 alloy after ECAP was evinced by the SAED pattern which is shown in Fig. 9(b).On the other hand, the tension stress mode in the upper surface of the core may lead to the formation of cracks and segmentation.Owing to the low workability of WE43 magnesium alloy at room temperature (Ref 44), conventional ECAP processing leads to cracking and full segmentation of the material.The hydrostatic back pressure which is provided by core-sheath configuration could affect the segmentation of the WE43 core during ECAP.The effect of back pressure on the stress and strain distributions and deformation inhomogeneity is inspected in the following section.
Based on the literature (Ref 45, 46), lower part of the core may accommodate lower strain compared to upper parts.Moreover, Shahmir and his colleagues (Ref 26), who studied the flow behavior of Titanium using core-sheath ECAP, have associated this phenomenon to bending of the core.However, it has been demonstrated that magnitude of shear deformation increases with friction coefficient.Similarly, Fig. 11(b) demonstrates that the distribution of strain relies on the distance to the coreÕs top surface in Core 1.Nevertheless, after optimization of the core-sheath configuration, the distribution of the plastic strain was relatively homogeneous in deformed area of the Core 7, as shown in Fig. 11(c).
4.2.1 Influence of Back Pressure on the Principal Stress.Figure 12 represents the variation of maximum principal stress and back pressure imposed on the upper surface of the core workpieces (Path 1).Since no second punch was considered in the exit channel in simulations, the source of imposed back pressure was only the friction force between the billet and the inner surface of the die exit channel.Introducing back pressure on the billet significantly decreases tensile stress during ECAP (Ref 33).Higher negative back pressure values were calculated for both workpieces as the distance to the tip of the core increased.Owing to the higher YTS of the sheath and shorter length of core in Billet 7 concerning Billet 1, imposed back pressure on Core 7 was notably increased compared to Core 1. Variation of maximum principal stress on the Core 1 demonstrates that remarkable areas of the workpiece experience tensile stress during deformation.Tensile stress varies from 20 to 170 MPa on the upper surface of Core 1. Conversely, principal stresses were fully compressive over the entire length of the Core 7. Certainly, the adequate magnitude of back pressure changes the mode of tensile stress to a fully compressive stress distribution (Ref 43) on the Core 7. As a consequence, fully compressive stress distribution led to crack closure on the top surface of Core 7 which is shown in Fig. 8(d), while imposed tensile stress caused catastrophic segmentation of Core 1 which is shown in Fig. 8(a).Besides, a smaller differential of principal stress along the length of the workpiece demonstrates that plastic deformation was more homogeneous during ECAP of Core 7 concerning Core 1.
4.2.2Influence of Back Pressure on Equivalent Plastic Strain.To assess the influence of back pressure on the deformation homogeneity during ECAP at room temperature, the equivalent plastic strain was captured along various paths of core workpieces.The distribution of equivalent plastic strain along longitudinal and radial paths on core workpieces is shown in Fig. 13.From Fig. 13(a), the magnitude of the equivalent plastic strain along the length of Core 1, significantly varied from Path 1 to Path 3.Moreover, Path 3 is the first to enter the MDZ in both cores and clearly shows that there is a strong accumulated deformation on the top surface of workpieces.During ECAP, the workpiece typically experiences more severe plastic deformation at the internal corner, which causes the maximum plastic strain on the top surface.Indeed, imposed plastic strain increases from the bottom to the top surface of the core 1.Interestingly, the distribution of PEEQ along longitudinal paths of the Core 7, shown in Fig. 13(b), exhibited a similar pattern.However, fluctuation of the PEEQ along a single path was slightly more pronounced in Core 7. The same trend of strain distribution along the diameter of billet for different kinds of materials such as ZK60 magnesium alloy

Deformation Homogeneity of ECAP
A systematic investigation of deformation homogeneity during ECAP processing at room temperature was conducted using both experimental microhardness values and numerical results of equivalent plastic strain in the deformed zone of the core.Interestingly, the trend of PEEQ along the radial path of cores shown in Fig. 13(c) was confirmed with microhardness measurements which are shown in Fig. 10(b).This agreement between experimental and FEM results clearly showed that 3D modeling of core-sheath ECAP was accurate.Figure 10(  where HV and PEEQ denote values of microhardness and equivalent plastic strain, respectively.The deformation inhomogeneity indices for core workpieces are all listed in Table 4. HII and SII for Core 1 were obtained at 0.21 and 1.340, respectively.Owing to applying fully compressive stress on Core 7, both inhomogeneity indices decreased two times.This indicates that optimization of core-sheath configuration aiming to increase hydrostatic back pressure led to a significant increase in deformation homogeneity.Consequently, a combination of imposing fully compressive stress and homogeneous deformation inhibited catastrophic segmentation of the Core 7 during ECAP at room temperature.

Conclusions
''This study was aimed at development of the optimal microstructure of the WE43 alloy for core-sheath ECAP at room temperature.To achieve this, the influence of solution treatment on the strain rate sensitivity (SRS) of the alloy was assessed through comprehensive compression tests.Following this, the deformation homogeneity of the solution-treated WE43 alloy was thoroughly studied using systematic stepwise FEM simulations and experimental ECAP conducted at room temperature.Thereby, to modify the core size and sheath material, the effects of back pressure on the microstructural Journal of Materials Engineering and Performance evolution and deformation uniformity of the alloy were profoundly investigated.The key findings were as follows: • Compression tests revealed that the high-volume fraction and large average size of Mg 14 YNd 2 , Mg 41 Nd 5 , and Mg 24 Y 5 precipitates resulted in intense SRS in the as-extruded alloy.• According to the simulations of ECAP, higher YTS of the sheath, along with a shorter length and diameter of the core, drastically increased the magnitude of the back pressure.
• The deformation homogeneity of ECAP increased with the increasing magnitude of the back pressure.
• Inhomogeneity indices of microhardness and equivalent plastic strain decreased by 33% and 50%, respectively, indicating improved deformation homogeneity of ECAP processing.''

Funding
Open Access funding provided by Lib4RI -Library for the Research Institutes within the ETH Domain: Eawag, Empa, PSI & WSL.

Fig. 1
Fig. 1 (a) Schematic of a core-sheath billet; (b) the 3D FEM model of core-sheath ECAP; (c) schematic of longitudinal and radial paths of the core; (d) steps of optimization of billet configuration via systematic simulations

Fig. 5
Fig. 5 Strain rate sensitivity exponent (m) of the WE43 alloy over quasi-static range

Fig. 6
Fig. 6 Distribution of maximum principal stress across the middle longitudinal cross section of the WE43 core workpiece with various billet configurations

Fig. 7
Fig. 7 Engineering tensile stress-strain curves of sheath materials

Fig. 11 Fig. 12
Fig. 11 Distribution of the equivalent plastic strain on WE43 workpiece across longitudinal and radial cross sections of (a) and (b) Core 1, (c) and (d) Core 7

Table 2
Details of precipitates in the WE43 alloy

Table 4
Inhomogeneity indices of microhardness and equivalent plastic strain