Materials and Design 30 (2009) 4192–4199

Contents lists available at ScienceDirect

Materials and Design

journal homepage: www.elsevier .com/locate /matdes

Analysis of the polypropylene mechanical behaviour response:Experiments and numerical modeling

H.M. Meddah a,*, N. Selini b, M. Benguediab b, M. Bouziane Mohamed b, M. Belhamiani b

a Department of Mechanical Engineering, University Mustapha Stambouli of Masacra, Masacra, Algeriab Department of Mechanical Engineering, University of Sidi Bel Abbes, Bp 89, Cité Ben M’hidi, Sidi Bel Abbes 22000, Algeria

a r t i c l e i n f o

Article history:Received 26 February 2009Accepted 26 April 2009Available online 8 May 2009

Keywords:PolypropyleneSemi-crystallineMechanical behaviourFEM

0261-3069/$ - see front matter � 2009 Elsevier Ltd. Adoi:10.1016/j.matdes.2009.04.043

* Corresponding author.E-mail address: hmmeddah@yahoo.fr (H.M. Medd

a b s t r a c t

The security requirements in the industrial world incite an ever deeper understanding of the behaviourand the fracture of polymeric materials used as structural parts of the passenger compartment. We arelooking at a polypropylene commonly used in this field in order to identify the physical processes respon-sible for their mechanical properties. The mechanical characterization of the response of the polymerunder simple and complex strain relies on a unique method of combining performing numerical analysistechniques. The behaviour of polypropylene with large deformations dissipative involves several pro-cesses. Its consequences on the mechanical properties of materials are significant. The analysis of theseresults to emphasis that the plasticity of the polymer involves addition mechanical properties. Takentogether, these observations can lay the groundwork for a thermodynamic modeling of the behaviourlaw to this class of polymer. The contribution of this approach was demonstrated by experimental andnumerical modeling of the polypropylene mechanical behaviour.

� 2009 Elsevier Ltd. All rights reserved.

1. Introduction

The semi-crystalline polymers are materials with a complexmicrostructure consisting of a disordered amorphous phase vis-cous nature of a phase and crystalline structure. The coexistenceand the interaction of these two phases of very different naturesare at the root of the complexity of their macroscopic behaviourthat could come within the purview of such behaviour elastovisco-plastic [1–7]. It must be seen in the point of microscopic view;polymer material is as complex as each of the three levels of micro-structure plays a role in the deformation. A wide spherolites, crazesmay be formed at interspherolitics. Meanwhile, spherolites may bedeformed, leading to the destruction of tapes. In terms of crystal-line lamellae and the amorphous phase, we can see a stretch ofthe whole or shear, which will be translated obviously on a smallerscale by slipping chains amorphous phase.

Several avenues of study of the rupture have come to our atten-tion in the literature. The first takes place at the scale of themechanics of continuous media. Position of initial defect (crackor discontinuity), this approach can predict the evolution of the lat-ter depending on the condition of loading and led a mechanicalcharacterization of quantities at time of the rupture. Other ap-proaches include phenomenological look simple mathematicalmodels to bring sufficient conditions for the beginning of failure.

ll rights reserved.

ah).

The expression of the criteria as well as the descriptors set gamecan be different depending on the size of plastic processes develop-ing in the material.

Polypropylene is a polyolefin catalyst obtained by linear stereo-specifique of propene. This type of catalyst allows the synthesis ofisotactic polypropylene (most recently syndiotactique), which onlyshows the properties required a structural use.

Our work takes place in this context, and is being conducted inpartnership with major players in the industry.

The objective is to portray the different ways of possible studyand choose the most appropriate approach of description of ourpolypropylene, the establishment of the constitutive laws and offracture criteria relevant in a finite element simulation of polypro-pylenes structure behaviour usually used in industry. We will firstpresent some information that will glimpse the significance of theeffects combines the formulation and implementation of the poly-propylene behaviour law.

After a presentation of the background and the general princi-ples, we show the performance of a numerical approach.

2. Theoretical principle of polymers deformation

The modeling of polymers is based on the first step the Voigtand Maxwell reological model types. These models allow tocombine with the viscoelasticity and viscoplasticity, the twobehaviours which is identified in polymers. In the case of semi-crystalline polymers, there are different behaviours of crystalline

H.M. Meddah et al. / Materials and Design 30 (2009) 4192–4199 4193

and amorphous phases. The models employed a unique equationfor different behaviours of two phases [8]. Other models, basedon the equation of Eyring [9] can response rubber elasticity ofthe amorphous phase as far as deformation of the crystalline phase[10]. Other models have been chosen to introduce solely on themodeling of crystalline texture [11–13]. The polymer is then con-sidered as a cluster comprised of polycrystalline crystallites ran-domly assigned. The crystalline phase is supposed to haveviscoplastic behaviour; the crystalline lamellae are distorted byshearing, parallel or perpendicular to the chains [14–16]. In orderto take into account the contribution of the amorphous phase tothe plastic deformation implies an undervaluation plastic hard-ness, particularly in shearing. This will be checked that quantityof amorphous phase will be significant in polymer.

3. Experiment

The analysis of polymer behaviour shows that a relevant mod-eling behaviour must be based on a mechanical approach, or atleast their mechanical effect.

3.1. Materials

A commercial polypropylene ‘‘PP” homopolymer supplied byGoodfellow is used. The number-average molecular weight andthe weight-average molecular weight are about Mn = 25 Kg/moland Mw = 180 Kg/mol, respectively. The crystal content is about55%. The glass transition temperature is about �20 �C and themelting temperature is almost 170 �C. Since the glass transitiontemperature of the amorphous phase is lower than ambient tem-perature, it is in the rubbery state [17].

3.2. Behaviour law parameters

3.2.1. Experimental methodsThe polypropylene material has initially studied starting from

tensile tests. Experimental uniaxial tensile tests are conducted todetermine the polypropylene constitutive law. The specimen wastaken from a plate. The geometry is corresponded to ASTM D638

Fig. 1. Specimens geometry on tension test.

M1A standard (Fig. 1) such as the overall length is of 100 mm,the width of the barrel is of 35 mm, and in melts of notch is of10 mm.

True axial stress–strains were locally measured using a video-extensometer connected to the Instron operating system to workon a wide field of speeds 0.1, 0.01, and 0.001 mm/s. The mechanicalproperties of polypropylene withdrawn of the tests are presentedin Table 1.

Young modulus (E) is given graphically to leave elastic slopes oftensile curves. It can be then considered like apparent because ittakes account of the viscosity of material. In calculations by finiteelements, the elastic module chosen will be the apparent module.

The Poisson’s ratio (m) is given to leave slope of longitudinaldeformation curve in function of the side deformation on the samespecimen’s. In the zone where the deformation remain homoge-neous (before necking), the Poisson’s ratio average value is 0.4.

Just like graphical determination of the Young modulus, deter-mination of (r0) employ serious problems precise details on thetrue axial stress–strain curve. The stress (r0) represents limit be-tween the field perfectly elastic and the beginning of plasticity.Determination of this limit is difficult to obtain, because of the vis-cosity presence.

The different material properties can be drawn from the curve(Fig. 2). Curves who illustrate, respectively, the behaviour frequentat ambient temperature under uniaxial request slow of a semicrystalline polymer reveal three principal steps:

Step 1: the beginning of the curve presents a quasi linear highslope at the origin. This part is called viscoelastic response. Asoftening almost absent, for which the yield stress present atransition round progressive.Step 2: deformation becomes irreversible, necking develops. Thisstep is associated partly with unfolding macromolecules underapplied the deformation effect. The necking stabilizes and theminimal diameter of specimen does not decrease almost anymore. Shoulders necking are propagated towards ends of thespecimen. This propagation is translated by a weak variationof load.

Table 1Mechanical properties of the polypropylene.

de=dt ðS�1Þ E (MPa) ry (MPa) t Density

0.1 2240 52 0.4 0.90.01 2140 420.001 2100 38

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.70

10

20

30

40

50

60

70

80

90

b3r3+b4r

2+b5rϕ1.r. exp(-b1r)ϕ2 . (1-exp(-b2r))

ϕ3

Step 3Step 2Step 1

True

axi

al S

tress

(MPa

)

True axial Strain

0,001 /S0,01 /S0,1 /S

Fig. 2. Tensile true stress–strain curves of polypropylene.

2080

2100

2120

2140

2160

2180

2200

2220

2240

2260

0 0,02 0,04 0,06 0,08 0,1

dε /dt (S-1)

Youn

g's

Mod

ulus

E (M

Pa)

32

34

36

38

40

42

44

46

48

50

52

Yiel

d St

ress

σ0(M

Pa)

Young's Modulud (MPa)Yield Stress (MPa)

Fig. 3. Young’s modulus and yield stress versus strain rate for tensile experiments.

y = 13,655 x - 50,919

-8

-7

-6

-5

-4

-3

-2

-1

03.58 3.68 3.78 3.88 3.98

Ln (σ0)

Ln (d

ε/d

t)

Fig. 4. Logarithmic strain rates versus logarithmic yield stress for tensileexperiments.

4194 H.M. Meddah et al. / Materials and Design 30 (2009) 4192–4199

Step 3: this step, qualified of beginning of structural hardeningand truth similarly dependent on the orientation chains accord-ing to the strain principal direction.

According to Fig. 2, for deformations fewer than 3–5%, one no-tices more the strain rate is high, more the stress is significant.The softening is marked less, it is followed of a long plate and fin-ishes by a hardening. There is the same behaviour for all speeds.The calculation of young modulus is done in the linear part (elas-tic), the end of this limit represent the yield stress, the latter is lo-cated thus before the peak of stress ry (with approximately from2% of axial strain).

One finds the same behaviour, more the strain rate is high, andmore the elasticity module and the stress with the peak is too high(Fig. 3).

From these results the logarithmic strain rate as a function ofthe logarithmic yield stress plot can be obtained and it is shownin Fig. 4. The slope of this straight line corresponds to the of prob-ability logarithmic magnitude for the tensile tests it was found tohave a value of 13.6. Intersect of this line gives an estimation ofthe pre-exponential factor. Distribution description of probabilitywill be done better with a threshold.

4. Numerical implementation

The finite element method is used widely in order to study andcalculation, after division of system elements, the unknownvariables to transform partial differential equations in algebraicequations. These equations can be modelled, with simplifyingassumptions, the problems of solid mechanics [18–22].

This work is dedicated with the presentation of behaviour lawvia an implementation in Abaqus ‘‘UMAT” within the framework

of one numerical simulation. For a first exploration of the potentialof this local approach, we simplify the model by assuming behav-iour isotropic to large deformation and hardening beyond thethreshold.

4.1. Basic formulation of constitutive equations

In this section a continuum mechanics formulation is used[23,24]. Assumptions of linearity elastic behaviour of the materialimpose on the potential qw to be a quadratic invariant tensor ofthe deformations (Eq. (1)). w is then a linear combination of the

first invariant: eeI

� �2 ¼ ee� �2

and of the second invariant eeII ¼

12 ee� �2

qw ¼ 12

k eeI

� �2 þ 4leeII

� �ð1Þ

k and l are the Lame coefficients.The property of the elastic strain introduces the shear elastic

module G and the elastic modulate of compression hydrostatic K.The expression potential of energy free is written (Eq. (2)):

qw ¼ 12

3K � 2G3

eeI

� �2 þ 4GeeII

� �

qw ¼ 12

K eeI

� �2 þ 4Ge0eII� �

ð2Þ

where e0e represents the deviatory elastic strain.While making intervened the young’s modulus E and the Pois-

son’s ratio m. Thermodynamic potential takes the expression givento the Eq. (3)

qw ¼ 12

tEð1þ tÞð1� 2tÞ ee

I

� �2 þ 2E

ð1þ tÞ eeII

� �ð3Þ

The law is obtained by the Eq. (4):

re ¼ q@w@ee¼ K ee

� �1þ 2Ge0e ð4Þ

The constitutive law, during the plastic deformation, rise partic-ularization of the free energy potential and of the potential of dis-sipation thus that choice criterion of plasticity.

The results obtained regarding to the large deformations showthat a phenomenon of the work is proclamation by a hardening.The elastic behaviour and work hardening are the different physi-cal natures and can exist separately. Indeed, hardening can utilizeslips on the level of the chains, contrary with the elastic strain.These considerations allow free energy in the partition form of en-ergy dependent on the elastic strain we and hardening energy wr

(Eq. (5))

qw ¼ qweðeeÞ þ qwrðrÞ ð5Þ

The expression hardening force rise phenomenological observa-tions behaviour of material. It allows reproducing the stress–straincurve of simple tensile request. Thus, this expression allows takingin account the existence of a threshold (Q 3), the hook observeswith the plasticity starting (Q2�(1-exp(-b2r))), the stress plate(Q1�r�exp(�b1r)) as well as hardening in large deformations(b3�r3 + b4�r2 + b5�r). These expressions approach to the expressionsused by Tillier in multiplicative model of the elastoplastic behav-iour law with stress with the plastic deformation [25]. The generalexpression Eq. (6) for qwr:

qwr ¼ bðrÞbðrÞ ¼ � Q1

b1r þ 1

b1

� �exp �b1rð Þ þ Q 2 r þ expð�b2rÞ

b2

� �þ b3

4 r4 þ b43 r3 þ b5

2 r2� �(

ð6Þ

Fig. 5. Typical meshing of specimens.

H.M. Meddah et al. / Materials and Design 30 (2009) 4192–4199 4195

In order to accommodate the stress hardening during the load-ing, we choose the following expression for R following previousstudies [26].

R associated to r can be calculated by applying the law [7].

r ¼ q @w@ee

X ¼ �q @w@ep

R ¼ q @w@r

8>>><>>>:

ð7Þ

X: tensor of the second order of components Xij.R: load associated to the isotropic hardening variable.

R ¼ q@w@r¼ q

@wr

@r¼ b0ðrÞ ð8Þ

R ¼ ðQ 1 � r � expð�b1rÞÞ þ tðQ 2 � ð1� expð�b2rÞÞÞþ ðb3r3 þ b4r2 þ b5rÞ þ Q3Þ ð9Þ

where Q3, Q2, b2, Q1, b1, b3, b4, and b5 are material constants.The formalism was implemented in the commercial code ABA-

QUS using the UMAT subroutine making a finite element modelingof the structure behaviour. We have an experimental databaseincluding a uniform tensile simple test on smooth specimens. Sim-ple tests are sufficiently well instrumented to be used as a basis ofparameters identification of polypropylene behaviour law. How-ever, all our parameters are identifiable to leave of the first threetests. The state law and the evolution laws are fully coupled. Animplicit Newton–Raphson scheme is used to solve the set of themodel equations.

We consider the simplest analysis (uniaxial tensile). Given themodel imperfections, we justify anyway the contribution of itscomponents piezo-dependent of plasticity criterion.

The numerical analysis of tensile tests of notched specimenswill be then used for validation modeling in complex situation(Fig. 1).

4.2. Meshing

A specimen’s geometry was studied by symmetry; only 1/4 ofthis one is meshed to represent the full structure. In the notchedzone, the mesh is refined using element quadratic with reducedintegration 6-node modified quadratic plane strain triangle‘‘CPE6 M”, contrary at the smooth zone where this one is coarserto using element quadratic 8-node biquadratic plane stress quadri-lateral, reduced integration ‘‘CPS8R”. The total number of nodes isof 905 and 280 elements. Tests on our specimen were carried outimposed displacement with the nodes of head. Numerically it isnot possible to return directly such a condition of loading in calcu-lation. Initially, longitudinal displacement of specimen was mea-sured during a rate test of variable diametrical reduction. Fig. 5indicates the meshing used in the geometry.

5. Results and discussion

The finite elements resolution allowed us to analysis numeri-cally and to determine the fields distribution of deformations ina specimens ‘‘in the notch vicinity” obtained for analysis the tensiletests with imposed displacement concerning to four deferent ratiorates reports, 0.2, 0.6, 1, and 1.4 mm.

Fig. 6 presents the evolutions and the localization of the longi-tudinal deformations in the useful zones and homogenized. It isnoted that the deformations are concentrated in notch bottom, thisconcentration decreases far from this zone, in addition, thisconcentration grows with the increasing of displacement. Thedeformed zone is smaller that the useful zone, it is homogeneous

in one transverse section; however the deformations distributionis not homogeneous on the side near to the defect, and is subjectedwith stresses of amplitude relatively weak. The deformationsstrongest are localised at the notched bottom of which the ampli-tude decrease with the far from measurement that one movesaway of this zone. If the yield stress of material exceeds, a plasticzone in the vicinity bottom of notch appears.

Fig. 7 presents the variation of the equivalent stress of Von-Mises in terms of time realized by the finite elements methodand taken in a specimens centre (the most requested part) forthe same analysis of tensile tests with four reports displacementratios 0.2, 0.6, 1, and 1.4 mm. The results are logical because onenotice an evolution of equivalent stress all in increasing displace-ment. For a displacement of 0.2, a linear evolutions can not be ex-ceeded the yield stress of material. On the other hand, for otherdisplacements, one notices formation of the plasticity with a hookappearance observed with starting of the plasticity thus thathardening.

For our taking into account of the sensibility and of the imposeddisplacement effect, one presents the stress–strain curves variationin function of time taken in the specimen centre (Figs. 8a–d). Of ourmodel, we have drawn stress–strain curves for our material on allrange of measurement.

One observes that in imposing an increasing of displacement,show us an increase of stress maximum thus that of strain.

Concerning the general aspect of the curves; two results inter-esting are to mention. The first concern the presence of a tensilehook on the level end maximum end, for three displacements(0.6, 1, and 1.4 mm) this hook indicates an effect of material struc-ture; the tensile hook indicates the rapid formation of constraintwhich supports the majority elongation and the effort.

The second point interesting relates to the plate of plastic defor-mation obtained after the hook traction. The load remains aboutconstant and characterized by one new-hardening. This result indi-cates a strong plastic flow associated an orientation and a slip of

Fig. 6. Longitudinal strain fields.

4196 H.M. Meddah et al. / Materials and Design 30 (2009) 4192–4199

the chains. In the end, fibres are strongly stretched; they create anadditional resistance then. The transverse deformations are how-ever under estimated in majority cases. One can give in question

the isotropy assumption, the dilation over-estimated in transversedirection be opposed with the reduction of section involving theleast transverse deformation in the numerical simulation. This

εεyy

(d = 1mm)

0

10

20

30

40

50

60

0

0,02

0,04

0,06

0,08

0,1

0,12

0,14

0,16

0,18

0,2

σyy

(MPa

)

0 0.2 0.4 0.6 0.8 1 1.2

Time (S)

Fig. 8c. Stress–strain curves as a function of time (d = 1 mm).

0 0.2 0.4 0.6 0.8 1 1.2Time (S)

(d = 1.4mm)

0

10

20

30

40

50

60

70

0

0,05

0,1

0,15

0,2

0,25

0,3

σyy

(MPa

)

εyy

Fig. 8d. Stress–strain curves as a function of time (d = 1.4 mm).

0

10

20

30

40

50

60

0 0.2 0.4 0.6 0.8 1 1.2

Time (S)

σeq

(Von

-Mis

es) M

Pad=0.2 mm

d=0.6 mm

d=1 mm

d=1.4 mm

Fig. 7. Evolution of the equivalent stress as a function of time.

(d= 0.2mm)

0

5

10

15

20

25

30

σyy

(MPa

)

0

0,01

0,02

0,03

0,04

0,05

0,06

0,07

0,08

0,09

0,1

εyy

0 0.2 0.4 0.6 0.8 1 1.2

Time (S)

Fig. 8a. Stress–strain curves as a function of time (d = 0.2 mm).

εεyy

(d = 0.6mm)

0

5

10

15

20

25

30

35

40

45

0

0,01

0,02

0,03

0,04

0,05

0,06

0,07

0,08

0,09

0,1

σyy

(MPa

)

0 0.2 0.4 0.6 0.8 1 1.2

Time (S)

Fig. 8b. Stress–strain curves as a function of time (d = 0.6 mm).

H.M. Meddah et al. / Materials and Design 30 (2009) 4192–4199 4197

explanation could be applied on the results of the longitudinaldeformation, an isotropic model can to lead to over-estimate eyy

to compensate under estimate of exx (Figs. 9a–d).The numerical model reproduced with less of 13% meadows the

developments of load and displacement on the ligament untilappearance of plasticity (Figs. 10a–d). As does, the loads so thattheir sensitivities of strain rate are correctly predicted.

(d = 0.2mm)

0.007

0.0075

0.008

0.0085

0.009

0.0095

0.01

0.0105

Ligament (mm)

εyy

-0.008

-0.0075

-0.007

-0.0065

-0.006

-0.0055

-0.005

-0.0045

-0.004

-0.0035

-0.003

εxx

0 1 2 3 4 5 6

Fig. 9a. Evolution of the transverses deformations as a function of the ligament(d = 0.2 mm).

(d= 0.6mm)

0.04

0.045

0.05

0.055

0.06

0.065

0.07

-0.06

-0.055

-0.05

-0.045

-0.04

-0.035

-0.03

Ligament (mm)

εxx

εyy

0 1 2 3 4 5 6

Fig. 9b. Evolution of the transverses deformations as a function of the ligament(d = 0.6 mm).

(d =1mm)

0.115

0.116

0.117

0.118

0.119

0.12

0.121

0.122

0.123

0.124

0 1 2 3 4 5

Ligament (mm)

-0.116

-0.115

-0.114

-0.113

-0.112

-0.111

-0.11

-0.109

-0.108

-0.107

-0.106

εyy

εxx

Fig. 9c. Evolution of the transverses deformations as a function of the ligament(d = 1 mm).

εεyy

Ligament (mm)

(d= 1.4mm)

0.155

0.16

0.165

0.17

0.175

0.18

0.185

0 1 2 3 4 5-0.175

-0.17

-0.165

-0.16

-0.155

-0.15

-0.145εx

x

Fig. 9d. Evolution of the transverses deformations as a function of the ligament(d = 1.4 mm).

(V = 0.2 /s)

2

2.5

3

3.5

4

4.5

5

5.5

6

6.5

7

Load

(N)

0.00E+00

5.00E-03

1.00E-02

1.50E-02

2.00E-02

2.50E-02

3.00E-02

3.50E-02

4.00E-02

4.50E-02

5.00E-02

Dis

plac

emen

t (m

m)

Ligament (mm)0 1 2 3 4 5 6

Fig. 10a. Load–displacement curve as a function of the ligament (V = 0.2/s).

(V = 0.6 /s)

4

5

6

7

8

9

10

11

12

13

0 1 2 3 4 5 6

Ligament (mm)

Load

(N)

0,00E+00

5,00E-02

1,00E-01

1,50E-01

2,00E-01

2,50E-01

3,00E-01

3,50E-01

4,00E-01

4,50E-01

5,00E-01

Dis

plac

emen

t (m

m)

Fig. 10b. Load–displacement curve as a function of the ligament (V = 0.6/s).

(V = 1 /s)

4

5

6

7

8

9

10

11

12

13

14

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5Ligament (mm)

Load

(N)

0.00E+00

1.00E-01

2.00E-01

3.00E-01

4.00E-01

5.00E-01

6.00E-01

7.00E-01

8.00E-01

Dis

plac

emen

t (m

m)

Fig. 10c. Load–displacement curve as a function of the ligament (V = 1/s).

(V = 1.4 /s)

4

6

8

10

12

14

16

0 1 2 3 4 5Ligament (mm)

Load

(N)

0,00E+00

1,00E-01

2,00E-01

3,00E-01

4,00E-01

5,00E-01

6,00E-01

7,00E-01

8,00E-01

9,00E-01

1,00E+00

Dis

plac

emen

t (m

m)

Fig. 10d. Load–displacement curve as a function of the ligament (V = 1.4/s).

4198 H.M. Meddah et al. / Materials and Design 30 (2009) 4192–4199

The results are located enough logically, the pace of the curvesload–displacement along the ligament are appreciably similar tothe four tests. The role of reinforcing loads mineral could leave towait a dissipation of energy more significant of material and thusa significant variation between responses at various rates. Thecurves similarities attest the importance effect of structure on thisanalysis, limiting the behaviour polymers expression.

The curves show a peak of load occurring front plasticity ofmaterial like attest Fig. 6 contours of deformations.

The tests with higher rate lead on levels of displacement andload clearly more significant what led with a consumption of en-ergy more significant.

6. Conclusions

The objective of this study allows us to propose a mechanicalcharacterization of polypropylene. We are interested in develop-ment of a suited model to describe the behaviour of this materialclass.

Experimental results gave useful information about the re-quired true stress–strain behaviour of the polypropylene basedon a non-contact method.

H.M. Meddah et al. / Materials and Design 30 (2009) 4192–4199 4199

In order to take into account the realistic behaviour in finite ele-ment simulations, the set of material parameters was identified onhardening and dilatational curves obtained by video-controlledtensile tests. The extrusion stress–strain evolutions were numeri-cally investigated.

The numerical shutter consisting then with exploration of thepotential of a behaviour law, based on one thermodynamic ap-proach employee in our computer code.

For simplicity of calculation, this type of modeling gave anpromising results in simple situation, and it is one tool adaptedwell with the study phenomena more complex like for examplethe fracture in a polymer structure.

Our model fact of intervening parameters identifiable on thesimple basis tests are in monotonous traction.

The unit our tests of analysis in term sizes total is accompaniedof a local sizes representation acceptable even in the case of stronglocalizations. Thus, the mode product the dissymmetry stress–strain and allows to envisage the evolutions of the transversedeformations during calculation thus that localizations of deforma-tions in the presence of the notches. Taking into account of a plas-ticity piezo-dependent is overall satisfactory.

References

[1] Kichenin J. Comportement thermomécanique du polyéthylène: application auxstructures gazières. Thèse de Doctorat de l’école polytechnique; 1992.

[2] Paquin A, Berveiller M. Différentes voies pour la modélisation micromécaniquedu comportement élastoviscoplastique des polymères semi-cristallins. LesCahiers de Rhéologie GFR-Rhéologie et structure 1996;15(1):23–8.

[3] Ouakka A, Dang Van K, Boytard K. Approche locale de la mécanique de larupture appliqué aux matériaux élasto-visco-plastique. 12éme Congrèsfrançais de mécanique AUM 1995;1:301–4.

[4] Ouakka A, Dang Van K, Gueugnaut D, Blouet P. An assessment of the defectsdamages in polyethylene Gaz pipes. In: 10th Int. conf. on deformation, yield

and fracture of polymers. Cambridge: The Chameleon Press Ltd.; 1997. p. 557–60.

[5] Kinloch AJ, Young RJ. Fracture behaviour of polymers. London: Chapman andHall; 1995.

[6] Colak OU. Modeling deformation behavior of polymers with viscoplasticitytheory based on overstress. Int J Plasticity 2005;21:145–60.

[7] Colak OU, Dusunceli N. Modeling viscoelastic and viscoplastic behavior of highdensity polyethylene (HDPE). J Eng Mater Technol 2006;128:572–8.

[8] Bahadur S. Strain hardening equation and the prediction of tensile strength ofrolled polymers. Polym Eng Sci 1973;13:266–72.

[9] Eyring H. Viscosity, plasticity and diffusion as examples of absolute reactionrates. J Chem Phys 1936;4:283–91.

[10] G’Sell C, Jonas JJ. Yield and transient effects during the plastic deformation ofsolid polymers. J Mater Sci 1981;16:1956–74.

[11] Dahoun A. Comportement plastique et textures de déformation des polymèressemi-cristallins en traction uniaxiale et en cisaillement simple. Thèse, InstitutNational Polytechnique de Lorraine, Nancy; 1992.

[12] G’sell C, Hiver J-M, Dahoun A. Int J Solid Struct 2002;39:3857.[13] Bai SL, Wang M. Polymer 2003;44:6537.[14] Zairi F, Aour B, Gloaguen JM, Nait-Abdelaziz M, Lefebvre JM. Comput Mater Sci

2006;38:202.[15] Aour B, Zairi F, Gloaguen JM, Naıit-Abdelaziz M, Rahmani O, Lefebvre JM. Int J

Mech Sci 2008;50:589.[16] Zairi F, Aour B, Gloaguen JM, Naıit-Abdelaziz M, Lefebvre JM. Scripta Mater

2007;56:105.[17] Zairi F, Aour B, Gloaguen JM, Naıit-Abdelaziz M, Lefebvre, JM. Polym Eng Sci

2008. doi:10.1002/pen.21042.[18] Oden JT. Finite elements of non linear continua. McGraw Hill Company; 1972.[19] Dhatt G, Touzot G. In: Maloine SA, editor. Une présentation de la méthode des

éléments finis, Deuxième edition. Université de Compiègne; 1984.[20] Renon N. Simulation numerique par elements finis des grandes deformations

des sols. Application a la Scarification. These de Doctorat de l’Ecole des Minesde Paris; 2002.

[21] Temimi-Maaref N. PhD thesis. Ecole des Mines de Paris, Paris; 2006.[22] Zienwicz OC, Taylor RL. In: AFNOR, editor. La méthode des éléments finis –

formulation de base et problèmes linéaires; 1991.[23] Germain P. Cours de mécanique des milieux continus. Paris: Masson; 1973.[24] Lemaitre J, Chaboche J-L. Mech Solid Mater. New York: Cambridge Univ. Press;

1990.[25] Tillier Y. PhD thesis. Ecole des Mines de Paris, Paris; 1998.[26] G’Sell C, Jonas JJ. J Mater Sci 1979;14:583.