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Advances in Materials Science and Engineering
Volume 2019, Article ID 5682585, 6 pages
Research Article

A New Flow Line Function for Modeling Material Trajectory and Textures in Nonequal-Channel Angular Pressing

1Mechanical Engineering Group, Faculty of Engineering, University of Kurdistan, PB 416, PC 66177-15175, Pasdaran Boulevard, Sanandaj, Kurdistan Province, Iran
2Université de Lorraine, CNRS, Arts et Métiers ParisTech, LEM3, F-57000 Metz, France
3Laboratory of Excellence on Design of Alloy Metals for Low-Mass Structures (DAMAS), Université de Lorraine, Lorraine, France

Correspondence should be addressed to A. Hasani; moc.liamg@namrainasah

Received 8 October 2018; Accepted 26 November 2018; Published 9 January 2019

Academic Editor: Ivan Gutierrez-Urrutia

Copyright © 2019 A. Hasani et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


One of the most applied severe plastic deformation processes is ECAP (equal-channel angular pressing) which is suitable to produce ultrafine-grained metallic materials with high mechanical performance. A variant of the ECAP process was proposed in 2009, which consists in reducing the diameter of the exit channel of the die; it is named the nonequal-channel angular pressing (NECAP) process. A flow line function was also proposed to describe the material flow and the deformation field during NECAP. In the present work, an improved version of that flow function is presented containing two additional parameters compared to the previously proposed function. The new parameters permit to control precisely the shapes and the positions of the flow lines. The new flow function was applied to 90° NECAP of commercially pure aluminum to characterize the deformation field and the extent of the plastic deformation zone. The crystallographic texture evolution is also simulated using the new function. Excellent agreements with experiments were obtained for both the flow line trajectories and the crystallographic texture.

1. Introduction

Severe plastic deformation (SPD) processes have been recognized in the last decades as the most feasible and convenient methods to achieve submicron grain-sized bulk materials with superior strengths [1]. Among the most studied SPD techniques, equal-channel angular pressing (ECAP) [2] is one of the most promising and practicable processes to achieve ultrafine-grained (UFG) microstructures [3]. The ECAP process fulfills practically almost all expectations of an SPD process: effective grain refinement, improved strength, and quite homogeneous deformation in the product. One deficiency of ECAP is that several passes are needed to achieve the desired microstructure. It is therefore practical to reduce the number of ECAP passes to obtain the same stage of grain refinement. For this purpose, a modified ECAP process has been proposed [4, 5], in which the thickness of the exit channel (c) is smaller compared to the width of the entry channel (p); it is named nonequal-channel angular pressing (NECAP). A similar method but with inverted ratio between the entrance and exit channel dimensions has been already employed in the continuous confined strip shearing process of sheets [6, 7]. The disadvantage of NECAP is that only one pass can be done. Nevertheless, if the outgoing channel is much smaller, very large strains can be obtained in a single pass. This is quite evident from the shear strain formula developed for 90° NECAP in references [4, 5]:

For example, for an exit channel 20 times smaller than the entry channel, the shear deformation is . It is actually possible to carry out such an experiment successfully [8]. At such a high strain, the material can reach the limiting steady stage with minimum grain size, replacing 10 ECAP steps.

Equation (1) gives only the total value of the strain if the deformation is approximated by ideal simple shear acting on the plane of intersection between the two channels. The deformation field, however, is more complex and not restricted to a single plane within the die [9]. For describing the deformation field, the flow line technique was applied in reference [9] for ECAP and was extended to the 90° NECAP process in references [4, 5], where the following flow function was proposed:where defines the starting coordinate of a chosen flow line initially passing the position (Figure 1) and is a parameter that controls the shape of the flow line in the deformation zone, basically accounting for the rounding of the flow line. The flow function defined in equation (2) was applied in reference [4] to describe the material trajectory and the texture evolution in commercially pure aluminum. It will be shown in the present work that, by introducing two additional parameters, this flow function can be significantly improved to obtain much better results for both the material trajectory and the texture evolution.

Figure 1: (a) Geometry for positioning the initial and end points of the flow lines in an NECAP die. The effect of the (b) parameter on the shape of the flow line that enters at and (c) parameter. The geometry of the NECAP die is and .

2. The Modified Flow Line Function

In order to increase the precision of the function presented in references [4, 5] and described above, a modification of the flow function is proposed here for the deformation process in an NECAP die. The modified flow function is the following:

The two additional parameters are and (compared to equation (2)). Each of them has a specific role in controlling the shape of the flow line; the parameter is adjusting the exit position of the flow line for a fixed , while permits to shift the entire flow line parallel to the entry line for a fixed value. The reference system is defined the same way as for equation (2); the origin of the coordinate system is fixed at the lower left corner of the die (Figure 1). In the particular case when and are equal to 1, equation (3) returns equation (2).

The equation of the flow line is obtained for a flow line entering at the position when is constant: . Along the flow line, plastic deformation starts at an initial point. The coordinates of this point have to satisfy equation (3), leading to the following relation:

Thus, while can be chosen free in the range , the choice of depends on the outgoing channel thickness and also on the parameter . For , the flow line begins at the same position as the upper part of the exit channel, while for , it begins at a higher position. For , it is situated below. Examples are shown in Figure 1(b). Therefore, one can position the beginning of the flow line readily in accordance with the experimental flow line, which is not possible in the previous flow function, where the starting point of the flow line has to be at the same vertical distance from the bottom of the channel: at the distance of the diameter of the exit channel.

The effect of the parameter is different: for a fixed value, the exit part of the flow line is shifted upwards when and shifted down when (Figure 1(c)). So, with the help of the parameter, the exit part of the flow line can be located at the proper position according to the experimental flow line. Therefore, the value of the parameter can be obtained from equation (3), by using the vertical position (Figure 1(a)) of a selected flow line:

In the following, the deformation field will be determined from the new flow function. Using the principles of two-dimensional fluid mechanics, an admissible velocity field can be defined from the flow function as follows:

The parameter can be determined from the incoming velocity of the material at the initial point , where :

Subsequently, the velocity gradient components, , are obtained by partial derivation of the velocity field (equation (6)). The nonzero components are as follows:

The other components of the velocity gradient tensor are null. This velocity gradient tensor can be directly used in a polycrystal plasticity code to impose the NECAP deformation on the polycrystal incrementally. During such a calculation, the polycrystal has to stay on the flow line, so its displacement can be calculated incrementally using the velocity field given by equation (6). In this way, the evolution of the crystallographic texture can be simulated by updating the orientations of the crystals that compose the polycrystal during its passage along the flow line. After identifying the three parameters of the flow function given in equation (3), two results can be obtained at the same time:(i)The shapes of the flow lines can be depicted for any starting point using equation (3)(ii)The evolution of the crystallographic texture can be simulated using the velocity gradient defined by equation (8)

3. Analysis of the Flow Lines in NECAP of Aluminum

Using the procedure presented in Section 2 above, the new flow function was applied on the same NECAP experiment as in reference [4], and the new results are displayed in Figure 2(a) in comparison with the previous one, shown in Figure 2(b). The parameter values are given in Table 1. As can be seen in Figure 2, the new flow function proposed in equation (3) describes the trajectory shapes very precisely, while the previous function (equation (2)) shows significant differences in the plastic strain region, with increasing deviations towards the bottom of the die.

Figure 2: Experimental flow lines (circles, taken from reference [4]) for NECAP of Al, p/c = 2, together with the fits obtained by the newly proposed flow line function (a) (equation (3)) and by the previous function (b) (equation (2)) identified by continuous lines. The plastic deformation zone is displayed by dash-dot lines, which is limited by the zone situated between 1% and 99% of the total strain in one pass. The dotted line identifies the position of the maximum strain rate.
Table 1: The parameter values and the total von Mises strains for the three flow lines shown in Figure 2.

Another difference between the previous and present flow functions is that the size of the plastic deformation zone is larger with the new function. This can be verified in Figures 2(a) and 2(b) where the isolines identified with 1% and 99% show the position of the starting of the plastic zone and its ending, respectively. Here, the percentage is expressed with respect to the total accumulated plastic strain that a material element experiences during its passage in the die. The total accumulated plastic strains along the three indicated flow lines in Figure 2(a) are given in Table 1. They are slightly smaller than those for the previous flow function (where they were 1.165, 1.164, and 1.165, respectively). There is no significant difference for the locations of the maximum strain rates which are indicated with dotted lines in Figure 2.

4. Crystallographic Texture

Crystal plasticity calculations—for texture evolution along a trajectory line—can be done directly using velocity gradient expressions (equation (8)) as input in a polycrystal plasticity code. Successful modeling of the texture development can provide support for the proposed flow function, so we have carried out texture simulations using the velocity gradient defined in equation (8) for the Al NECAP testing originally published in reference [4]. The viscoplastic self-consistent (VPSC) model was employed [10] in its version further tuned with the help of finite element results in references [11, 12].

The material had some relatively weak starting texture (Figure 3(a)), comprising mostly from the cube component, which was discretized into 2000 grain orientations. The deformation texture after one NECAP pass experiment is shown in Figure 3(b).

Figure 3: {111} pole figures of the initial texture (a) and the deformation texture after one NECAP pass (b). SPN is the shear plane normal. The key figure of the ideal orientations of shear textures of the NECAP process is presented in (c) for p/c = 2. Circles in (c) represent the cube orientation.

The 12 {111}<110> type slip systems were used with a strain rate sensitivity index of 0.166. The interaction parameter (α) in the VPSC localization equation was taken as 0.7 (see the original publication for the meaning of α [11]). Strain hardening was simulated using the Zhou et al. [13] approach. However, the simulated textures turned out to be very slightly sensitive to hardening. The reason for this was that the sample was already in the hardened state before NECAP, so not much strain hardening could take place during further deformation by NECAP. The shape of the grains was already of pancake type before testing, flattened in the TD plane. Using the experimental data, the principal axes of the grains were taken as 1.5, 1.5, and 0.444 (relative values), in the ED, ND, and TD directions, respectively. Both the preceding flow function and the present new one were employed to simulate the deformation texture after one NECAP pass for the p/c = 2 geometry for flow line number 2. The simulation results are displayed in Figure 4. The plotting of the simulated textures was done for a Gaussian spread of 10° around each orientation using the JTEX software [14].

Figure 4: Simulated textures in {111} pole figure using the old flow line function (a) and the newly proposed one (b), in comparison with the experiment (c). SPN indicates the ideal shear plane normal, and C is the exact position of one of the {111} reflections of the C texture component.

As can be seen in Figure 4, the simulation reproduces the experimental features faithfully with the new function. Two peaks were selected to identify the position of the texture in Figure 4: the shear plane normal position, where several ideal orientations coincide, and the C orientation at the periphery of the pole figure (see the positions of the ideal components in Figure 3(c)). The peak positions are in exact positions compared to the experimental texture in Figure 4(a). However, when the previous flow function is used, a deviation of about 7° is observed in Figure 4(b); the simulated texture is rotated in the anticlockwise direction. There is a significant difference also in the relative peak intensities; the new flow function reproduces better the relative peak intensities.

5. Conclusions

The main objective of this work was to propose a new flow line function for a more precise modeling of the deformation field in nonequal-channel angular pressing. The modification implies two additional parameters; each of them has a geometrical meaning. The new function was tested on NECAP processing of pure Al. The presented flow line function can be used not only for NECAP: the formation of chips during a machining process is similar to NECAP, but also for a better description of the machining process. The previous flow function approach was already employed in machining [15]. The new function should lead to better results.

The following main conclusions can be drawn from the results:(1)The proposed flow function can describe the material flow much better than the previous one presented in references [4, 5]. It is capable of capturing the shape of the whole trajectory of a material point passing through the plastic deformation zone. The new flow function was successfully fitted to the experimental flow lines obtained by a 90° NECAP test on aluminum [4]. It has been found that the plastic deformation zone in the NECAP process was wider than the one predicted by a previous study [4].(2)The new flow function is capable of describing precisely the textures that develop during the deformation of the material along the flow line in NECAP.

Data Availability

All data, including the flow line function, the velocity gradient file, the measured texture data, and the simulated textures, are available by directly contacting Prof. Laszlo S. Toth or Dr. Arman Hasani at the following e-mail addresses: or

Conflicts of Interest

The authors declare that they have no conflicts of interest.


This work was supported by the French State through the program “Investment in the Future” operated by the National Research Agency (ANR) and referenced by ANR-11-LABX-0008-01 (LabEx DAMAS).


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