Steady plastic flow of a polymer during equal channel angular extrusion process: experiments and numerical modeling.
Among various forming processes, equal channel angular extrusion (ECAE) allows to introduce very large plastic strains in the material without modifying the geometry of the sample. The process was first introduced in 1981 by Segal et al.  to refine the microstructure and then change the mechanical properties of the extruded material. Since then, this technique has attracted a lot of experimental, theoretical, and numerical investigations essentially focused on metallic materials in recent years [1-7]. A quick review of these investigations suggests this process originally devoted to metals provides a potential way for refining the microstructure of polymers. However both structure and polymer behavior are radically different from that of metals. The behavior of a polymer is obviously linked to its microstructure which can exhibit amorphous or semicrystalline forms. In comparison with metals, the yield strain of polymers is much larger. Furthermore, the yield behavior of polymers is significantly affected by hydrostatic pressure, strain rate, and temperature. This makes it difficult to transfer the findings from metals to polymers. Therefore, the benefits of ECAE for polymers must be accurately investigated. To date, very little attention was paid to the behavior of polymers extruded by ECAE [8-23]. This method was first applied to a polymer by Sue and Li . The authors have shown the effectiveness of the method in altering the morphology of a linear low-density polyethylene. The application of the process to a high-density polyethylene and a polypropylene was analyzed by Campbell and Edward . It was shown that bending plays a significant role in the extrusion of these two polymers. The resultant molecular orientation was investigated by X-ray diffraction. More recently, the morphological changes of a polypropylene subjected to ECAE were examined by Phillips et al. . Sue et al.  also observed bending of extruded polycarbonate samples. The extrusion temperature effect on bending was highlighted. The enhancement of the fracture resistance of extruded polycarbonate samples was pointed out by Li et al.  and Xia et al. . The dynamic mechanical behavior of an extruded semicrystalline polyethylene terephthalate was investigated by Xia et al. . An improvement of bending and torsional storage moduli was found. The structure modification of the same material was studied by utilizing X-ray diffraction and scanning electron microscopy in [15-17]. The effect of ECAE on the mechanical response of polymethylmethacrylate was investigated by Weon et al. . The most important result is an enhancement of both the rigidity and the fracture toughness. Creasy and Kang [19, 20] examined short glass fiber/polyacetal composites processed by ECAE. The technique was used to control the fiber length and orientation. The method was also used to modify the aspect ratio and orientation of clay nanoparticles in nylon-6/clay nanocomposites by Weon and Sue . Numerical investigations were recently achieved to understand the influence of key parameters (die geometry, friction, ram speed, extrusion temperature, number of extrusion sequences, and processing route) on the viscoplastic deformation behavior of polymers during ECAE process [22, 23].
ECAE impact on microstructure and on the enhancement of mechanical and physical properties of polymers reported in the very few investigations available in literature support the effectiveness of this process. The change in microstructure after ECAE is mainly related to three interdependent factors: (a) initial mechanical and physical characteristics of the material, (b) tool geometry and (c) process conditions (ram speed, extrusion temperature, lubrication,...). To achieve the desired properties after extrusion, a good knowledge of deformation mechanisms in the material is required. For this purpose, prediction of polymer behavior during ECAE is a major step to achieve process optimization.
The goal of this article is to investigate experimentally and numerically the behavior of a polypropylene processed by ECAE. The experimental load-ram displacement and deformation behavior of the polymer during ECAE are examined. The flow behavior and damage accumulation during tensile deformation were locally measured by a video-controlled technique. The tensile tests were conducted under a constant local true strain rate by regulating the cross-head speed and the results were used as input data for the modeling. The ECAE process was modeled by a 2D numerical simulation. The Gurson-Tvergaard damage formulation was also used to describe the material behavior. This approach allows the extrusion load, the strain, stress, and damage fields during ECAE to be evaluated. The effect of the use of a back-pressure during ECAE on damage development in the sample is also studied.
A commercial polypropylene (PP) homopolymer supplied by Goodfellow is used. The number-average molecular weight and the weight-average molecular weight are about [M.sub.n] = 25 kg/mol and [M.sub.w] = 180 kg/mol, respectively. The crystal content is about 55%. The glass transition temperature is about -20[degrees]C and the melting temperature is about 170[degrees]C. Since the glass transition temperature of the amorphous phase is lower than ambient temperature, it is in the rubbery state. PP plates with a thickness of 10 and 2 mm were obtained by compression molding. Samples for ECAE and videotraction tests were taken along the same direction of the plates, then surfaced on the cutting facets and polished.
The PP samples were processed by ECAE. The methodology is quite simple. It consists in a solid state extrusion of the sample through two channels with the same cross-section and presenting an elbow (Fig. 1). The sample is placed into the entrance channel and is pressed into the exit channel under the action of a ram, without varying its original cross-section. Since the sample moves as a rigid body inside the channels, the deformation is expected to be simple shear, uniformly distributed across a thin layer and located inside the elbow. In Fig. 1, the geometrical parameters [PHI], [PSI], and r are respectively the channel angle, the outer corner angle and the inner radius which here are chosen to be [PHI] = 135[degrees], [PSI] = 34[degrees], and r = 2 mm. The lengths of the entrance and exit channels are, respectively, 75 mm and 50 mm. The two channels have a cross-section area of 10 X 10 [mm.sup.2]. An electromechanical Instron (model 5800) testing machine was used to extrude the sample through the angular die. PP plates were machined into prismatic samples of 70 X 10 X 10 [mm.sup.3] (Fig. 2a). Note that the widths and thicknesses of the channels and the sample are the same, but with a clearance fit.
To determine the plastic flow behavior of the material, uniaxial tensile tests were achieved. True stress, true strains and volumetric strain were locally measured using a video-extensometer connected to the Instron operating system. The strain measurement device allows to capture the relative displacement of barycenters of seven markers made on the front side of the tensile sample. In this technique, strains are measured in the axial direction but also in the transverse direction in a representative volume element (RVE) strictly localized in the plastic instability region . Closed-loop regulation of the local axial strain at a constant rate is achieved in real time by the system. Once the components of the strain tensor are known, the true plastic dilatation strain [E.sub.kk.sup.p] can be calculated from:
[FIGURE 1 OMITTED]
[E.sub.kk.sup.p] = [E.sb.kk] - (1 - 2v)[E.sub.11.sup.e], (1)
[FIGURE 2 OMITTED]
where [E.sub.11.sup.e] = [[summation].sub.11]/E is the elastic axial strain, v is the Poisson's ratio, and E is the Young's modulus. It is assumed that the sample deforms isotropically in the two transverse directions (i.e., [E.sub.22] = [E.sub.33]). The true (Cauchy) axial stress [SIGMA.sub.11] was calculated from the total load F and the current cross-section area S:
[[summation].sub.11] = F/S. (2)
To deviate from the incompressibility assumption, the current cross-section area S was calculated as:
S = [S.sub.0]exp(2[E.sub.22], (3)
where [S.sub.0] is the initial cross-section area.
For the video-controlled tensile tests, samples presenting a specific geometry (Fig. 2b) were prepared. Indeed, a sufficiently large radius of curvature was retained to localize the deformation while keeping a low triaxiality in the zone of interest.
The load evolution with ram displacement during the ECAE pass under various ram speeds is plotted in Fig. 3. At the beginning of extrusion, the load quickly increases with ram displacement. Then, a significant drop followed by a fairly constant plateau value is observed. The plateau (steady-state) is linked to the uniformity of the deformation band size. The pressing speed only affects the peak load and has no major influence on the load during the steady plastic flow. Interrupted tests, also given in Fig. 3, show the deformation behavior of the sample during the process. When the material crosses the elbow, whitening is observed at the bottom surface of the sample. In literature, this phenomenon is generally associated with nucleation and propagation of voids with size higher than the light wavelength (typically 0.6 [micro]m). This whitening can be then attributed to damage occurring in the material and can be only activated in tension. Therefore, one may conclude that bending rather than shearing mechanism seems to act in this region. Campbell and Edward  have shown that varying the extrusion temperature from 0 to 110[degrees]C has no effect on the bending of high-density polyethylene and polypropylene. In a previous work dealing with an amorphous glassy polymer (polycarbonate) processed by ECAE, no cavitation was highlighted . The state of the amorphous phase and the intrinsic mechanical behavior (initial yield strain, hardening,...) significantly act on the mechanical behavior of the polymer sample during ECAE, and then on the propensity to develop shearing.
[FIGURE 3 OMITTED]
In Fig. 4, true axial stress and void volume fraction are plotted versus true axial strain. In this section, we are only focused in experimental data given by solid lines. The void volume fraction is determined using relation (1) and assuming plastic incompressibility of the material ligament between voids and absence of voids in the virgin material. Shear yielding is the dominant mechanism of plastic deformation in the PP material. However, damage constitutes an important part of the apparent deformation. Furthermore, the onset of damage does not coincide with the onset of the nonlinearity observed on the stress-strain curve.
[FIGURE 4 OMITTED]
The microstructural evolution is obviously linked to the plastic strain achieved in the extruded sample. A plastic strain value estimate can be provided by analytical solutions developed in the literature [1, 2]. However, they are founded on assumptions which are too restrictive (rigid perfectly plastic material) and not realistic regarding the complex behavior (e.g., damage development, strain hardening,...) of the studied material. So, a finite element (FE) analysis seems the most relevant way to investigate the deformation mechanisms in the extruded material.
FE Modeling of ECAE Process
FE simulations were performed by using MSC.Marc FE code to predict the behavior of the sample when it is extruded through the channels. The MSC.Marc software contains a series of integrated programs that facilitate analysis of nonlinear engineering problems (in the fields of structural mechanics, heat transfer,...) in one, two or three dimensions. The reader can refer to Ref.  for further information. The sample with dimensions of 70 X 10 X 10 [mm.sup.3] and the experimental tool we have designed were modeled for 2D plane strain simulations. The plane strain assumption is reasonable since measurements on the deformed sample reveal that the strain in the out-of-plane direction (thickness direction) is negligible. The meshing of the sample was carried out by means of 140 X 20 four-node plane strain elements. This mesh size was adopted because previous FE analyses [22, 23] have shown that it gives a good compromise between computation time and precision of the results. Note that automatic refinement of the meshing was used to accommodate large strains. The interaction between the sample and the tool was modeled by a hard-contact behavior under frictionless conditions. Indeed, previous FE analyses [22, 23] have shown that the friction has a slight effect in this case. The ram was moved vertically by a displacement boundary condition. A finite strain formulation was adopted to take into account the large deformation behavior.
Basic Formulation of Constitutive Equations
Since damage influences the plastic flow of the material, it must be included in the numerical analysis. During the last three decades, the development of ductile damage models for metallic materials has been the subject of considerable attention. A few of recent studies have attempted to model the ductile damage behavior of polymers. Most of them used the well-known Gurson yield surface . The Gurson model was used in its original form [28, 29] or in a modified form to account for the pressure dependence [30-32] or elasticity  of the solid ligaments between voids. This modeling is justified by micromechanical considerations of void evolution. In this work, the material is assumed to be isotropic and the strain rate dependence is not addressed. The elastic behavior is governed by the hypoelastic law and the plastic deformation is assumed to be governed by the yield function of Gurson-Tvergaard [27, 34] from which the plastic constitutive response is derived:
[[PHI].sub.p] ([SIGMA], [bar.[sigma]], f) = [3/2][[[SIGMA]':[SIGMA]]'/[bar.[sigma].sup.2]] + 2[q.sub.1]f cosh ([1/2][q.sub.2][[[SIGMA].sub.kk]/[bar.[sigma]]]) - [1 + ([q.sub.1]f)[.sup.2]] = 0. (4)
where [[SIGMA].sub.kk] and [SIGMA]' are the first invariant and deviatoric part of the macroscopic Cauchy stress tensor [SIGMA], respectively, f is the current void volume fraction, [q.sub.1], [q.sub.2] are parameters introduced by Tvergaard  and [bar.[sigma]] is the effective stress of the material ligament between voids which was found to be well described by a power law of the form:
[bar.[sigma]] = 30 + 44[bar.[epsilon].sup.p.sup.1.6], (5)
where [bar.[epsilon].sup.p] is the effective plastic strain.
The rate of increase of the void volume fraction is given by:
[dot.f] = [dot.f.sub.n] + [dot.f.sub.g], (6)
in which [dot.f.sub.n] and [dot.f.sub.g] are the nucleation and growth rate of voids given respectively by:
[dot.f.sub.n] = [[f.sub.N]/[[s.sub.N][square root of (2[pi])]]]exp [-[1/2]([[bar.[epsilon].sup.p] - [[epsilon].sub.N]]/[s.sub.N])[.sup.2]][dot.bar.[epsilon].sup.p] (7)
[dot.f.sub.g] = (1 - f)[dot.E.sub.kk.sup.p], (8)
where [f.sub.N] is the volume fraction of void nuclei, [[epsilon].sub.N] is the mean strain for void nucleation, and [s.sub.N] is the standard deviation.
Identification of the required parameters is achieved by means of void volume fraction evolution. Checking the initial void volume fraction to zero, the void volume fraction function of the axial strain is fitted by optimizing the set of damage parameters. The best fitting was obtained for the following values: [q.sub.1] = 1.2, [q.sub.2] = 1.9, [[epsilon].sub.N] = 0.1, [s.sub.N] = 0.01, [f.sub.N] = 0.01. The determination of the elastic parameters completes the identification procedure: E = 2100 MPa and v = 0.4. Figure 4 shows a nice agreement between the constitutive model and experimental results.
FE results concerning evolution of the applied load during the ECAE process are shown in Fig. 5. A fairly good agreement is obtained with experimental results when considering the maximum load and the global shape. However, a significant difference in the plateau load is clearly highlighted. This is probably the consequence of the viscous effects which are not taken into account in the constitutive law. The simulation without damage effect provides an overestimate of the peak load.
[FIGURE 5 OMITTED]
Figure 6a shows the distribution of the accumulated equivalent plastic strain at the end of the ECAE operation. Both end sides of the sample undergo inhomogeneous plastic strain. A stationary deformation field where plastic strain is uniform can be clearly seen. In this steady plastic flow region, the variation of equivalent plastic strain across the width at the middle of the sample is shown in Fig. 6b. Because of the outer corner angle and the small corner gap, the plastic strain is lower around the bottom surface. The strain distribution is relatively uniform more than 70% of the sample. Uniform mechanical properties are therefore expected in this zone. In Fig. 6b, the mean normal stress achieved across the sample width both in the deformation zone (crossing plane) and in the steady-state are plotted against distance from the bottom. When the sample is in the crossing plane, the zone next to the outer corner angle is subjected to tension whereas the zone closer to the inner radius undergoes compressive mean stress. The trend is somewhat reversed in the steady-state. The high-compressive state at the top surface achieved at crossing plane probably leads to a densification phenomenon which compensates for the moderate tensile state during the steady plastic flow. Therefore, the possibility to develop damage at the top of the sample is reduced. Because of a compressive state of stress imposed after the deformation zone, the damage magnitude decreases.
Sample behavior in the die and material damage distribution are presented from the beginning up to the end of the process in Fig. 7. Steps of sample evolution are correlated with the load-ram displacement curve given in Fig. 5 and can be compared with the photographs of interrupted real experiments given in Fig. 3. As expected, the void volume fraction development corresponds to peak ram load and is mainly located at the bottom region.
Figure 8 presents the gradual evolution of void volume fraction for five material points along the cross-section in the middle of the sample. At the bottom of the sample (node A), the void volume fraction starts to increase when the material crosses the deformation zone but as the bottom undergoes compression in the steady-state phase void closure is observed. For the other material points, the same trend is observed with a peak relatively less pronounced than that of node A. Indeed, the void volume fraction increases first and reaches a steady-state corresponding saturation of the void/plastic strain level. Moreover, one can observe that the voids decrease while approaching the top of the sample and become insignificant at node E.
Since damage formation is attributed to bending of the sample, a back-pressure was applied in the simulations of ECAE. When the sample crosses the elbow, the back-pressure moved back in the exit channel at the bottom of the sample. Figure 9 presents the effect of back-pressure on void volume fraction magnitude and localization at the end of the process. It influences in a significant manner the damage magnitude in the steady-state region. As shown in Fig. 10 for node C, a back-pressure of about 100 N seems to be sufficient to avoid damage during the steady plastic flow. Furthermore, the filling of the corner is complete; the mechanisms of bending are disappeared and the shearing of the sample is then expected. That is reinforced by the disappearance of the exit channel gap between the exit channel and the top surface of the sample. At the end of the sample, a damaged zone is developed before the application of the back-pressure during the process. It is important to note that the plastic strain distribution along the sample width is not affected by the back-pressure. The resulting polymer microstructure should not be the same, but in such a way specimen integrity is definitely preserved.
[FIGURE 6 OMITTED]
[FIGURE 7 OMITTED]
[FIGURE 8 OMITTED]
[FIGURE 9 OMITTED]
A promising metal forming technique, namely ECAE, was applied to a typical polymer (polypropylene). Experimental results gave useful informations about the required load and deformation behavior of the polymer during ECAE. Because of a tensile state of stress in the bottom area of the sample a whitening phenomenon, attributed to voiding damage, was observed in this region. To take into account the realistic behavior in FE simulations, the material was described by the well-known Gurson-Tvergaard model. The set of material parameters was identified on hardening and dilatational curves obtained by video-controlled tensile tests. The extrusion load, strain, stress, and damage evolution during ECAE were numerically investigated. The numerical load-ram displacement curve supplied a quite good approximation of the experiments but overestimated the load in the steady plastic flow. Application of a back-pressure during extrusion seems to be a useful way to avoid void formation in the sample. The predictions suggest that uniform microstructure more than 70% of the sample width could be obtained. Microstructural observations and mechanical evaluation of the extruded samples are in progress.
[FIGURE 10 OMITTED]
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F. Zairi, (1) B. Aour, (1) J.M. Gloaguen, (2) M. Nait-Abdelaziz, (1) J.M. Lefebvre (2)
(1) Laboratoire de Mecanique de Lille (UMR CNRS 8107), USTL, Polytech'Lille, Avenue P. Langevin, 59655 Villeneuve d'Ascq Cedex, France
(2) Laboratoire de Structure et Proprietes de l'Etat Solide (UMR CNRS 8008), USTL, Batiment C6, 59655 Villeneuve d'Ascq Cedex, France
Correspondence to: F. Zairi; e-mail: firstname.lastname@example.org
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|Author:||Zairi, F.; Aour, B.; Gloaguen, J.M.; Nait-Abdelaziz, M.; Lefebvre, J.M.|
|Publication:||Polymer Engineering and Science|
|Article Type:||Technical report|
|Date:||May 1, 2008|
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