Analysis of deformation characteristics of Asymmetric Gradient Extrusion in preparing ultrane grained bulk materials

A novel method to prepare ultra-ne grained bulk materials, which is named Asymmetric Gradient Extrusion (AGE), is proposed in this article. In AGE, the cross section of extrusion channel is rectangle and two inclined planes are stagger arranged along extrusion direction. To realize repeating extrusion, the thickness of workpiece is restricted to equal the width of channel’s outlet. The deformation characteristics of AGE was rst theoretically analyzed by slip line eld. Then, these results were veried and supplemented by nite element analysis. The results reveal that deformation characteristics of workpiece in channel is largely related to the two inclined planes. Two independent deformation zones (TIDZ) can be formed with increasing distance between the two inclined planes. In addition, the accumulated strain generated in the TIDZ is complementary in value. Furthermore, the shearing effects of workpiece subjected in one pass extrusion has been improved. A more uniform strain distribution and higher shearing effect can be generated in AGE, which make it as a very promising method to produce ultra-ne grained bulk materials.


Introduction
Ultra-ne grained (UFG) materials, which exhibit excellent mechanical properties, have attracted great interesting during the past decades [1] . Important contributions in this context have been made by the development of so-called "top-down" techniques. Severe plastic deformation (SPD) method as one of these techniques is of more interest as it provides UFG or even bulk nanostructured materials (BNMs) with no contamination or porosity [2] .
Distinguished with other methods, most of SPD techniques realize grain fragment by repetitive processes.
Direct extrusion is a traditional metal forming method, which can generate a large strain, so that it has the potential to prepare ultra ne grained materials. The possibility of using extrusion to prepare ultra ne grained materials has been veri ed [20,21] . Using high ratio extrusion (HRE) [22] or hydrostatic extrusion with an extra high ratio [23] , favorable results in grain re nement were reported. However, these extrusion techniques all change the shape of the material, making it di cult to repeatedly extrude material with the same die. Moreover, the uniformity of strain distribution inside the material after extrusion is poor.
Therefore, some modi cations should be made to overcome these obstacles in extrusion to realize grains fragment.
Based on the conventical direct extrusion, this article proposes a new SPD method, which named Asymmetric Gradient Extrusion (AGE), to produce ultra-ne grained bulk materials. Comparing with traditional extrusion processes, a much more uniform strain distribution can be realized with this new SPD technique. The deformation characters of billet in one pass of AGE is theoretically analyzed by slip line eld. In addition, nite element analysis was proceeded to verify the theoretical results and the advantage of this SPD technique in producing ultra-ne grained materials is also emphasized and discussed.

Principle Of The Asymmetric Gradient Extrusion Technique
In most cases of extrusion, the die apertures are of symmetry character in geometry. Accordingly, deformation zones are formed symmetrically in workpiece and the aimed reduction in cross-section of workpiece can be realized. However, the variation in cross-section brings a huge di culty in re-extruding workpiece with the same die. To solve this issue, some amendments have been introduced in AGE. Firstly, the cross-section of die aperture and initial billet are all designed with rectangle pro le which are shown in Fig. 1(a). Therefore, the deformation of workpiece in channel can be treated approximatively as a plane strain case. The illustration of AGE process on a plane is displayed in Fig. 1(b). To make sure the shape of workpiece unchanged after extrusion, the thickness of initial billet is constrainedly equal to the width of channel outlet. As shown in Fig. 1(b), the extruded workpiece has the same shape and dimensions with its initial counterpart. Therefore, the repetitive extrusion of workpiece can be realized, and an aimed strain can be gained in workpiece with several cycles.
Another modi cation has been introduced in AGE is the asymmetric distribution of two extrusion bevels in channel which are generally symmetric designed in conventional extrusion. A vertical distance h is arranged between the two extrusion bevels, which has great effect on the deformation behaviors of workpiece in extrusion. Depends on the extrusion bevels, two deformation zones (DZs) would be formed, and the parameter h de nes the independence or interaction of this two DZs. The in uence of parameter h would be studied and discussed in the next parts. Besides h, the angles α and β of two extrusion bevels also can be assigned independently. These two parameters are largely related to the characters of deformation elds around the two extrusion bevels. However, the same angles are used in this article. The in uence of the two angles and parameters optimization of AGE will be dealt with in our next work.

Mechanism Of Grain Fragmentation
According to the characteristics of metal ow in channel, the workpiece can be divided into ve parts, which are schematic illustrated in Figure 2. In region , the material, assumed to be rigid, moves downward with a constant velocity V , which is equal to the punch velocity. The material undergoes plastic deformation in region and region . Depends on the value of parameter h assigned, the two deformation regions and can be interacted in region or totally independent. However, to realize our goal, the parameter h should be large enough to separate the two plastic deformation regions. Therefore, the material in region is ow rigidly. In region , the material moves to the exit of the die with constant velocity V without any further deformation. The volume of the material does not change before and after extrusion, so the speed relationship of the material in region and region can be expressed as V = (b/a)V . Region is the rst deformation zone, which is separated from region I and region III by entry and exist surfaces Γ 1 and Γ 2 . They belong to the slip line eld of rst deformation zone (DZ-1) and correspond to the beginning and end of deformation. Similarly, Γ 3 and Γ 4 are the entry and exist surfaces of the second deformation zone (DZ-2), which belong to the slip line eld of DZ-2.
Slip line eld (SLF) analysis is a traditional theoretical method to analyze metal forming characteristics.
With the SLF solution, the extrusion mechanism through wedge-shaped die has been well studied by many scholars [20,21] . Due to the similarity of die apertures, the SLF of AGE can be deduced which is illustrated in Figure 3. As discussed above, two independent deformation zones (DZs) should be formed in AGE, and correspondingly there are two isolated SLFs distributed along channel vertical direction.
As shown in Figure 3, the vertical distribution of these two SLFs reveals that the parameter h largely de nes the behavior between these two DZs. In order to form two independent DZs, a large value should be used for h. Due to the SLFs of these two DZs are very similar in characteristic, only the SLF of DZ-1 is discussed in detail below. The geometry of DZ-1 slip line con guration is de ned by the eld angles λ 1 , φ 1 , and θ 1 and the die angle α. The value of λ 1 is dependent on the friction condition at the interface between the die and the work material. As for a frictionless condition, the value of λ 1 equals π/4. The parameter φ 1 re ects the range of reductions for which the eld is valid. When the extrusion ratio tends to its maximum value (for which θ 1 =0), the SLF tends to its limit. When θ 1 →0, the point E 1 →D 1 , and the point C 1 →F 1 so that the slip line eld A 1 E 1 C 1 F 1 B 1 can be approximated with a triangular one A 1 B 1 C 1 . The kinematically admissible velocity eld associated with the SLF from DZ-1 becomes a triangular velocity eld as shown in gure 4(a). In this eld, the material moves parallel to the inclined wall with velocity V a .
In each point on the surfaces Γ 1 and Γ 2 the velocities V and V a suddenly change the direction.
For an arbitrary point P on AC, its corresponding point on BC is Q. As shown in Figure 2, the velocity at point P is affected by both the region velocity V and region velocity V a . The velocity decomposition of V and V a along the AC tangential and normal directions is shown in the gure 4(a). Due to the continuity of metal ow, V and V a on the AC normal direction must be equal, that is V an_AC =V n . However, as illustrated in the Figure 4(a), the velocities of V and V a on the AC tangential direction have different values, V at_AC >V t . The differences between V at_AC and V t de ne the velocity discontinuities, giving rise to tangential stress τ AC . Similarly, at point Q on BC, the velocity decomposition of region velocity V a and region velocity V in the BC normal and tangential directions is also displayed in the Figure 4(a). In addition, the velocity V at_BC and V t in the BC tangential direction are also different. Therefore, a stress τ BC along the BC tangential direction is formed at point Q.
The tangential stress distribution on the boundary of the two deformation zones in AGE is schematic illustrated in Figure 4(b). When an elongated and thin enough grain (obtained from a previous AGE pass) crosses one velocity discontinuity surface along which tangential stress exceeds strength in pure shear (τ > k), the grain strength will be overcome and the fragmentation process starts. Accordingly, the speed discontinuous surface plays an important role in grain fragmentation. The larger the number of discontinuous surfaces, the more effective the fragmentation is. Compared with traditional extrusion, AGE forms two independent DZs, increasing the number of discontinuous surfaces. Moreover, the shear stresses on the discontinuous surfaces of the two DZs are complementary. The uniformity of the shear stress formed by AGE is greatly enhanced. To verify the deformation characters of AGE as discussed in section 2, nite element analysis is proposed in this section. In order to highlight the in uence of parameter h, three analysis models with different value for h were established. These three analysis models are named Model-1, Model-2 and Model-3 respectively. Corresponding to the parameters of AGE as illustrated in Figure 1(b), their values used in the three nite element models are listed in Table 1. The rst model which named Model-1 is symmetry extrusion as the value 0 is used for h. In addition, the other models which named Model-2 and Model-3 are all asymmetry extrusion. However, a higher value h has been used in Model-3 which leads to generating two independent DZs. Distinguish with Model-3, the two DZs are interacted in Model-2 as a small value is assigned for h.

Finite Element Analysis
The commercial nite element software DEFORM was used for simulation. As discussed in section 1, all the simulations are proceed based on plane strain assumption. The punch and die are all set as rigid, and an elastic-plastic model is used for workpiece. A material AL-1100 was chose for workpiece from the DEFORM material library. Shear friction model with a coe cient 0.12 was used for all the contact surfaces. The punch moves along channel vertical direction with a velocity of 1mm/s, and a room temperature 20 o C was assigned for all simulations. Workpieces used in the three analysis modes with the same geometry dimensions and meshes, which are all meshed by mapping method and plane strain elements.

Characters of the deformation zones
Variation of mesh shape can be regarded as an indirect re ection of deformation characteristics. The results of mesh distortion with the three analysis models in extrusion process are shown in Figure 5. The signi cant in uence of parameter h on the interaction manners of DZs is schematically veri ed. In Model-1, two DZs are formed symmetrically about the vertical center of workpiece. With increasing h, the two DZs are gradually separated on the extrusion direction (Model-3). In addition, two independent DZs would be generated as a larger value has been assigned to h (Model-2). As these two independent DZs forming, a dead metal zone where materials ow rigidly is generating simultaneously. In terms of the total area of DMs in workpiece, Model-2 takes the rst place in value. It can be deduced that AGE (Model-2) is more e cient in workpiece deformation than other conventional extrusions under one pass.
The results of velocity eld in DZs of workpiece for these three analysis models are displayed in Figure 6.
As discussed in section 2, due to constancy of volume, materials at die outlet should be of the same velocity as an identical moving condition implied on punches in the three analysis models. Corresponding to the results shown in Figure 5, Model-1 which is a conventional symmetric extrusion has the smallest velocity variation area among the three models. Through transforming the symmetrical structure into vertical separation of channel, a larger velocity eld can be formed as displayed in Model-2 and Model-3.
However, the Model-2 which is an AGE emphasized in this article produces two independent velocity variation areas. It is meaningful that materials can be shaped more smoothly in AGE, and forming defects such as cracks can be largely restrained.
Slip line eld in workpiece can be re ected by the simulation results of equivalent strain rate which are displayed in Figure 7. According to the results, a higher strain rate has been generated at the entrance and exit of each deformation zones. With a larger parameter h is assigned, two independent DZs are formed in AGE (Model-2). It means that materials would be suffered shear effects with four times in one pass. In other words, more works can be impacted on grains fragment. Through AGE, the e ciency of extrusion on grains re nement has be largely improved. With simulation results of equivalent strain rate, the correctness of slip line eld as analyzed and discussed in section 2 has been veri ed.

Comparison of extrusion results
The contour plot of equivalent strain distribution in workpiece after extrusion is shown in Figure 8. The results show that for the symmetric extrusion of Model-1, the formed equivalent strain eld also presents a symmetric distribution. In addition, the equivalent strain eld formed by asymmetric extrusion, Model-2 and Model-3, is larger than that by symmetric extrusion in value. However, Model-2 which forms two independent plastic deformation zones, has more uniform distribution of equivalent strain than Model 3.
The results of equivalent stress distribution along the midline OP of workpieces are shown in Figure 9.
Corresponding to the characteristics of deformation, the equivalent strain formed by Model-1 is symmetrically distributed. The equivalent strain in two asymmetric extrusion, Model-2 and Model-3, is larger than that in symmetric extrusion with Model-1. In addition, the equivalent strain distribution formed by Model 2 is the most uniform, and its equivalent strain uctuation is less than 0.05. It should be emphasized that for asymmetric extrusion, the extrusion angles of two deformation zones can be independently adjusted to obtain higher and more uniform equivalent strain results.
The statistical results of the equivalent strain in workpieces after extrusion is shown in Figure 10. It can be seen from the results that the equivalent strain distribution of Model-2 is the most concentrated. The average values of equivalent strain of the three models are 0.518, 0.638 and 0.607 respectively. All the analysis results show that the asymmetric extrusion with two independent plastic deformation zones, Model-2, can form large and uniform equivalent strain in workpiece. Therefore, the advantage of asymmetric extrusion proposed in this paper in the preparation of ultra ne-grained materials has been veri ed.

Conclusions
A novel method for preparing ultra ne-grained materials which named asymmetry direct extrusion is presented in this article. In addition, the deformation characters of this process were analyzed by slip line eld and nite element method. Compared with the traditional extrusion method, AGE can form two independent deformation zones. Moreover, these two deformation zones are complementary. The distance between the two deformation zones is a key parameter. Too small distance will cause mutual interference between the two deformation zones and deteriorate the uniformity of deformation. The results of strain analysis after extrusion show that, compared with other extrusion methods, the internal strain after asymmetric extrusion is large and uniform. Asymmetric Gradient Extrusion has shown great advantages in the preparation of bulk ultra ne-grained materials.

Declarations Ethical Approval
The article follows the guidelines of the Committee on Publication Ethics (COPE) and involves no studies on human or animal subjects.

Consent to Participate
Not applicable. The article involves no studies on humans.

Consent to Publish
Not applicable. The article involves no studies on humans.

Authors Contributions
Junkai Fan and Chengpeng Wang conceived and designed the simulation process; Junkai Fan and Wu Zhao performed the simulations; Junkai Fan analyzed the data and wrote the paper; Wei Liu reviewed the paper. Equivalent strain distribution along OP line Figure 10 Statistical results of the equivalent strain in workpieces after the extrusion