Jtam-A4.dvi


JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

54, 1, pp. 263-275, Warsaw 2016
DOI: 10.15632/jtam-pl.54.1.263

ANALYSIS OF PLASTIC DEFORMATION OF SEMI-CRYSTALLINE

POLYMERS DURING ECAE PROCESS USING 135◦ DIE

Benaoumeur Aour, Ali Mitsak

Laboratory of Applied Biomechanics and Biomaterials, Department of Mechanical Engineering, 0ran, Algeria

e-mail: ben aour@yahoo.fr

In this paper, analysis of plastic deformation of high density polyethylene (HDPE) and po-
lypropylene (PP)during an equal channel angular extrusion (ECAE)process is investigated.
The effects of ram speed, number of passes, processing route and temperature are tested
experimentally using a 135◦ die. The results show that the pressing force decreases with an
increase in the number of passes and reaches a saturation state rapidly for routesA andC
compared to routesBA andBC. Furthermore, it is found that the reduced curvature of the
extruded samples is obtained by route C, however, the maximum warping is obtained by
route A. A slight influence of temperature on the reduction of the warping is observed on
the extruded samples. In order to predict the plastic strain inside the extruded samples, an
elastic viscoplastic model is identified using compressive tests at different strain rates and
coupled with the finite element method (FEM). A good correlation is found between the
numerical modeling and experimental findings. FEM results show that the PP samples di-
splay a higher level of plastic strain compared toHDPE samples. However, almost the same
degree of strain heterogeneity is obtained for both polymers. Finally, in order to reduce the
warping and improve the strain homogeneity, a controlled back-pressure with small corner
angle is expected to be an adequate solution.

Keywords: ECAE, polymers, finite element analysis, plastic strain, back-pressure

1. Introduction

Equal channel angular extrusion (ECAE) is an innovative process to improve physical and
mechanical properties ofmaterials by severe plastic deformation (SPD)without alteration of the
geometric shape of workpiece.Moreover, since the cross-section of theworkpiece is notmodified
after extrusion, the process can be repeated several times, and by changing the orientation of
the workpiece between consecutive extrusions, stylish microstructures can be developed in the
extrudedmaterials. Up to now, the majority of research and development on ECAE have been
conducted onmetallic materials (Segal, 1995; Iwahashi et al., 1996; Valiev and Langdon, 2006).
However, for polymeric materials, little work is available to address the mechanical behaviour
during ECAE process (Sue and Li, 1998; Campbell and Edward, 1999; Li et al., 2000; Weon et
al., 2005; Wang et al., 2006).
According to our knowledge, this processwas first applied to polymers by Sue andLi (1998).

They showed that the ECAE process is effective in altering the morphology of a linear low
density polyethylene (LLDP). Sue et al. (1999) reported that for ECAE to be effective, it is
necessary that the extrusion be held at temperatures slightly below the glassy transition in the
case of polycarbonate (PC).For the samepolymer,Li et al. (2000) confirmed that themechanical
properties can be tailored by extruding thematerial via various processing routes and a different
number of passes.
The effect ofmolecular anisotropy on the impact strength of polycarbonate (PC)was exami-

ned byXia et al. (2001a). They found that the enhancement of the impact resistance is directly



264 B. Aour, A. Mitsak

related to the changes inmolecular orientation induced by the ECAEprocess. According toXia
et al. (2001b), the crystallinity andmolecular orientationwere identifiedas two important factors
affecting the dynamicmechanical properties of the ECAE-oriented semicrystalline polyethylene
terephthalate (PET).An improvement of the bending and torsional storagemoduluswas found.
Creasy andKang (2005) studied fibre fracture during theECAEprocess of short fibre-reinforced
thermoplastics. They found that the fibre length can be controlled and oriented by setting the
process temperature below the melting point of the polymer crystallites. On the other hand,
the effect of different ECAE routes on the tensile, fracture toughness, flexural, and ballistic
impact properties of polymethylmethacrylate (PMMA)was investigated byWeon et al. (2005).
A fruitful discussion was reported by Wang et al. (2006) on lamellar formation and relaxation
in simple sheared polyethylene terephthalate (PET) using the in-situ time resolved synchrotron
Small-Angle X-ray Scattering (SAXS) technique. Recently, numerical and experimental investi-
gations were achieved to highlight the effects of themain geometrical and processing parameters
on the viscoplastic behaviour of polymers during the ECAE process (Zäıri et al., 2008; Aour et
al., 2009; Bouaksa et al., 2014).

The findings presented above show that the ECAEprocess is an effective tool for the impro-
vement ofmechanical properties of polymersby inducingmolecular orientation inbulkpolymers.
This feature enables ECAE to have useful applications for the fabrication of many anti-impact
components, such as fighter-jet canopies, vehicle structures, windshields, and anti-theft trans-
parencies (Xia et al., 2001a). Furthermore, the ECAE technique can be easily incorporated into
the conventional polymer processing setup without much capital investment by attaching, for
example, a conventional injection unit to the entrance channel, which can potentially be used
for extruding pipes, tubes, rods, sheets, plates and other profiles with significantly improved
physical andmechanical properties (Sue et al., 1999).

In this paper, an experimental andnumerical investigation of plastic deformationof two semi-
crystalline polymers (HDPE and PP) during the ECAE process using 135◦ die is presented. In
order to achieve this objective, the paper is organised as follows. The experimental procedure
is discussed in Section 2. Section 3 is focused on the presentation of the experimental results
obtained for the effects of processing routes, number of passes and temperature. Section 4 is de-
voted to describe the elastic viscoplastic constitutive law and its identification using compressive
tests at different strain rates. Section 5 is reserved for the presentation of the FEM results. A
particular attention is made on the effect of the back-pressure on the homogeneity and the level
of the plastic strain distribution into the extruded samples. Finally, some concluding remarks
are given in Section 6.

2. Experimental procedure

2.1. ECAE device

After an optimization study of various geometrical parameters (Aour et al., 2008), anECAE
devicewith a channel angleΦ=135◦ anda corner angle θ=34◦ has beendesigned andmanufac-
tured (Fig. 1a).Thedie consists of two square channels of cross-sectional area 10.1mm×10.1mm,
which allows one to apply four routes (A, BA, BC and C) as shown in Fig. 1c. The lengths of
the entrance and exit channels are respectively 75mm and 50mm. An electromechanical Istron
5800 testing machine has been used to extrude the samples through the angular die.

2.2. Processing routes

Figure 1b shows a set of material axes referred to the sample, which is useful in describing
the different routes. TheX-direction is the zero strain direction (transverse direction: TD), the



Analysis of plastic deformation of semi-crystalline polymers... 265

Fig. 1. (a) Photograph of the ECAE device; (b) diagram of the sample showingmaterial axes used to
describe different routes; (c) schematic illustration of the processing routes:A,BA,BC andC

flow direction (FD) is taken as the Y -direction, and the Z-direction (normal direction: ND) is
normal to the plane in which the sample shear occurs. In this study, four different routes are
investigated (Fig. 1c):

• RouteA: the sample is re-inserted in the same orientation as the previous pass.

• RouteBA: the sample is rotatedalternatively by+90
◦ and−90◦ aroundtheY -axis between

two successive passes.

• RouteBC: the sample is rotated by +90
◦ around the Y -axis after each pass.

• RouteC: the sample is rotated around the Y -axis by 180◦ and then re-extruded.

2.3. Materials and samples preparation

Two semi-crystalline polymers (high density polyethyleneHDPEand polypropylenePP) ha-
ve been selected for this study.These polymers have been supplied by theGoodfellowCompany.
The crystal content is about 70% for HDPE and 55% for PP. ECAE samples of 10mm×10mm
cross-section and 70mm in length have been cut from commercially plates in the same direction,
then surfaced simultaneously on the cutting facets and polished. The HDPE and PP samples
have been respectively annealed in vacuum at 120◦C and 85◦C for 2h.

In this work, four parameters are studied experimentally: the ram speed, number of passes,

processing route and temperature effect. The extrusion tests have been performed without lubri-

cation.

3. Experimental results

3.1. Effect of ram speed

Figure 2 illustrates the influence of the ram speed on the evolution of the pressing force using
a 135◦ die in the case of HDPE (Fig. 2a) and PP samples (Fig. 2b). Three different values of
ram speeds (0.7, 0.07 and 0.007mm/s) have been tested without lubrication.
It can be observed that the pressing force required for extrusion increases with an increase

in the ram speed for both polymers. Indeed, when the ram speed is increased from 0.7 to
0.007mm/s, the maximum force required for extrusion of HDPE samples varies from 932 to
1212N (i.e., a difference of 280N), however, for PP samples, a difference of 432N is noticed.
Furthermore, at the stage of the steady state of the plastic flow, different trends are revealed
for eachmaterial. In the case of HDPE, the pressing force remains almost constant, however, in



266 B. Aour, A. Mitsak

Fig. 2. Variation of the pressing force versus ram displacement in one ECAE pass using 135◦ die in the
case of (a) HDPE and (b) PP samples

the case of PP, a slight increase is observed. This can be attributed to the flexibility of HDPE
which is higher than that of PP.

3.2. Effect of the processing route and number of passes

The advantage of the ECAE process, in addition to maintaining constant cross-section of
the extruded sample throughout the process, it is possible to generate a number of dissimilar
deformation histories and create various forms of molecular orientations if multiple passes with
a suitable selection of processing routes are carried out. It was demonstrated by Li et al. (2000)
that well-controlled morphology can lead to great improvements in physical and mechanical
properties of the extruded polymer both along and perpendicular to the extrusion direction. In
this subsection, the samples are processed via four ECAE processing routes using a 135◦ die
at room temperature and a ram speed of 0.70mm/s. In order to make a comparison between
the different routes, the evolution of themaximumpressing force versus the number of passes is
plotted in Fig. 3. It can be seen that, for routes A and C, the pressing force decreases with an
increase in the number of passes, however in the case of routesBA andBC, a periodic variation
is noticed for HDPE samples, and a random variation is highlighted for PP samples. These
variations explain that thematerials have different strengths in each direction due to anisotropy
andmobility of the crystalline lamellae inside the bulkmaterial with respect to ECAE loading.
Moreover, it can be observed that the routesA andC reach their saturation values after almost
four passes,while the routesBA andBC require a highnumber of passes to achieve its saturation
state.

Fig. 3. Variation of themaximum force versus the number of passes during extrusion through a 135◦ die
with different routes for (a) HDPE and (b) PP samples



Analysis of plastic deformation of semi-crystalline polymers... 267

Thewarpingof the extrudedsamplesduetovariousprocessingrouteshasbeenalsoquantified
in this experimental part. Figure 4 shows pictures ofHDPEsamples that have undergone sixteen
passes of ECAEby different processing routes. The obtained results for HDPE andPP samples
are listed in Table 1. The maximum curvature of the sample (warping) has been quantified by
measuring the height of the sample before and after the ECAE process. For both polymers, it
has been found that themaximumwarping is always obtained by routeA, however theminimum
warping is obtained by routeC. Furthermore, thewarping obtained for PP samples is quite high
than that of HDPE samples.

Fig. 4. HDPE samples extruded at room temperature after 16 passes using a 135◦ die with different
processing routes

Table 1.Maximum values of the curvature obtained by different routes using a 135◦ die and a
length of 70mm after 16 passes on HDPE and PP samples

Extruded
material

Height before Height after 16 Curvature
Route ECAE ECAE passes: Cu=Ha−Hb

Hb [mm] Ha [mm] [mm]

A
HDPE 9.85 16.50 6.65
PP 9.77 17.20 7.43

BA
HDPE 10.17 15.00 4.83
PP 9.29 13.50 4.21

BC
HDPE 10.13 13.50 3.37
PP 9.13 13.25 4.12

C
HDPE 9.72 13.00 3.28
PP 9.65 13.15 3.50

3.3. Effect of temperature

According to Sue et al. (1999), the warping is generated due to the existence of residual
stress and the concurrent stress relaxation process on the extruded samples. Moreover, it is
believed that the stress relaxation process can be greatly accelerated at elevated temperatures.
Consequently, in order to highlight the temperature effect on the warping reduction by stress
relaxation, the ECAEprocess has been carried out onHDPE samples at different temperatures
T = {25◦C,40◦C,60◦C} via routes A and C. The obtained results after sixteen passes are
illustrated in Table 2.

It can be seen that a slight reduction of warping is obtained even with several passes and at
elevated temperatures.Noting that, in the case ofPC samples, a significant reduction ofwarping
was found by Sue et al. (1999) via elevation of the extrusion temperature. However, in the case
of HDPE and PP, it is advised to test other parameters such as the use of back pressure which
will be the subject of the last Section.



268 B. Aour, A. Mitsak

Table 2.Maximum values of the curvatures of HDPE samples obtained after 16 passes using a
135◦ die via routesA andC at T = {25◦C,40◦C,60◦C}

Tempe-
rature
[◦C]

Height before Height after 16 Curvature
Route ECAE ECAE passes: Cu=Ha−Hb

Hb [mm] Ha [mm] [mm]

25 9.85 16.50 6.65
A 40 9.87 16.50 6.63

60 9.80 15.00 5.20

25 9.72 13.00 3.28
C 40 9.72 13.00 3.28

60 9.75 12.50 2.75

4. Elastic-viscoplastic constitutive model

The constitutive equations governing the behaviour of polymers under the ECAE process lo-
adingsmust take into account complex phenomena such as viscoplasticity, hardening, relaxation
and strain memory effect. These phenomena were studied by many authors basing on physi-
cal (Arruda et al., 1995; Ahzi et al., 2003; Bouaksa et al., 2014) or purely phenomenological
(Chaboche, 1997; Ho andKrempl, 2002; Colak, 2003) considerations. In this paper, a phenome-
nological constitutive model based uponChaboche’s model is applied (Lemaitre andChaboche,
1994; Ambroziak and Klosowski, 2006). This model incorporates the initial linear response, the
non-linear behavior and the rate-dependent yield stress.

4.1. Constitutive equations

One of the fundamental principles that all constitutive equations have to satisfy is the prin-
ciple of objectivity. Tensor rates used in the constitutive equations need to be objective. A
corotational objective rate of a tensor M is denoted by

M̂= Ṁ+MΩ−ΩM (4.1)

where Ṁ is the material rate with respect to the basis of M. M̂ is the objective rate of M,
and Ω is a skew-symmetric spin tensor. A well-known objective rate is the Jaumann rate. It is
obtained by settingΩ=W in Eq. (4.1)

σ̂= σ̇+σW−Wσ (4.2)

where σ̂ is the objective rate of the Cauchy stress tensor σ based upon the spin tensorW.

The strain rate tensor D is decomposed into an elastic partDe and a viscoplastic partDvp

as follows

D=De+Dvp (4.3)

The elastic strain rate tensorDe is given by the hypoelastic law

D
e =C−1σ̂ (4.4)

whereC is the fourth-order isotropic elastic modulus tensor

Cijkl =
E

2(1+ν)

[
(δikδjl+ δilδjk)+

2ν

1−2ν
δijδkl

]
(4.5)



Analysis of plastic deformation of semi-crystalline polymers... 269

with E, ν and δ are respectively Young’s modulus, Poisson’s ratio and the Kronecker-delta
symbol.
The viscoplastic strain rate tensorDvp can be written by

D
vp =

3

2
ṗ
σ
′−X′

J(σ−X)
(4.6)

whereJ(σ−X) is a distance in the stress space. For amaterialmeeting theVonMises criterion,
we use

J(σ−X)=

√
3

2
(σ′−X′) : (σ′−X′) (4.7)

where σ and X are the stress and back stress tensors, and σ′ =σ− tr(σ)/3I and X′ are the
stress and back stress deviatoric tensors in the stress space, respectively. ṗ is the equivalent
viscoplastic strain rate written as

ṗ=
〈J(σ−X)−R−k

K

〉n
(4.8)

The brackets are defined by 〈w〉 = wH(w), where H(w) is the Heaviside function (H(w) = 0
if w < 0, H(w) = 1 if w ­ 0). k is the yield stress at zero plastic strain, K is the viscoplastic
resistance, n is the rate sensitivity coefficient and R is the isotropic internal stress or the drag
stress.
The strain hardening of thematerial is described by isotropic and kinematic hardening rules

which allow both the expansion and translation of the yield.
The isotropic hardening rule is defined by

Ṙ= b(R1−R)ṗ with R(0)= 0 (4.9)

whereR1 is the boundary of isotropic hardening and b defines the rate at which the size of the
yield surface changes as the plastic straining develops.
Equation (4.9) may be replaced by its integrated form as (Lemaitre and Chaboche, 1994)

R=R1[1− exp(−bp)] (4.10)

The nonlinear kinematic hardening is defined from the linear-Ziegler rule by adding the recall
term as shown in the evolution of the back stress tensor below (MSC.Marc, 2005)

Ẋ=
C

R+k
(σ−X)ṗ−γXṗ with X(0)= 0 (4.11)

whereC and γ are twomaterial constants. γ=0 stands for the linear-kinematic rule.
The evolution law given by (4.11) may be formulated in terms of the objective rate of the

back stressX, say X̂, as follows (Bruhns, 2009)

X̂=K(τ,X,κ) :Dvp (4.12)

whereK(τ,X,κ) is a 4th order tensor-valued constitutive function, τ is theKirchhoff stress and
κ is a scalar internal variable.

4.2. Identification of the material parameters

The material parameters (E,k,K,n,b,R1,C,γ) of the elastic-viscoplastic constitutive law
have been identified from a least-square regression fitting using the experimental data of com-
pression tests on HDPE and PP specimens at room temperature and under different strain
rates (Aour, 2007). The values of the identified parameters for the studied polymers are listed
in Table 3. Figure 5 shows a fairly good agreement between the identified constitutive model
and experimental stress-strain curves of HDPE and PP. Indeed, the constitutive law is able to
reproduce three main features of the behaviour: the linear elastic response, the rollover to yield
and the post-yield response.



270 B. Aour, A. Mitsak

Table 3.Values of material parameters for HDPE and PP

Parameter Unit
Values for Values for
HDPE PP

E MPa 500 1100

ν – 0.38 0.4

k MPa 10 10

K MPa 15.6 30.2

n – 5.2 6.9

b – 40 65

R1 MPa 10 18

C MPa 50 15

γ – −1.1 −3.2

Fig. 5. Stress-strain curves obtained by compression tests and the constitutive model for (a) HDPE and
(b) PP at room temperature and different strain rates

5. Finite element results

In order to predict the plastic deformation behaviour of HDPE and PP samples during the
ECAEprocess, finite element simulations have been carried out using the softwareMSC.Marc CO

under plane-strain conditions. The die geometry, sample dimensions and processing conditions
have been taken similar to those used in the experimental study. The sample has been meshed
with 2800 four-node isoparametric elements. Thedie and the ramhave been assumed to be rigid.

5.1. Estimation of the pressing force during ECAE

Figure 6 shows a comparison between the experimental pressing force-ram displacement
curves and the finite element results using different friction coefficients for the extrusion of
HDPE andPP samples through a 135◦ die at a ram speed of 0.70mm/s. The friction conditions
between the tooling and the samples are modelled using Coulomb’s friction law. As shown in
Fig. 6, the FEM results are closer to the experimental data when the friction coefficient is equal
to 0.075 for HDPE and 0.025 for PP. It is worth noting that the damage mechanisms, which
occur at the elbow of the die (plastic deformation zone), have not beenmodelled in the present
constitutive model. Thus, the comparison is only made for the stage of steady state of the
material flow during the ECAE process.

5.2. Estimation of the equivalent plastic strain

Figure 7 shows the equivalent plastic strain contour plots of HDPE and PP samples during
the ECAE process with a ram speed of 0.70mm/s considering the friction coefficients which



Analysis of plastic deformation of semi-crystalline polymers... 271

Fig. 6. Comparison between the experimental curve and FEM results using different friction coefficients
in the case of (a) HDPE and (b) PP

gave the best agreement with the experimental results, i.e., f =0.075 for HDPE and f =0.025
for PP. It can be observed that the plastic strain is not uniform along width of the samples
for both polymers. It should be noted that the effective plastic strain generally decreases from
the top surface to the bottom surface of the samples. This can be attributed to the presence
of the bendingmechanisms since the inner part of the sample flows faster than the outer part.
In other words, the deformation mechanism in the bottom region is rather bending than shear.
Furthermore, it can be seen that the fronts of the extruded samples have not undergone a high
level of plastic deformation. This is mainly due to the filling status of the channel when the
sample passed through the elbow.

Fig. 7. Equivalent plastic strain contours for ECAE of (a) HDPE and (b) PP samples using a 135◦ die

In order to quantify the degree of strain homogeneity inside the sample, the distribution of
the equivalent plastic strain along the sample width at the steady state region is presented in
Fig. 8. It can be seen that the equivalent plastic strain in the PP sample is higher than that
of the HDPE sample. However, almost the same degree of strain heterogeneity is obtained for
both polymers, since the variation factor is 24% for HDPE and 22% for PP. Noting that the
variation factor denoted by V is defined as the ratio of the standard deviation s to the average
equivalent plastic strain εpave (Aour et al., 2006)

V =
s

ε
p
ave
=
1

ε
p
ave

√√√√ 1
N

N∑

i=1

(ε
p
i −ε

p
ave)2 ·100

% (5.1)

where ε
p
i is the equivalent plastic strain value at a given integration point along the sample

width, εpave is the arithmetic average of the equivalent plastic strain values computed on N
integration points.



272 B. Aour, A. Mitsak

Fig. 8. Distribution of the equivalent plastic strain in HDPE and PP samples extruded by a 135◦ die

5.3. Equivalent plastic strain rate

It is recognized that the stress-strain behaviour of polymers is strongly dependent on the
strain rate due to the viscoplastic nature of polymers (Ward and Hadley, 1995). In order to
highlight the spatial variation of the plastic strain and the trends in the degree of homogeneity,
the strain rate distributionwithin the plastic deformation zone (PDZ) is addressed here. Indeed,
themore the plastic deformation rate is uniform along the shear plane, the greater is the degree
of homogeneity of the plastic deformation into the sample. Figure 9 shows the distribution of the
equivalent plastic strain rate ε̇p at an intermediate stage of the ECAE process using a 135◦ die
for HDPE and PP samples. It can be seen that the distribution of ε̇p inside the PDZ is neither
uniformnor symmetrical with respect to the shear plane, which justifies the heterogeneity of the
plastic strain distribution. Furthermore, it can be observed that this later decreases significantly
from the inner corner to the outer corner. This aspect can be allotted to the coupled effect of
the viscoplastic behaviour and geometrical features of the die.

Fig. 9. Distribution of the equivalent plastic strain rate inside (a) HDPE and (b) PP samples during the
ECAE process using a 135◦ die

5.4. Effect of the back-pressure

In order to improve the degree of the plastic strain homogeneity and reduce the warping
of the samples, the application of back-pressure seems to be a suitable solution. It consists
in applying a constant load to the sample front at the exit channel using a second ram. The
obtained results for the equivalent plastic strain distributionalongwidthof theHDPEsampleby
applying different values of back-pressure are shown in Fig. 10. It can be seen that a significant
improvement in plastic strain homogeneity is obtained when the corner angle θ = 5◦ and the



Analysis of plastic deformation of semi-crystalline polymers... 273

back-pressure P = −500N (Fig. 10a), however, when θ = 34◦, a slight effect is highlighted
(Fig. 10b). Indeed, when θ = 5◦, the variation factor is reduced by 11% (from V(P=0N) = 28%
to V(P=−500N) = 17%), however, when θ = 34

◦, the variation factor is reduced by 4% (from
V(P=0N) = 24% to V(P=−500N) = 20%). Consequently, in order to improve the plastic strain
homogeneity, it is advised to use a low outer corner angle with an adequate back-pressure. This
allows one to promote shearing deformations and reduction of the bendingmechanisms.

Fig. 10. Effect of back-pressure on the evolution of equivalent plastic strain in the case of a 135◦ die

6. Conclusion

In this study, the effects of ramspeed, processing route, numberof passes, temperature andback-
pressure onHDPEandPPbehaviour during theECAEprocess using a 135◦ die are investigated
by experimental testing and finite element modeling. The following conclusions can be drawn:

• It is found that the pressing force required for extrusion increases with an increase in the
ram speed, and the pressing force of PP samples is about 200Nhigher than that ofHDPE.

• For both polymers, the significant reduction of warping is obtained by route C, whereas,
the maximumwarping is obtained by routeA.

• It is found that the pressing force decreases significantly with an increase in temperature,
while a slight reduction of warping is observed as the extrusion temperature is increased.

• A good agreement is noticed between the experimental curves and the FEM results when
the friction coefficients are equal to 0.075 for HDPE and 0.025 for PP. This allows one to
carry out the ECAE process without a lubricant.



274 B. Aour, A. Mitsak

• It is found that PP samples display a higher level of plastic strain than HDPE samples.
However, almost the same degree of strain heterogeneity is obtained for both polymers.

• The distribution of the equivalent plastic strain rate inside PDZ is found to be neither
uniform nor symmetrical about the shear plane.

• It is expected that the use of a low corner angle with an adequate back-pressure can
effectively reduce the warping and improve the strain homogeneity as well as the level of
shear deformation.

Acknowledgements

The authors thank Profs.MoussaNait-Abdelaziz, Fahmi Zäıri and Jean-Michel Gloaguen of of Poly-

tech’Lille (France) for their assistance in the experimental tests andnumerical simulations.The comments

from the reviewer are also greatly acknowledged.

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Manuscript received September 22, 2014; accepted for print August 14, 2015