Large-strain viscoelastic–viscoplastic constitutive modeling of semi-crystalline polymers and model identification by deterministic/evolutionary approach Hemin Abdul-Hameed ⇑ , Tanguy Messager ⇑ , Fahmi Zaïri ⇑ , Moussa Naït-Abdelaziz ⇑ Univ Lille Nord de France, F-59000 Lille, France Université Lille 1 Sciences et Technologies, Laboratoire de Mécanique de Lille (LML), UMR CNRS 8107, F-59650 Villeneuve d’Ascq, France article info Article history: Received 2 August 2013 Received in revised form 18 March 2014 Accepted 21 March 2014 Keywords: Semi-crystalline polymers Visco-hyperelasticity Viscoplasticity Identification procedure abstract Above the glass transition temperature, a semi-crystalline polymer can behave like an elastomer or a stiff polymer according to the crystal content. For a reliable design of such polymeric materials, it is of prime importance to dispose a unified constitutive modeling able to capture the transition from thermoplastic- like to elastomeric-like mechanical response, as the crystal content changes. This work deals with poly- ethylene materials containing a wide range of crystal fractions, stretched under large strains at room temperature and different strain rates. A large-strain viscoelastic–viscoplastic approach is adopted to describe the mechanical response of these polymers. In order to identify the model parameters, an ana- lytical deterministic scheme and a practical, ‘‘engineering-like’’, numerical tool, based on a genetic algo- rithm are developed. A common point of manipulated constitutive models is that the elementary deformation mechanisms are described by two parallel resistances; one describes the intermolecular interactions and the other deals with the molecular network stretching and orientation process. In a first approach, the semi-crystalline polymers are considered as homogeneous media; at each crystal content, the semi-crystalline polymer is thus considered as a new material and a new set of model parameters is provided. In a second approach, the semi-crystalline polymer is seen as a two-phase composite, and the effective contribution of the crystalline and amorphous phases to the overall mechanical response is inte- grated in the constitutive model, which allows simulating the transition from thermoplastic-like to elas- tomeric-like mechanical response. In this case, one set of model parameters is needed, the only variable being the crystal volume fraction. The identification results obtained using deterministic and numerical methods are discussed. Ó 2014 Elsevier B.V. All rights reserved. 1. Introduction Nowadays, thermoplastic polymers are widely used for engi- neering applications. However, the prediction of their mechanical behavior remains a complex task since these materials have a highly non-linear stress–strain relationship depending on several external (strain rate, temperature) and structural (entanglement, cross-linking, crystal content, crystalline lamellae size, lamellae distribution) factors. Besides, large plastic strains may be locally reached in polymer components. Over decades many constitutive models have emerged to predict the stress–strain behavior of amorphous thermoplastic polymers both in the glassy and rubbery states and many works have been done to improve the constitutive models to adapt and respond to different material types under dif- ferent conditions, as reviewed in many works [1–22]. The physi- cally-based constitutive models for the glassy amorphous polymers are inspired from the early work of Haward and Thackray [23] founded on the observation of a large recoverable extension under glass transition points. In these constitutive models, a visco- plastic dashpot for the intermolecular interactions is connected to a non-linear spring to simulate the alignment of the polymer chains at large strains. To predict the stress–strain behavior of semi-crystalline polymers, many authors [24–33] used purely phe- nomenological constitutive models. Inherent to the structure of these models is the absence of linking to the microstructure which prevents the understanding and prediction of crystal content effects on the overall mechanical response [34–42]. Recently, http://dx.doi.org/10.1016/j.commatsci.2014.03.043 0927-0256/Ó 2014 Elsevier B.V. All rights reserved. ⇑ Address: Laboratoire de Mécanique de Lille, Université Lille 1 Sciences et Technologies, Cité Scientifique, F-59650 Villeneuve d’Ascq, France. Tel.: +33 3 20 51 57 78. E-mail addresses: [email protected](H. Abdul-Hameed), [email protected](T. Messager), [email protected](F. Zaïri), [email protected](M. Naït-Abdelaziz). Computational Materials Science 90 (2014) 241–252 Contents lists available at ScienceDirect Computational Materials Science journal homepage: www.elsevier.com/locate/commatsci
Hemin Abdul-Hameed ⇑, Tanguy Messager ⇑, Fahmi Zaïri ⇑, Moussa Naït-Abdelaziz ⇑Univ Lille Nord de France, F-59000 Lille, FranceUniversité Lille 1 Sciences et Technologies, Laboratoire de Mécanique de Lille (LML), UMR CNRS 8107, F-59650 Villeneuve d’Ascq, France
a r t i c l e i n f o a b s t r a c t
Article history:Received 2 August 2013Received in revised form 18 March 2014Accepted 21 March 2014
Above the glass transition temperature, a semi-crystalline polymer can behave like an elastomer or a stiffpolymer according to the crystal content. For a reliable design of such polymeric materials, it is of primeimportance to dispose a unified constitutive modeling able to capture the transition from thermoplastic-like to elastomeric-like mechanical response, as the crystal content changes. This work deals with poly-ethylene materials containing a wide range of crystal fractions, stretched under large strains at roomtemperature and different strain rates. A large-strain viscoelastic–viscoplastic approach is adopted todescribe the mechanical response of these polymers. In order to identify the model parameters, an ana-lytical deterministic scheme and a practical, ‘‘engineering-like’’, numerical tool, based on a genetic algo-rithm are developed. A common point of manipulated constitutive models is that the elementarydeformation mechanisms are described by two parallel resistances; one describes the intermolecularinteractions and the other deals with the molecular network stretching and orientation process. In a firstapproach, the semi-crystalline polymers are considered as homogeneous media; at each crystal content,the semi-crystalline polymer is thus considered as a new material and a new set of model parameters isprovided. In a second approach, the semi-crystalline polymer is seen as a two-phase composite, and theeffective contribution of the crystalline and amorphous phases to the overall mechanical response is inte-grated in the constitutive model, which allows simulating the transition from thermoplastic-like to elas-tomeric-like mechanical response. In this case, one set of model parameters is needed, the only variablebeing the crystal volume fraction. The identification results obtained using deterministic and numericalmethods are discussed.
� 2014 Elsevier B.V. All rights reserved.
1. Introduction
Nowadays, thermoplastic polymers are widely used for engi-neering applications. However, the prediction of their mechanicalbehavior remains a complex task since these materials have ahighly non-linear stress–strain relationship depending on severalexternal (strain rate, temperature) and structural (entanglement,cross-linking, crystal content, crystalline lamellae size, lamellaedistribution) factors. Besides, large plastic strains may be locallyreached in polymer components. Over decades many constitutive
models have emerged to predict the stress–strain behavior ofamorphous thermoplastic polymers both in the glassy and rubberystates and many works have been done to improve the constitutivemodels to adapt and respond to different material types under dif-ferent conditions, as reviewed in many works [1–22]. The physi-cally-based constitutive models for the glassy amorphouspolymers are inspired from the early work of Haward and Thackray[23] founded on the observation of a large recoverable extensionunder glass transition points. In these constitutive models, a visco-plastic dashpot for the intermolecular interactions is connected toa non-linear spring to simulate the alignment of the polymerchains at large strains. To predict the stress–strain behavior ofsemi-crystalline polymers, many authors [24–33] used purely phe-nomenological constitutive models. Inherent to the structure ofthese models is the absence of linking to the microstructure whichprevents the understanding and prediction of crystal contenteffects on the overall mechanical response [34–42]. Recently,
Fig. 1. Schematic illustration of the strain multiplicative decomposition.
242 H. Abdul-Hameed et al. / Computational Materials Science 90 (2014) 241–252
Ayoub et al. [42] proposed a physically-based constitutive modelfor semi-crystalline polymers integrating explicitly the crystal con-tent in the mathematical formulation. The constitutive model isbased on the Boyce et al. [7] approach, which was later extendedby Ahzi et al. [36] to capture the strain-induced crystallization ofthe initially amorphous polyethylene terephthalate. Inherent tothe model structure is the assumption that the resistance in thesemi-crystalline polymer to deformation is the sum of elastic–viscoplastic crystalline and amorphous intermolecular resistancesand a visco-hyperelastic network resistance. The Ayoub et al.[42] constitutive model was able to capture the transition fromthermoplastic-like to elastomeric-like mechanical response ofpolyethylene, as the crystal content changes. The authors [42]identified the model parameters using an analytical deterministicscheme. The deterministic procedure uses a precise algorithm ofidentification and calculates a unique set of model parameters[17]. However, the application of this method demands anadvanced expertise in model formulation and wide experience inthe understanding of experimental material behavior. Moreover,during the cycle of model-experiments comparison and model(re)formulation, the complexity of material response leads oftento an excessive number of parameters which do not necessarilyall have the reason to be in the model.
The identification procedure is the main barrier of handling theconstitutive models. The difficulty of model parameters identifica-tion is proportional with the complexity of the constitutive modelitself; it is time wasting. An efficient technique of parameters iden-tification is vital to a utility of constitutive models. Thus, develop-ing a software solution for model parameters identification allowstime and cost effective solution and keeps constitutive modelsmuch more useful [43–53]. In the present work, we intend todevelop and evaluate a numerical tool which allows to identifythe overall constitutive model variable sets, directly from stress–strain curves, to contrast with the analytical deterministicapproach. Moreover, such identification procedures should be reli-able, useful and convenient for a large number of users. Thenumerical tool proposed in this work is dedicated to non-special-ists of mathematical optimization heuristics, thus to exhibiting apractical ‘‘engineering-like’’ tool design. A great deal of researcheshave been conducted on the model parameters identification formetals, but few researches have been carried out to determinethe model parameters for polymers, especially at large strains.
The present work is focused on the constitutive modeling ofsemi-crystalline polymers but also on the problem of modelparameters identification. A genetic algorithm (GA) based optimi-zation procedure is designed to determine the parameters oflarge-strain viscoelastic–viscoplastic constitutive models, and theresults determined by GA compared to results of an analyticaldeterministic scheme. The application is performed on polyethyl-ene with a wide range of crystal fractions including thermoplasticelastomer and stiff thermoplastic polymer mechanical responses.To illustrate the interest, reliability and usefulness of the proposednumerical identification tool, two constitutive models are retained.One [7] supposes the semi-crystalline material as homogeneousand the other [42] considers it as heterogeneous. The robustnessof both constitutive models is examined. A secondary objective isto revise the Ayoub et al. [42] constitutive model in order to reducethe required model parameters.
The present paper is organized as follows. In Section 2, we reviewlarge-strain viscoelastic–viscoplastic constitutive modelingapproaches. In Section 3, we present the analytical deterministicscheme. In Section 4, we present the computational formulation ofthe problem and the designed GA-based identification tool. In addi-tion the robustness, reproducibility and uniqueness of the solutionsare discussed. In Section 5, the identification results are presentedand discussed. The concluding remarks are detailed in Section 6.
In this section, the main elements of two recently developedconstitutive models for thermoplastic polymers are summarized.The first one (referred to the BSL model for ‘‘Boyce–Socrate–Llanamodel’’) considers the material as homogeneous while the secondone (referred to the MBSL model for ‘‘modified BSL model’’) treatsthe material as heterogeneous by distinguishing amorphous andcrystalline phases. These constitutive models satisfy the contin-uum mechanics rules within the context of the large-strain visco-elastic–viscoplastic framework. A common point of theseconstitutive models is that the resistance to deformation in thesemi-crystalline polymers is the sum of a resistance A describingthe intermolecular interactions and a resistance B describing themolecular network stretching and orientation process. The inter-molecular resistance is composed of a linear elastic spring in serieswith a viscoplastic damper and the molecular network resistance iscomposed of a non-linear spring in series with a viscous damper.
As a point of departure, a summary of the finite strain kinematicframework is given. The key quantity is the deformation gradientdefined by: F = ox/oX where x is the position of a material pointin the current configuration and X is the position of this materialpoint in the initial configuration. Note that all tensors are denotedby bold-face symbols.
Due to the model structure, the Taylor assumption prevails, i.e.the intermolecular deformation gradient FA and the network defor-mation gradient FB are equal to the total deformation gradient F:
F ¼ FA ¼ FB ð1Þ
Note that for the MBSL constitutive model the effective contri-bution of the crystalline and amorphous phases to the overallintermolecular resistance are also integrated with the Taylorassumption:
FA ¼ FaA ¼ Fc
A ð2Þ
where the superscripts a and c denote the amorphous and crystal-line phases, respectively.
Following the Lee [54] decomposition, schematically illustratedin Fig. 1, the deformation gradient tensors can be further decom-posed multiplicatively into elastic (network) and viscoplastic(flow) parts as:
FA ¼ FeAFp
A ð3Þ
FB ¼ FNB FF
B ð4Þ
where the superscripts e, p, N and F denote the elastic, viscoplastic,network and flow parts, respectively. Note that the decompositiongiven in Eq. (3) is also applicable to crystalline and amorphousphases in the case of the MBSL constitutive model.
According to the polar decomposition, the deformation gradienttensors can be further decomposed into stretch and rotationmovements:
FA ¼ VeARe
AVpARp
A ð5Þ
H. Abdul-Hameed et al. / Computational Materials Science 90 (2014) 241–252 243
FB ¼ VNB RN
B VFBRF
B ð6Þ
The rate kinematics LA ¼ _FAF�1A for resistance A and LB ¼ _FBF�1
B
for resistance B are described by the velocity gradients:
LA ¼ _FeAFe�1
A þ FeA
_FpAFp�1
A Fe�1
A ¼ LeA þ Lp
A ð7Þ
LB ¼ _FNB FN�1
B þ FNB
_FFBFF�1
B FN�1
B ¼ LNB þ LF
B ð8Þ
in which the dot expresses the time derivative. The plastic and flowparts Lp
A and LFB of the velocity gradients may be written as:
LpA ¼ Fe
A_Fp
AFp�1
A Fe�1
A ¼ DpA þWp
A ð9Þ
LFB ¼ FN
B_FF
BFF�1
B FN�1
B ¼ DFB þWF
B ð10Þ
where DpA and DF
B are the rates of inelastic deformation and, WpA and
WFB are the inelastic spins which are assumed, without loss of gen-
erality, equal to zero. In addition to be irrotational, the viscoplasticflow is assumed incompressible, i.e. det Fp
A ¼ det FFB ¼ 1.
The plastic and flow deformation gradients FpA and FF
B are com-puted by integrating the following formula derived from Eqs. (9)and (10):
_FpA ¼ Fe�1
A DpAFe
AFpA ð11Þ
_FFB ¼ FN�1
B DFBFN
B FFB ð12Þ
in which the rates of inelastic deformation DpA and DF
B must beprescribed.
In the case of the MBSL constitutive model, two rates of inelasticdeformation Dp c
A and Dp aA must be specified for the crystalline and
amorphous domains, leading to two distinct viscoplastic and elas-tic deformation gradients for the two phases. The elastic and net-work deformation gradients Fe
A and FNB are obtained using Eqs. (3)
and (4).
2.1. Boyce–Socrate–Llana (BSL) constitutive model
The first constitutive model presented in this paper is that pro-posed by Boyce et al. [7]. This model is based on two basic resis-tances, as can be seen in Fig. 2.
2.1.1. Resistance A: intermolecular interactionsThe intermolecular resistance is constituted by a linear spring in
series with a viscoplastic damper. The intermolecular Cauchystress TA is related to the stretch part of the elastic deformationgradient Ve
A by the following constitutive relationship:
TA ¼1Je
A
CeA lnðVe
AÞ ð13Þ
A: I
nter
mol
ecul
arre
sist
ance B
: Netw
ork resistance
Linear E, v
NonlinearG, s,
NonlinearCr , Nr
NonlinearC
StiffnessNetwork orientation
Molecular relaxation
Viscoplasticity
A: I
nter
mol
ecul
arre
sist
ance B
: Netw
ork resistance
Linear E, v
NonlinearG, s,
NonlinearCr , Nr
NonlinearC
StiffnessNetwork orientation
Molecular relaxation0γ
Viscoplasticity
Fig. 2. Schematic representation of the BSL constitutive model.
in which JeA ¼ det Fe
A is the elastic volume change, lnðVeAÞ is the Hen-
cky strain and CeA is the fourth-order elastic stiffness tensor
expressed, in the isotropic case, as follows:
ðCeAÞijkl ¼
E2ð1þ mÞ ðdikdjl þ dildjkÞ þ
2m1� 2m
dijdkl
� �ð14Þ
in which two parameters are involved: the Young’s modulus E andthe Poisson’s ratio m. The term d represents the Kronecker-deltasymbol.
The viscoplastic strain rate tensor DpA is described by the follow-
ing flow rule:
DpA ¼ _cp
A
T0Affiffiffi2p
sA
ð15Þ
where T0A ¼ TA � traceðTAÞ=3I is the deviatoric part of TA,sA ¼ ðT0A � T
0A=2Þ1=2 is the effective stress and _cp
A is the viscoplasticshear strain rate:
_cpA ¼ c0 exp �DG
kh1� sA
s
� �� �ð16Þ
in which three parameters are involved: the pre-exponential factorc0, the activation energy DG and the athermal shear strength s. Theterms k and h denote the Boltzmann’s constant and the absolutetemperature, respectively.
2.1.2. Resistance B: network stretching and orientation processThe molecular network resistance is constituted by a non-linear
spring in series with a viscous damper. The molecular networkCauchy stress TB is expressed as a function of the elastic deforma-tion gradient FN
B using a relationship involving an inverse Langevinfunction L�1 [7]:
TB ¼1JN
B
Cr
3
ffiffiffiffiffiffiNrp
�kNB
L�1�kN
BffiffiffiffiffiffiNrp� �
BN � ð�kNB Þ
2I
h ið17Þ
in which two parameters are involved: the initial hardening modu-lus Cr and the limiting chain extensibility
ffiffiffiffiffiffiNrp
. The term JNB ¼ det FN
B
is the network volume change, I is the identity tensor and �kNB is
given by: �kNB ¼ ½traceðBNÞ=3�1=2
in which BN ¼ ðJNB Þ�2=3
FNB ðF
NB Þ
T.
The flow strain rate tensor DFB is described by the following flow
rule:
DFB ¼ _cF
BT0Bffiffiffi2p
sB
ð18Þ
where T0B ¼ TB � traceðTBÞ=3I is the deviatoric part of TB,sB ¼ ðT0B � T
0B=2Þ1=2 is the effective stress and _cF
B is the flow shearstrain rate:
_cFB ¼ CðkF
B � 1Þ�1sB ð19Þ
in which one parameter is involved: the viscous parameter C. Theterm kF
B is given by:
kFB ¼ ½traceðBFÞ=3�1=2
in which BF ¼ FFBðF
FBÞ
T
The overall Cauchy stress tensor T is the sum of the intermolec-ular Cauchy stress tensor TA and the network Cauchy stress tensorTB:
T ¼ TA þ TB ð20Þ
The intermolecular part of the BSL model involves five parame-ters and its network part three parameters, totaling eightparameters.
A:
Inte
rmol
ecul
ar r
esis
tanc
e
B: N
etwork resistance
Linear Ea , va
Nonlinear
ΔGa , sa ,
NonlinearCr , Nr
NonlinearC
Nonlinear
ΔGc , sc ,
Linear Ec , vc
00
Fig. 4. Schematic representation of the MBSL constitutive model.
244 H. Abdul-Hameed et al. / Computational Materials Science 90 (2014) 241–252
2.2. Modified Boyce–Socrate–Llana (MBSL) constitutive model
Semi-crystalline materials may be considered as heterogeneousmaterials which comprise both amorphous and crystallinedomains coupled in a rather complex manner. In order to treatthe large-strain viscoelastic–viscoplastic response of polyethylenematerials containing a wide range of crystal fractions and a rub-bery amorphous phase, Ayoub et al. [42] proposed a micromechan-ical approach inspired from that initially proposed by Ahzi et al.[36] for the strain-induced crystallization of initially amorphouspolyethylene terephthalate above the glass transition temperature.The constitutive model proposed by Ayoub et al. [42] introducesthe crystal volume fraction as variable and is able to capture thetransition from thermoplastic-like to elastomeric-like response ofpolyethylene when the crystal content vcv decreases. This materialcharacteristic is illustrated in Fig. 3a in which it can be observedthat a decrease in crystal content leads to a decrease in initial elas-tic stiffness, a more gradual rollover to yield, a decrease both inyield stress and in strain hardening slope. From a micromechanicalpoint of view, the semi-crystalline material may be modeled as atwo-phase composite, as depicted in Fig. 3b, constituted by crystal-line and rubbery amorphous domains. The micromechanicshomogenization scheme will allow reaching the macro-scalebehavior. Both crystalline and amorphous domains are supposedto participate to intermolecular interactions by acting in parallel.The rheological representation of the model is given in Fig. 4.
2.2.1. Resistance A: intermolecular interactionsThe resistance A represents the contribution of both crystalline
and amorphous phases and effect of both phases can be treated in acomposite framework. The overall intermolecular Cauchy stress TA
is given by [42]:
TA ¼ ðvcvÞbTc
A þ ð1� vcvÞbTa
A ð21Þ
where vcv is the crystal volume fraction, and b is a material constantto account for interaction phenomena. The b constant has been esti-mated in [42] by inverse method using polyethylene systems over awide range of crystal fractions and a value of 3.8 has been found. Tc
A
and TaA are the crystalline and amorphous Cauchy stresses, respec-
tively, related to the corresponding stretch part of the elastic defor-mation gradient Ve c
A and Ve aA by the following constitutive
relationships:
TcA ¼
1Je c
A
Ce c lnðVe cA Þ ð22Þ
(a
(b
Fig. 3. Structure–response relationship and micromechanics-based modeling: (a) micromechanical response above the glass transition temperature and (b) micromechanical t
TaA ¼
1Je a
A
Ce a lnðVe aA Þ ð23Þ
in which Je iA ¼ det Fe i
A is the elastic volume change, lnðVe iA Þ is the
Hencky strain and Ce iA is the fourth-order elastic stiffness tensor
of the crystalline and amorphous phases; the exponent i denotesthe phase under consideration (crystalline c or amorphous a).Assuming both phases isotropic, Ce i
A can be expressed as:
ðCe cA Þijkl ¼
Ec
2ð1þ mcÞðdikdjl þ dildjkÞ þ
2mc
1� 2mcdijdkl
� �ð24Þ
ðCe aA Þijkl ¼
Ea
2ð1þ maÞðdikdjl þ dildjkÞ þ
2ma
1� 2madijdkl
� �ð25Þ
in which four parameters are involved: the Young’s moduli Ec andEa, and the Poisson’s ratios mc and ma.
The viscoplastic strain rate tensors of the crystalline portionDp c
A and the amorphous portion Dp aA are given by:
Dp cA ¼ _cp c
A
T0 cAffiffiffi2p
scA
ð26Þ
Dp aA ¼ _cp a
A
T0 aAffiffiffi2p
saA
ð27Þ
where T0 cA and T0 a
A are the deviatoric parts of TcA and Ta
A, respectively,sc
A ¼ ðT0 cA � T
0 cA =2Þ1=2 and sa
A ¼ ðT0 aA � T
0 aA =2Þ1=2 are the effective stres-
ses and, _cp cA and _cp a
A are the flow shear strain rates which follow thesame expressions:
)
)
structure of semi-crystalline polyethylene materials and corresponding large-strainreatment using the volume fraction concept.
Table 1Identified parameters by deterministic approach for the BSL constitutive model.
H. Abdul-Hameed et al. / Computational Materials Science 90 (2014) 241–252 245
_cp cA ¼ c0 exp �DGc
kh1� sc
A
jsc
� �� �ð28Þ
_cp aA ¼ c0 exp �DGa
kh1� sa
A
sa
� �� �ð29Þ
in which five parameters are involved: the pre-exponential factorc0, the activation energies DGc and DGa, and the athermal shearstrengths sc and sa. The couple of parameters DGa and sa capturebarrier to molecular chain segment rotation in the amorphousphase whereas the couple of parameters DGc and sc capture barrierto crystallographic shear in the crystalline phase. The term j, intro-duced in Eq. (28), is a scale factor taking into account the effect ofcrystal morphology (depending on crystal content) on the crystalflow shear strain rate. In order to reduce the number of modelparameters, it is worth noticing that the shear strengths sc and sa
are considered to be constant with the plastic stretch, which consti-tutes a valuable difference with the MBSL constitutive model ini-tially proposed by Ayoub et al. [42].
2.2.2. Resistance B: network stretching and orientation processIt is assumed that the strain hardening response is dominated
by molecular orientation rather than crystallographic orientation.The resistance B is also constituted by a non-linear spring in serieswith a viscous damper and the same equations listed earlier in Sec-tion 2.1.2 are used. The sum of the intermolecular Cauchy stressand the network Cauchy stress gives the overall Cauchy stress inthe semi-crystalline polymer. The MBSL model requires ten inputdata for its intermolecular part and three others for its networkpart, totaling thirteen parameters.
3. Deterministic identification of model parameters
In this section, the methodology to identify the BSL and MBSLparameters following a classical analytical deterministic methodis detailed: The approach consists of successively identifying thedifferent branches of the constitutive model following a ‘‘step-by-step’’ methodology. The application is performed on polyethyl-ene materials at different crystal volume fractions, and stretched atdifferent strain rates. The details of the experiments (microscopicand macroscopic mechanical characterizations) can be found else-where [42].
3.1. Identification of BSL model parameters
The initial elastic stiffness E is obtained from the initial slope ofthe stress–strain curve. The pre-exponential factor c0 is prescribedto a value of 1.75 � 106 s�1 [7] which will be taken in all what
Fig. 5. Shear yield stress as a function of the normalized strain rate.
follows. Fixing c0 to this value, the activation energy DG and theshear strength s are determined using the following relation:
sy;A ¼skhDG
ln_cp
A
c0
� �þ s ð30Þ
in which _cpA and sy,A were approximated as
ffiffiffi3p
_e and ry=ffiffiffi3p
, respec-tively, _e being the applied strain rate and ry the experimentallymeasured yield stress. The shear yield stress being plotted as a func-tion of the normalized strain rate for the three polyethylene mate-rials (Fig. 5), the parameters DG and s are simultaneouslydetermined by a linear regression method using Eq. (30). The net-work parameters, Cr, Nr and C, are the outcome of a fitting procedureon the strain hardening. The parameter values determined by this
True strain
(c)
0
5
10
15
20
25
30
35
0 0.5 1 1.5 2
0 0.5 1 1.5 2
True strain
Tru
e st
ress
(M
Pa)
Simulation
0.01 s-1
0.005 s-1
0.001 s-1
0.0005 s-1
0.0001 s-1
o Experimental
Fig. 6. Stress–strain curves of the BSL constitutive model following the determin-istic identification results for different crystal volume fractions: (a) vcv = 0.72, (b)vcv = 0.30, and (c) vcv = 0.15.
Table 2Identified parameters by deterministic approach for the MBSL constitutive model.
Ec (MPa) Ea (MPa) DGc (J) DGa (J) sc (MPa) sa (MPa)
4490 4.7 1.25 � 10�19 2.12 � 10�19 78.84 0.55
246 H. Abdul-Hameed et al. / Computational Materials Science 90 (2014) 241–252
method are listed in Table 1. The comparison between experimentaland simulated stress–strain curves can be found in Fig. 6. The BSLmodel is able to capture the experimental observations, over therange of crystallinities and strain rates under investigation, to a sat-isfactory extent.
3.2. Identification of MBSL model parameters
Fig. 7 presents the parameters E, DG, j and s as a function of thecrystal volume fraction, the trends being captured by non-linearregressions. Following the MBSL decomposition of amorphousand crystalline domains, it is possible to focus on the specific cor-responding parameters:
fEa;DGa; sag for vc ¼ 0fEc;DGc; scg for vc ¼ 1
ð31Þ
The parameter values of amorphous and crystal domains aregiven in Table 2. Keep in mind that a key assumption of thisapproach is that the properties of crystalline and amorphousdomains are the same whatever the crystal content. That is a
0
1000
2000
3000
4000
0 0.25 0.5 0.75 1
Volume fraction of crystal phase
E (
MP
a)
Deterministic data
Constitutive model
1.0E-19
1.5E-19
2.0E-19
2.5E-19
0 0.25 0.5 0.75 1
Volume fraction of crystal phase
G (J
)
Deterministic data
Nonlinear regression
1.E-13
1.E-07
2.E-07
3.E-07
4.E-07
5.E-07
0 0.25 0.5 0.75 1
Volume fraction of crystal phase
C (
MP
a.s)
-1
Deterministic data
Nonlinear regression
8.38.3 )(4500)1(5.4 cvcvE χ+χ−=
)69.4exp(108.721025.1 -2019cvG χ−×+×=Δ −
)4.5exp(101.78109.97 -09-10cvC χ×+×=
0
1000
2000
3000
4000
0 0.25 0.5 0.75 1
Volume fraction of crystal phase
E (
MP
a)
Deterministic data
Constitutive model
1.0E-19
1.5E-19
2.0E-19
2.5E-19
0 0.25 0.5 0.75 1
Volume fraction of crystal phase
G (J
)
Deterministic data
Nonlinear regression
1.E-13
1.E-07
2.E-07
3.E-07
4.E-07
5.E-07
0 0.25 0.5 0.75 1
Volume fraction of crystal phase
C (
MP
a.s)
-1
Deterministic data
Nonlinear regression
8.38.3 )(4500)1(5.4 cvcvE χ+χ−=
)69.4exp(108.721025.1 -2019cvG χ−×+×=Δ −
)4.5exp(101.78109.97 -09-10cvC χ×+×=
Fig. 7. Parameter evolutions plotted as a fu
consequence of the micromechanics homogenization conceptswhich see the semi-crystalline materials as two-phase composites.An important difference between crystalline and amorphousparameters may be observed. The network parameters, Cr, Nr andC, follow a monotonic evolution with the crystal volume fractionas shown in Fig. 7. The comparison between the model and theexperimental data is shown in Fig. 8. It can be observed that thetwo-phase MBSL constitutive model stays in good agreement withthe experimental data.
4. Numerical strategy of direct parameter identification
Contrary to the deterministic approach consisting in a ‘‘step-by-step’’ identification method, the present section details the
0
0.5
1
1.5
2
2.5
3
0 0.25 0.5 0.75 1
Volume fraction of crystal phase
Cr (
MP
a)
Deterministic data
Nonlinear regression
0
50
100
150
200
250
300
0 0.25 0.5 0.75 1
Volume fraction of crystal phase
Nr
Deterministic data
Nonlinear regression
))1023.6(1(37.2962.26 9 cvrC
χ−×−+−=
)1.7exp(2485.24 cvrN χ−+=
0
20
40
60
80
0 0.25 0.5 0.75 1
Volume fraction of crystal phase
s (M
Pa)
0
10
20
30
40
Deterministic data
Nonlinear regression
)5exp(54 cvχ−+=κ
)02.3exp(03.44.3 cvs χ+−=
0
0.5
1
1.5
2
2.5
3
0 0.25 0.5 0.75 1
Volume fraction of crystal phase
Cr (
MP
a)
Deterministic data
Nonlinear regression
0
50
100
150
200
250
300
0 0.25 0.5 0.75 1
Volume fraction of crystal phase
Nr
Deterministic data
Nonlinear regression
))1023.6(1(37.2962.26 9 cvrC
χ−×−+−=
)1.7exp(2485.24 cvrN χ−+=
0
20
40
60
80
0 0.25 0.5 0.75 1
Volume fraction of crystal phase
s (M
Pa)
0
10
20
30
40
Deterministic data
Nonlinear regression
)5exp(54 cvχ−+=κ
)02.3exp(03.44.3 cvs χ+−=
Deterministic data
Nonlinear regression
Deterministic data
Nonlinear regression
)5exp(54 cvχ−+=κ
)02.3exp(03.44.3 cvs χ+−=
nction of the crystal volume fraction.
H. Abdul-Hameed et al. / Computational Materials Science 90 (2014) 241–252 247
direct analysis of the whole stress–strain data, and enabling thedirect determination of the overall model parameter sets. Theexperimental curve fitting problem can be then formulated asan optimization search [17]: as discussed afterwards, it thusconsists in minimizing the discrepancies between experimentaland numerical overall stress–strain results. Such optimizationsearch problems, manipulating mixed variables, involve localoptima and then require global robust search procedures: aGA identification tool has thus been developed for the presentwork.
4.1. Formulation of the optimization problem
In this study, the experimental database available for a specificmaterial consists of NV stress–strain curves, NV being the number ofapplied strain rates. Each of these curves is deducted from a givennumber of experimental points, number noticed NP,L for the Lthvelocity, i.e. 1 6 L 6 NV. Hence, the behavior of a tested materialis described by a database of experimental stress values noticedTExp
L;K , where 1 6 L 6 NV and 1 6 K 6 NP,L.
(a)
0
20
40
60
80
100
120
True strain
Tru
e st
ress
(M
Pa)
(b)
0
20
40
60
80
100
True strain
Tru
e st
ress
(M
Pa)
(c)
0
10
20
30
40
0 0.5 1 1.5 2
0 0.5 1 1.5 2
0 0.5 1 1.5 2
True strain
Tru
e st
ress
(M
Pa)
Simulation
0.01 s-1
0.005 s-1
0.001 s-1
0.0005 s-1
0.0001 s-1
Simulation
0.01 s-1
0.005 s-1
0.001 s-1
0.0005 s-1
0.0001 s-1
o Experimental
o Experimental
o ExperimentalSimulation
0.01 s-1
0.005 s-1
0.001 s-1
0.0005 s-1
0.0001 s-1
Fig. 8. Stress–strain curves of the MBSL constitutive model following the deter-ministic identification results for different crystal volume fractions: (a) vcv = 0.72,(b) vcv = 0.30, and (c) vcv = 0.15.
Then, let us call TNumL;K ¼ TNum
L;K ðXÞ the corresponding stress values,deducted from a constitutive model. These numerical valuesdepend on the corresponding constitutive model variables set,i.e. X = {E; DG; s; Cr; Nr; C} for the BSL constitutive model andX = {Ec; Ea; DGc; DGa; sc; sa; Cr; Nr; C} for the MBSL constitutivemodel. X can be viewed as a design parameter set to optimize. Inthat way, the identification problem taken into considerationconsists to minimize the gap between the experimental TExp
L;K andnumerical TNum
L;K stresses, measured at the same strain rate. Follow-ing the approach presented by Pyrz and Zaïri [17], the identifica-tion problem consists to minimize the average discrepancy,normalized with respect to the NP,L points of the NV experimentalstress–strain curves. By this means, a reasonable objectivefunction, to be minimized, can be formulated as:
f ðXÞ ¼ 1NV
XNV
L¼1
1NP;L
XNP;L
K¼1
kTNumL;K � TExp
L;K k !
ð32Þ
It may be noticed that, as for the large majority of practicalengineering optimization problems [45,46], this non-linear formu-lation implies interdependent real and integer (i.e. continuous anddiscrete) variables.
4.2. Genetic algorithm identification tool
First, it could be underlined that the experimental databasesconsidered for the identification of model parameters induce largenumber of experimental points. Moreover, the mechanical behav-iors of the studied materials are strongly non-linear. Furthermore,the constitutive model parameters considered as design variables Xare both real and integer parameters, leading thus to a ‘‘mixed-variables’’ optimization problem. At last, and due to the constitu-tive model formulations detailed in Section 2, these variables arestrongly interdependent.
By this mean, such identification problems imply local optimaand non-convexity and large cardinality of the design space asmentioned by previous works of the literature [17,49,52]. Suchpractical difficulties are largely responsible for the interest grantedto global stochastic methods in engineering sciences from the lastdecades [43–46]. Indeed, evolutionary search, commonly usedtoday in engineering sciences, is naturally suited and convenientto treat with mixed variables and solve multi-modal and largeoptimization problems. Many works [46,53] have demonstratedthe reliability, the usefulness and efficiency of such procedures,allow to dealing with difficult, big size, high cardinality, continuousor discrete and non-convex ‘‘engineering-like’’ search problems. Inthat way, such allows to design numerical tool handy and dedi-cated to non-specialists of mathematical optimization heuristics[45,46]. A GA-based optimization procedure has been thusdesigned and computed for this study.
As can be reminded, GA belongs to the general class of evolu-tionary algorithms (EA): they are general purpose, stochasticsearch methods inspired by natural evolution [43]. The main ideaof such optimization procedures consists in processing at a timea fixed number of potential solutions X, called population. Eachof these individuals is characterized by its corresponding objectivefunction value f(X), called fitness. According to evolutionary theo-ries, the fittest (i.e. leading to the lowest fitness) individuals arelikely to form a new generation of solutions by recombining theirfeatures using a set of biologically inspired stochastic operators[43,44]: First, the selection step allows high probability to the fit-test individuals among the whole population to become parents.Next, the crossover operator recombines the genetic characteristics(i.e. the optimization parameters) of selected parents, thus produc-ing children individuals expected to improve the optimizationsearch. Some of the children are then arbitrary transformed by
248 H. Abdul-Hameed et al. / Computational Materials Science 90 (2014) 241–252
the mutation operator, preventing the population to become ahomogenous and to focus on a local optimum [45]. This simulatedreproduction procedure is renewed for a fixed number of genera-tions. At last, the GA yields the best individual (of minimal fitness)which found during this evolution process.
Numerous works of the literature, especially for optimizationwith discrete parameters, use binary encodings to represent theoptimization parameter set X [46]. The developed GA manipulatesdirectly real and integer variables. The corresponding genetic oper-ators have been chosen following recommendations of the litera-ture [44,46]. Their principles are schematically illustrated inFig. 9. As can be seen, the starting population of individuals X israndomly created. The fitness evaluation is allowed by couplingthe GA process with the experimental database and the consideredconstitutive model. Next, the selection step operates by tourna-ments between randomly chosen individuals. Besides, this selec-tion is elitist: the best individual of the current population isautomatically chosen in the following generation. The whole arith-metical crossover and the random uniform mutation are applied.
The genetic operators depend on probability levels, as remindedby the flowchart of Fig. 9. Accordingly to recommendations of theliterature and preliminary tests, crossover and mutation probabil-ity of PCross = 75 and PMut = 5, respectively, have been considered inthis study. The random point of crossover has been chosen to be0 6 q 6 1.5, using the boundary rebound technique [44,46]. Forthe numerical applications, the numbers of individuals and gener-ations have been chosen accordingly to the cardinalities of each ofthe treated identification problems and preliminary tests. The
Fig. 9. Flowchart of the numer
corresponding values NX and NG are detailed in the following sec-tion. Besides, it should be underlined that, due to the stochasticnature of the GA process, each identification problem studiedthereafter in this work has been analyzed by performing a mini-mum of 10 successive runs of the numerical tool depicted in Fig. 9.
5. Numerical identification results and discussion
The identification of constitutive model parameters has beenthen performed using the numerical approach, based on evolution-ary optimization, detailed in the previous section.
5.1. Preliminary numerical tests
First, in order to validate the developed numerical identificationtool, preliminary numerical tests have been carried out. The stress–strain curves under consideration have been numerically gener-ated using the BSL model: the objective is to re-identify the pre-scribed model parameters.
5.1.1. Parameter identification for static loadingThe first preliminary numerical tests performed take into con-
sideration the BSL constitutive model (see Section 2.1) to re-iden-tify the related model parameters for a given material butconsidering a static condition; the effect of viscosity in both inter-molecular and network resistances have been then neglected. Itthus allows minimizing the interdependence effects between
ical identification process.
Table 3Preset model parameters value used in Fig. 10.
E (MPa) s (MPa) Cr (MPa) Nr
1700 27 1.8 300
0
80
160
240
320
400
0 0.5 1 1.5 2 2.5 3
True strain
Tru
e st
ress
(M
Pa)
Intial data (BSL-generated)
Best result of GA-identification
Fig. 10. Initial (BSL-generated) and GA-identified numerical stress–strain curves.
Table 5Preset model parameters value used in Fig. 11.
E (MPa) DG (J) s (MPa) C (MPa s) �1 Cr (MPa) Nr
796 9.20 � 10�20 71 1.3 � 10�10 1.73 20
0
40
80
120
0 0.4 0.8 1.2 1.6 2
True strain
Tru
e st
ress
(M
Pa)
0.05 s-1
0.005 s-1
0.0005 s-1
Fig. 11. Initial (BSL-generated), best and average solution of GA-identified numer-ical stress–strain curves for different strain rates.
H. Abdul-Hameed et al. / Computational Materials Science 90 (2014) 241–252 249
model parameters. In that way, the material behavior is dependingonly on four parameters: E, s, Cr and Nr.
The prescribed fictive parameter values are presented in Table 3.Considering these parameters, a corresponding stress–strain curvehas been thus numerically generated from the reduced BSL consti-tutive model, as previously mentioned. This curve, comprisingNP,L = 400 points, is plotted in Fig. 10. As mentioned previously,the effect of strain rate is hence neglected, therefore NV = 1.
The robustness of the developed numerical identification tool(see Fig. 9) has been then evaluated by ‘‘re-identifying’’ the consti-tutive model parameters using this numerical stress–strain curvedepicted in Fig. 10. For each model parameter, the intervals ofsearch have been chosen. These intervals are detailed in Table 4.The numbers of generations and individuals of the GA process havebeen fixed to NG = 5000 and NX = 400, respectively.
For this evaluation, 100 successive runs of GA have been carriedout. Table 4 details the obtained results: first, the best solution,leading to the lowest fitness value f observed during the 100 runs.Moreover, the whole 100 identification results have been analyzedin terms of average values (and corresponding standard devia-tions), for each model parameter, as presented in Table 4.
As can be seen in Table 4, the identification procedure exhibits aperfect reproducibility: the model parameters (E, s, Cr, Nr) obtained(both for the best solution and for the average values) equate thechosen values given in Table 3. The corresponding standard devia-tions appear to be negligible. It thus demonstrates both the effi-ciency of the identification tool proposed and the uniqueness ofthe optimization solution. Moreover, it should be mentioned thatall of the stress–strain curves deducted from the GA-identifiedparameter sets are identical and match exactly the initial one gen-erated by using the BSL model as shown in Fig. 10.
5.1.2. Overall parameter identificationFor the second preliminary test set, the BSL constitutive model
has also been considered. However, the effects of viscosity have
Table 4Interval of search and re-identified values of the constitutive model parameters used in F
E (MPa) s (MPa)
Best solution 1700.003174 26.99999698Average value 1700.003174 26.99999684Standard deviation 7.94869 � 10�7 8.55143 � 10�7
Upper limit of search 3000 200Lower limit of search 200 10
been henceforth analyzed, unlike the previous tests discussed inSection 5.1.1 even the purpose being however similar: it consiststo re-identify model parameters using numerical stress–straincurves generated by the BSL model. The constitutive model param-eters to be re-identified are E, DG, s, Cr, Nr and C. It could be men-tioned that the intermolecular parameters DG and s (see Eq. (16))are interdependent with the parameter c0. In addition, networkparameters Cr and Nr are interdependent, as can be seen in Eq.(17), but both are related to the viscous parameter C.
Preset of constitutive model parameters are presented inTable 5. Considering these values, corresponding stress–straincurves have been thus numerically generated using the BSL model.In order to take into account the viscosity effects, three strain rateshave been considered (i.e. NV = 3): 0.05, 0.005 and 0.0005 s�1.These curves, plotted in Fig. 11, comprise NP,L = 400 points. Therobustness of the developed numerical identification tool (seeFig. 9) has been anew evaluated by ‘‘re-identifying’’ the constitu-tive model parameters using the stress–strain curves depicted inFig. 11. For each model parameter, the range of search have beenchosen sufficiently wide, as detailed in Table 6. The numbers ofgenerations and individuals of the GA process have been settledto NG = 15,000 and NX = 450, respectively.
20 successive runs of GA have been carried out. Table 6 detailsthe obtained results. For each material variable, the whole 20 iden-tification results have been also analyzed in terms of best solution(exhibiting the lowest fitness value f), average values (and corre-sponding standard deviations), as presented in Table 6. The resultshows non-uniqueness of identified parameters due to interdepen-dence of parameters above mentioned. However, and for the wholeparameter solutions, the corresponding stress–strain curves arealways in very good agreement with the initial ones, as the exam-ple depicted in Fig. 11.
250 H. Abdul-Hameed et al. / Computational Materials Science 90 (2014) 241–252
5.2. Parameters identification of BSL constitutive model
The BSL constitutive model treats materials as homogenousmedia. Thus, each PE material parameter must be identified sepa-rately; for each crystal content considered, a set of BSL modelparameters was found by using the developed numerical identifi-cation tool. The variables of the BSL constitutive model to be iden-tified are then X = {E; DG; s; Cr; Nr; C}. The model parameters to beidentified and their corresponding optimization ranges (i.e. upperand lower search limits) are listed in Table 7. Numbers of genera-tions and individuals are NG = 15,000 and NX = 490, respectively.For each of crystal content considered, 10 successive runs werealso performed. As noticed previously in Section 5.1, the identifica-tion tool appears to be robust, allowing an excellent reproducibilityof the results: for each run, the parameter sets obtained are leadingto similar fitness values. The best solutions of identified variablevalues are detailed in Table 7 for each crystal volume fractionconsidered.
Fig. 12 presents the simulated stress–strain curves, obtainedusing the identified parameters, and the experimental stress–strain curves. A good agreement between experiments and BSLconstitutive model can be observed. It is worth noticing that allthe GA-identified material parameters do not evolve monotoni-cally with respect to the crystal content, due to unrestricted limitof research. Thus, the numerical identification tool allows optimiz-ing parameters led to best fit with the experimental data comparedto the deterministic method.
. 2
(c)
0
10
20
30
40
True strain
Tru
e st
ress
(M
Pa)
0 0.5 1 1.5
True strain
Simulation
0.01 s-1
0.005 s-1
0.001 s-1
0.0005 s-1
0.0001 s-1
o Experimental
Fig. 12. Stress–strain curves of the BSL constitutive model following the GAidentification results for different crystal volume fractions: (a) vcv = 0.72, (b)vcv = 0.30, and (c) vcv = 0.15.
5.3. Parameters identification of MBSL constitutive model
As can be reminded, the MBSL constitutive model considerssemi-crystalline materials as heterogeneous media by taking thecrystal volume fraction as unique variable. Similarly at theapproach detailed in Section 5.2, the MBSL material parameterset X has been identified using the proposed identification proce-dure. The crystalline and amorphous Young’s moduli, Ec and Ea,and the interaction parameter j were not numerically identifiedbut fixed according to deterministic values. Thus, the optimizationproblem consists to identify the parameter set X = {DGc; DGa;sc; sa; Cr; Nr; C}.
The interdependencies between crystalline and amorphousphases play in different ways in the branches A and B of the MBSLmodel (see Fig. 4). The effect of the crystal volume fraction vcv canbe identified and then adjusted for the viscoplastic behavior
Table 8Identified parameters for the MBSL constitutive model and ranges of optimization search. The identification is performed using the experimental data of polyethylene with 0.72crystal volume fraction.
DGc (J) DGa (J) sc (MPa) sa (MPa) C (MPa s)�1 Cr (MPa) Nr
H. Abdul-Hameed et al. / Computational Materials Science 90 (2014) 241–252 251
(branch A) by opposition to the visco-hyperelastic response(branch B) where only macroscopic data can be analyzed. Theparameters of the viscoplastic branch A {DGc; DGa; sc; sa} and ofthe visco-hyperelastic branch B {Cr; Nr; C} have been identifiedusing merely the experimental data of polyethylene with 0.72crystal volume fraction. Table 8 indicates the upper and lowerbounds of the search of each corresponding parameter. Note thatthe parameters {Cr; Nr; C} evolve according to the deterministickinetics given in Fig. 7. Numbers of generations and individualsare NG = 15,000 and NX = 490, respectively.
10 successive runs of the identification tool have beenperformed. Table 8 provides the fittest result obtained for the
(a)
0
20
40
60
80
100
120
True strain
Tru
e st
ress
(M
Pa)
(b)
0
20
40
60
80
100
True strain
Tru
e st
ress
(M
Pa)
(c)
0
10
20
30
40
0 0.5 1 1.5 2
0 0.5 1 1.5 2
0 0.5 1 1.5 2
True strain
Tru
e st
ress
(M
Pa)
Simulation
0.01 s-1
0.005 s-1
0.001 s-1
0.0005 s-1
0.0001 s-1
Simulation
0.01 s-1
0.005 s-1
0.001 s-1
0.0005 s-1
0.0001 s-1
o Experimental
o Experimental
o ExperimentalSimulation
0.01 s-1
0.005 s-1
0.001 s-1
0.0005 s-1
0.0001 s-1
Fig. 13. Stress–strain curves of the MBSL constitutive model following the GAidentification results for different crystal volume fractions: (a) vcv = 0.72, (b)vcv = 0.30, and (c) vcv = 0.15.
parameter set X = {DGc; DGa; sc; sa; Cr; Nr; C}. Fig. 13 presents theidentification result in terms of stress–strain curves on the poly-ethylene containing 72% of crystal phase, and the model predic-tions for the two other crystal volume fractions (0.30 and 0.15).The global response is well reproduced by the model but it maybe also remarked that the strain rate dependence is not well cap-tured for the lowest crystal volume fractions.
6. Conclusion
This work was dedicated to the constitutive modeling of semi-crystalline polyethylene polymers and to strategies of parametersidentification. The large-strain viscoelastic–viscoplastic frameworkwas used to capture the thermoplastic/elastomeric transition inthe mechanical response. Two modeling strategies were used. Inthe first one, the semi-crystalline material is considered as a homo-geneous medium, and a set of model parameters is associated ateach crystal fraction. In the second one, a two-phase representa-tion of the semi-crystalline material is considered by distinguish-ing amorphous and crystalline domains, and only one set ofmodel parameters is required. The model parameters were identi-fied by providing two strategies: (i) the classical analytical deter-ministic method, proceeding by ‘‘step-by-step’’ parameteranalysis and (ii) a numerical identification tool, enabling to directlyidentify the whole parameter sets following an evolutionary opti-mization approach. For the two constitutive models, the identifiedparameter sets obtained led to stress–strain evolutions correctlymatching the experimental data.
The numerical identification methodology developed appears tobe a useful, simple and reliable technique. Indeed, the processallows to directly obtain the whole parameter sets, in contrast tothe deterministic one, which processes ‘‘step-by-step’’ to deter-mine successively the different parameters.
Moreover, using the two-phase model, the numerical identifica-tion approach appears to be predictive: The mechanical responseof polyethylene with different crystallinities can be deducted fromthe parameters identification of only one material crystal fraction.
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