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

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. +e new parameters permit to control precisely the shapes and the positions of the flow lines. +e 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. +e 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.


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].e 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 nonequalchannel 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].e 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. is 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 c � 20.05.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.e 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 x 0 defines the starting x coordinate of a chosen flow line initially passing the y � c position (Figure 1) and n is a parameter that controls the shape of the flow line in the deformation zone, basically accounting for the rounding of the flow line.e 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.

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.e modified flow function is the following: e two additional parameters are m and k (compared to equation ( 2)).Each of them has a specific role in controlling the shape of the flow line; the m parameter is adjusting the exit position of the flow line for a fixed k, while k permits to shift the entire flow line parallel to the entry line for a fixed m value.e 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 m and k are equal to 1, equation (3) returns equation (2).
e equation of the flow line is obtained for a flow line entering at the x 0 position when f is constant: f � (1 − (x 0 /p)) n .Along the flow line, plastic deformation starts at an initial (x 0 , y 0 ) point.e coordinates of this point have to satisfy equation (3), leading to the following relation: us, while x 0 can be chosen free in the range 0 ≤ x 0 ≤ p, the choice of y 0 depends on the outgoing channel thickness c and also on the parameter k.For k � 1, the flow line begins at the same position as the upper part of the exit channel, while for k < 1, it begins at a higher position.For k > 1, it is situated below.Examples are shown in Figure 1(b).
erefore, 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.
e effect of the m parameter is different: for a fixed k value, the exit part of the flow line is shifted upwards when m > 1 and shifted down when m < 1 (Figure 1(c)).So, with the help of the m parameter, the exit part of the flow line can be located at the proper position according to the experimental flow line.erefore, the value of the m parameter can be obtained from equation (3), by using the vertical position y E (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 f as follows: e λ parameter can be determined from the incoming velocity of the material v 0 at the initial point (x 0 , y 0 ), where Subsequently, the velocity gradient components, L ij � zv i /zx j , are obtained by partial derivation of the velocity field (equation ( 6)). e nonzero components are as follows: e other components of the velocity gradient tensor are null.is velocity gradient tensor can be directly used in a polycrystal plasticity code to impose the NECAP 2 Advances in Materials Science and Engineering deformation on the polycrystal incrementally.During such a calculation, the polycrystal has to stay on the ow line, so its displacement can be calculated incrementally using the velocity eld 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 ow line.After identifying the three parameters of the ow function given in equation ( 3), two results can be obtained at the same time: (i) e shapes of the ow lines can be depicted for any starting x 0 point using equation (3) (ii) e evolution of the crystallographic texture can be simulated using the velocity gradient de ned by equation ( 8)

Analysis of the Flow Lines in NECAP of Aluminum
Using the procedure presented in Section 2 above, the new ow 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).e parameter values are given in Table 1.
As can be seen in Figure 2, the new ow function proposed in equation (3) describes the trajectory shapes very precisely, while the previous function (equation ( 2)) shows signi cant di erences in the plastic strain region, with increasing deviations towards the bottom of the die.
Another di erence between the previous and present ow functions is that the size of the plastic deformation zone is larger with the new function.
is can be veri ed in Figures 2(a) and 2(b) where the isolines identi ed 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.
e total accumulated plastic strains along the three indicated ow lines in Figure 2(a) are given in Table 1.ey are slightly smaller than those for the previous ow function (where they were 1.165, 1.164, and 1.165, respectively).ere is no signi cant di erence for the locations of the maximum strain rates which are indicated with dotted lines in Figure 2.

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].e viscoplastic self-consistent (VPSC) model was employed [10] in its version further tuned with the help of finite element results in references [11,12].e material had some relatively weak starting texture (Figure 3(a)), comprising mostly from the cube component, which was discretized into 2000 grain orientations.
e deformation texture after one NECAP pass experiment is shown in Figure 3(b).
e 12 {111}<110> type slip systems were used with a strain rate sensitivity index of 0.166.e 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.e 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.e 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.
e simulation results are displayed in Figure 4. e plotting of the simulated textures was done for a Gaussian spread of 10 °around each orientation using the JTEX software [14].
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)).
e 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.ere is a significant difference also in the relative peak intensities; the new flow function reproduces better the relative peak intensities.

Conclusions
e 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.
e modification implies two additional parameters; each of them has a geometrical meaning.e new function was tested on NECAP processing of pure Al. e 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.
e previous flow function approach was already employed in machining [15].
e new function should lead to better results.2)) identified by continuous lines.e 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.e dotted line identifies the position of the maximum strain rate.(1) e proposed ow function can describe the material ow 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.e new ow function was successfully tted to the experimental ow 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) e new ow function is capable of describing precisely the textures that develop during the deformation of the material along the ow line in NECAP.National Research Agency (ANR) and referenced by ANR-11-LABX-0008-01 (LabEx DAMAS).

Figure 1 :
Figure 1: (a) Geometry for positioning the initial and end points of the ow lines in an NECAP die.e e ect of the (b) k parameter on the shape of the ow line that enters at x 0 10 mm and (c) m parameter.e geometry of the NECAP die is p 20 mm and c 10 mm.

Figure 2 :
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.e 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.e dotted line identifies the position of the maximum strain rate.

Figure 4 :
Figure 4: Simulated textures in {111} pole gure using the old ow 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} re ections of the C texture component.

Figure 3 :
Figure 3: {111} pole gures of the initial texture (a) and the deformation texture after one NECAP pass (b).SPN is the shear plane normal.ekey gure 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.

Table 1 :
e parameter values and the total von Mises strains for the three flow lines shown in Figure2.