Micromechanical modeling of isotropic elastic behavior of semicrystalline polymers

Acta Materialia 54 (2006) 1513–1523 www.actamat-journals.com

Micromechanical modeling of isotropic elastic behavior of semicrystalline polymers F. Be´doui, J. Diani *, G. Re´gnier, W. Seiler Laboratoire d’Inge´nierie des Mate´riaux (LIM, UMR CNRS 8006), ENSAM, 151 bd de l’Hoˆpital 75013 Paris, France Received 19 May 2005; received in revised form 10 November 2005; accepted 11 November 2005 Available online 20 January 2006

Abstract Considering semicrystalline polymers as heterogeneous materials consisting of an amorphous phase and crystallites, several micromechanical models have been tested to predict their elastic behavior. Two representations have been considered: crystallites embedded in a matrix and a layered-composite aggregate. Several homogenization schemes have been used in these representations. Firstly, comparisons between the models and experiments show that the micromechanics approach applies at this scale. Secondly, the results differ according to the rubbery or glassy state of the amorphous phase. Finally, the results suggest that the spherulitic mesostructure does not affect the material behavior while infinitesimal elastic strains are considered.
2005 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Polymers; Micromechanical modeling; Elastic behavior; Microstructure; X-ray diffraction

1. Introduction Semicrystalline polymers are increasingly being used as structural materials. Much work has focused on the prediction of the crystallinity and microstructure morphology of these materials [1–4], and several methods have been developed to characterize this microstructure by optical microscopy or scanning electron microscopy (SEM) after an etching processing [5] or by X-ray scattering [6]. Less work has been done in terms of relating the microstructure to the mechanical behavior. Semicrystalline polymers may be considered as heterogeneous materials, and micromechanical models can be used to estimate or predict their mechanical properties. Recent work [7–10] has explored this area, focusing on largestrain-scale elastoplastic or elastoviscoplastic behaviors. Very little work has been done at the small-strain level

*

Corresponding author. Fax: +33 1 44 24 69 90/62 90. E-mail address: [email protected] (J. Diani).

[11]. Even less is known about the elasticity of these materials. Previous work [12] has made it possible to understand a paradox involving polypropylene (PP) and polyethylene (PE) using micromechanical modeling. It has been shown that using a micromechanics approach may help explain why PP is more rigid than PE despite a lower crystallinity and less rigid amorphous and crystalline phases. Beyond this qualitative result, the applicability of micromechanical modeling for the quantitative prediction of the elastic properties of semicrystalline polymers remains an issue. In order to decide whether micromechanical modeling applies at the scale of the crystallites, three semicrystalline polymers are considered: an isotactic PP, a PE, and a polyethylene terephthalate (PET) crystallized by annealing. Firstly, the mechanical properties of the homogeneous materials were measured and the results considered in terms of crystallinity. Secondly, the microstructure was observed and the crystallite lamellae dimensions were estimated when possible. Finally, considering two schematic representations and several homogenization schemes, the estimates of the models were compared with experimental data.

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2005 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.actamat.2005.11.028

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2. Microstructure and mechanical properties

Table 1 Young’s modulus and crystalline volume for PP

2.1. Experimental results 2.1.1. Mechanical properties Three commercial homopolymers were chosen for this study: a PP (ELTEX PP HV 252) and a high-density PE (HD6070 EA) supplied by Solvay, and a PET (ARNITE 00D301) supplied by DSM. We are interested in the elastic isotropic behavior of these materials, and thus we focus on measurements and estimates of Young’s modulus. A few Poisson’s ratio values were also estimated. PP plates with a thickness of 1 and 3 mm were obtained by injection molding. Isotropic samples were taken from the core of the plates. The purpose of this operation was to obtain two different isotropic microstructures and crystallinities from the same material. The preparation consisted of keeping only 0.5 mm in the central part after a milling procedure. This operation was done to ensure an isotropic spherulitic microstructure for all samples, which leads to isotropic mechanical properties. PET plates with a thickness of 4 mm were injection molded in a cold mold regulated at a temperature of 10 C to get a fully amorphous material. Using the same procedure as that for the PP samples, isotropic samples of 0.5 mm were taken from the core of the plates. The amorphous samples were then annealed at 110 C for various times in order to obtain semicrystalline samples with different crystallinities. Tensile tests were performed using an Instron machine (model 4500) equipped with a uniaxial mechanical Instron extensometer. The crosshead speed was set to 1 mm/min according to the standard ISO 527. Samples were also characterized in terms of crystalline fraction, which was achieved by density measurements. All the crystallinities and the Young’s modulus values represent an average of three tests. The standard deviation of the Young’s modulus of the PET was calculated from eight measurements and is equal to 80 MPa; that of the PP was calculated from four measurements and is equal to 40 MPa. The standard deviation for the crystallinity is 2%. Tables 1 and 2 summarize the experimental values obtained. For the PE, experimental data were taken from the Solvay database (Table 3). To refine the comparison between the models and experiments, we also used three values of Poisson’s ratio. They were estimated according to the following equation from Young’s modulus measurements and bulk moduli determined from PVT data (see Ref. [12] for a detailed report):
 1 E m¼ 1
ð1Þ 2 3B For the PP of 57% of crystallinity, m is equal to 0.42; for the PE at 45% and 67% crystallinity, m is 0.49 and 0.46, respectively. 2.1.2. Microstructure Semicrystalline polymers are made up of crystalline lamellae embedded in an amorphous phase. The crystalline

1 mm PP plate 3 mm PP plate

Young’s modulus (MPa) (standard deviation = 40 MPa)

Crystalline volume (%) (standard deviation = 2%)

1740 1601

61 57

Table 2 Young’s modulus and crystalline volume for PET Young’s modulus (MPa) (standard deviation = 80 MPa) Crystalline volume (%) (standard deviation = 2%)

2600

2652

2838

3025

3390

0.5

2

25

32

41

Table 3 Young’s modulus and crystalline volume for PE (BP-Solvay database) Young’s modulus (MPa) Crystalline volume (%)

230 46

270 49

720 60

1000 65

1050 66

1150 70

1300 72

lamellae may be ordered into spherulite, ‘‘shish-kebab’’, or even bamboo microstructure [2–4]. Here, we are dealing with three isotropic semicrystalline polymers. The microstructure was observed using an optical microscope. In Fig. 1, images of two different samples of PP are shown. One obtains a different microstructure as regards the spherulite radius depending on processing conditions. Average spherulite radii of approximately 10 lm for the core of the 1 mm thick plates and 40 lm for that of the 3 mm thick plates are found from Fig. 1. Previous work [12] has demonstrated that one of the key morphological parameters governing the elastic behavior of the materials under investigation is the crystalline lamella shape ratio. To obtain this parameter, it is necessary to increase the contrast between the two phases (amorphous and crystalline) by etching the amorphous phase. Olley and Basset [5] have proposed an etching solution for polyolefins consisting of a mixture of sulfuric acid, orthophosphoric acid, and potassium permanganate. Details of the composition are reported in Table 4. PP samples were exposed to the solution for 14 h. For better results, samples were shaken during etching. The chemical etching allows the observation of the treated samples using optical microscopy or SEM. Fig. 2 shows two PP etched samples observed using an optical microscope; the amorphous phase appears dark while crystallites are white. Firstly, one can note that the average length of crystalline lamellae is much smaller than the spherulite dimension. Secondly, using Scion Image software developed by NASA [13], one can get a fair estimate of the average crystallite length. The average crystallite length in the core of the 1 and 3 mm thick plates were measured (Fig. 2). Despite the difference between the spherulite radii (10 lm for the plate of 1 mm thickness and 40 lm for the plate of 3 mm thickness), similar values of the average

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Fig. 1. Spherulite morphology in the core of injection molded isotactic PP plates: 1 mm plate thickness (left) and 3 mm plate thickness (right).

Table 4 PP etching solution composition [5] Component

Content (wt.%)

Content (vol.%)

q (g/cm3)

H2SO4 H3PO4 KMnO4

65.8 32.9 1.3

65 35 –

1.83 1.7 –

crystalline lamella lengths of approximately 1 lm were measured for both plates. As it is still difficult to extract an accurate value of the lamella thickness from Fig. 2, the thicknesses of the crystalline lamellae were determined

in our laboratory by small-angle X-ray scattering (SAXS) measurements. They are identical for both samples: about 12 nm [6]. This result is consistent with the literature as it is generally considered that lamella width should be about 10 times the thickness [14]. After annealing several amorphous samples of PET at high temperature (110 C), different crystallinities were obtained (25%, 32%, and 41%). A wide-angle X-ray scattering technique was used to characterize the lamella size for the different samples [15]. Measurements were performed using a Seifert diffractometer with Cu Ka radiation. A quantitative study of the crystalline lamella morphology

Fig. 2. Lengths of lamellae in isotactic PP spherulite. Left: 3 mm thick PP plate; right: 1 mm thick PP plate.

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was carried out through a 2h integration. Peak profiles were deconvoluted using ORIGIN software [16]. Crystalline peaks were analyzed as Pearson VII curves. The first crystallographic plane of interest is (105); its normal is close to the chain axis direction and consequently provides information on the lamella thickness. Planes (1 0 0) and (0 1 0) provide dimensions for lamella width and length, respectively. Crystal sizes along directions that are normal to these planes are calculated using the Scherrer relation: Lhkl ¼

kk cos hhkl  Dhhkl

ð2Þ

where Dhhkl is the angular width of the diffracted (hkl) plane band and k is a proportionality factor equal to 0.9. The parameter Dhhkl is usually considered as the width of the peak at its mid-height. The planes (105), (1 0 0), and (0 1 0) are recorded at 2h angles of 65.8, 33.8, and 24.9 for Cu Ka radiation [17]. Fig. 3 shows a 2h diffraction pattern for the sample of 32% crystallinity. Table 5 shows values obtained using Eq. (2) for crystalline lamella length, width, and thickness for the different crystallinities. For PE, the chemical etching method of Olley and Basset [5] did not give us accurate images to determine the lamella dimensions and only the lamella thickness can be obtained using SAXS [18]. The microstructure has been proved to be spherulitic, but lamella dimensions are difficult to estimate. Hence, micromechanical modeling is used in Section 4 to estimate lamella dimensions regarding the experimental values of Young’s modulus vs. crystallinity.

4

1.2x10

4

1.0x10

3

Intensity

Amorphous 3

6.0x10

3

4.0x10

(010)

(100) (110)

3

2.0x10

(105)

0.0 20

60

40

2.2.1. Crystalline phase The crystalline phase consists of polymer lamellae, which show a highly anisotropic behavior with a very high modulus along the chain axis. Elastic constants have been theoretically calculated for several materials and are reviewed by Ward [19]. PP chains have a helical conformation angle [20]. The theoretical stiffness tensor has been calculated by Tashiro et al. [21]: 1 0 7:78 3:91 3:72 0 0:9 0 B 3:91 11:55 3:99 0
0:36 0 C C B C B C B 3:72 3:99 42:44 0
0:57 0 C GPa CcPP ¼ B B 0 0 0 4:02 0
0:12 C C B C B @ 0:9
0:36
0:57 0 3:1 0 A 0

0

80

100

2θ Angle (˚ )

Fig. 3. Deconvolution of PET diffraction peaks for a sample of 32% crystallinity.

Table 5 PET crystalline lamella dimensions Crystalline volume (%)

Length (nm)

Width (nm)

Thickness (nm)

25 32 41

31 30 32

29 32 30

15 15 14


0:12

0

0

2:99 ð3Þ

This tensor is consistent with the value of 40 GPa along the chain direction that was observed in X-ray studies [22]. PE has a zigzag planar chain conformation. This conformation of the molecular chain influences the stiffness of the crystalline lamellae [20]. Due to its zigzag conformation, PE stiffness is highly dependent on temperature. The stiffness of PE lamellae was chosen as calculated by Choy and Leung [23] (see Ref. [12] for discussion): 1 0 7:0 3:8 4:7 0 0 0 B 3:8 7:0 3:8 0 0 0 C C B C B C B 4:7 3:8 81 0 0 0 c C GPa B CPE ¼ B ð4Þ 0 0 1:6 0 0 C C B 0 C B @ 0 0 0 0 1:6 0 A 0

Cr source

8.0x10

2.2. Mechanical properties of constitutive phases

0

0

0

0

1:6

PET lamella stiffness was experimentally measured by Hine and Ward [24]. A theoretical estimate was provided by Rutledge [25]. For most components, experimental values of the elastic stiffness tensor are comparable to the theoretical ones. However, a discrepancy is observed for the C33 stiffness tensor component, which may be the result of imperfect crystallites. Rutledge [25] noted that the C33 stiffness tensor component may drop drastically due to chain axis misorientations. A better experimental estimate of this component has been given by Matsuo and Sawatari [26], and the PET crystal stiffness tensor is given by 1 0 7:70 5:46 5:07 0 0 0 B 5:46 7:70 5:07 0 0 0 C C B C B B 5:07 5:07 118 0 0 0 C C GPa CcPET ¼ B B 0 0 0 1:62 0 0 C C B C B @ 0 0 0 0 1:62 0 A 0

0

0

0

0

1:12 ð5Þ

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2.2.2. Amorphous phase The PET transition temperature is around 80 C; hence, the material is in the glassy state at room temperature. Amorphous samples of PET underwent tensile tests, from which we determined a Young’s modulus of 2600 MPa (see Table 2) and Poisson’s ratio was estimated as 0.4. Purely amorphous samples of PP and PE are impossible to obtain at room temperature. As the amorphous phase of both bulk isotactic PP and PE is in a rubbery state, the shear modulus at plateau G0N is related to the molecular mass between entanglements Me by G0N ¼

qRT Me

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Table 6 Mechanical properties of amorphous phases

PET PP PE

Young’s modulus (MPa)

Bulk modulus (MPa)

Poisson’s ratio

2600 0.9 4.5

4333 2200 3000

0.4 0.49993 0.49975

moduli extrapolated from PVT data. The values characterizing the elastic mechanical behaviors of the amorphous phases are reported in Table 6 for the three materials. 3. Micromechanical modeling

ð6Þ

where q is the amorphous phase density, T the temperature, and R the ideal gas constant. In semicrystalline polymers, the amorphous phase is confined between crystalline lamellae and therefore the chain mobility is reduced [27]. The confinement can also be seen as a phenomenon whereby increasing the density of entanglements, and so decreasing the molar mass between entanglements Me, should increase the modulus. But there is no experimental evidence of a drastic change of the Young’s modulus in the literature. Some authors [10,11,28] have considered the possibility that the amorphous phase confinement does not significantly affect the rigidity of the amorphous phase and have considered it in a rubbery state. Other authors [14,29–31] have chosen a Young’s modulus one or two orders of magnitude higher than the rubbery state. The choice is not based on physical modeling or experimental evidence but on micromechanical modeling; the amorphous phase parameters are calculated by an inverse method using simplistic micromechanical models. For example, the Hashin upper and lower bounds are used assuming the amorphous and crystalline phases are both isotropic [30]. But the crystalline phase is highly anisotropic and the use of analytical expressions for the Hashin–Strikman bounds becomes invalid. The value of Young’s modulus of an amorphous phase obtained by micromechanical modeling depends greatly on the model used. Recently, Spitalsky and Bleda [32] have simulated the elastic properties of interlamellar tie molecules in semicrystalline PE by atomistic modeling. In their work, it appears that Young’s modulus of PE tie chains changes drastically for highly extended chains only. Dealing with elastic deformation, Young’s modulus should not be affected by the tied effect. Considering the lack of experimental and theoretical evidence of a drastic change of Young’s modulus of the amorphous phase due to the crystalline environment, it is reasonable to assume the amorphous phase is in a rubbery state. Me is chosen equal to 7 kg/mol for PP and 1.4 kg/ mol for PE [33], the amorphous phase density at room temperature is taken equal to 850 kg/m3 for PP and 855 kg/m3 for PE [34], and then G0N at ambient temperature is equal to 1.5 MPa for PE and 0.3 MPa for PP. Values of Poisson’s ratios for both polymers are estimated through the bulk

3.1. General theory Considering semicrystalline polymers are heterogeneous materials, micromechanical modeling is used to estimate the mechanical properties of equivalent homogeneous materials. Two material representations are commonly used for polymers. The first representation is the model arising from the Eshelby inclusion theory [35] for which the reinforcing phase of ellipsoidal shape is embedded in a matrix. In the present case, crystalline lamellae are the reinforcing phase while the amorphous phase is the matrix (Fig. 4(a)). In such a representation, one of the key parameters is the shape ratio of the ellipsoidal reinforcements. The second model, designed specially for polymers, is due to Lee et al. [7,8]. In this representation, the material is defined by an aggregate of layered two-phase composite inclusions. Each inclusion is represented by a layer of amorphous phase on top of a crystalline lamella (Fig. 4(b)). Since the materials considered in the present work are isotropic, we are dealing in the following with randomly dispersed inclusions in both representations. For both simplified material microstructures (Fig. 4), the equivalent homogeneous medium behavior is defined by the linear relation R¼C:E

ð7Þ

where R and E denote the macro stress and macro strain tensor: Z Z 1 1 R ¼ hriV ¼ rðxÞ dV ; E ¼ heiV ¼ eðxÞ dV ð8Þ V V V V Amorphous phase

a

b

1 2

3

Crystallite 1: Direction of lamella growth 2: Lamella width direction 3: Direction of the chains

3 2 1

Amorphous phase Crystalline lamella

Fig. 4. Material microstructure modeling: (a) schematic of inclusionmatrix; (b) aggregate of layered-composite inclusions.

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In the following, both microstructures presented in Fig. 4 will be considered separately, allowing us to review the possible homogenization schemes for each representation. 3.2. Crystalline phase randomly dispersed in amorphous matrix 3.2.1. Bounds The simplest bounds are Reuss and Voigt ones which assume that stresses [36] or strains [37] are identical in both phases. In case of a great difference between the elastic constants of the two phases, as for both PP and PE, one gets very wide bounds of limited usefulness. Closer bounds are due to Hashin and Strickman [38] by the use of variational formulation. In their homogenization scheme, inclusions are assumed to be surrounded by a reference medium Cref. The choice of the reference medium behavior defined by Cref depends on the nature of the bounds. If

Cref < Cc Cref > Cc

then CHS is a lower bound then CHS is an upper bound

ð9Þ

The difficulty rises from the choice of the reference medium. In our case, the reinforcing phase (crystalline lamellae) is highly anisotropic and the reference medium is not clearly defined. For the upper bound, the reference media have to satisfy: 8e; e : Cref : e P Max½e : RT : Cc : R : e R

R : Rotation tensor

e : Strain tensor

ð10Þ

sup

Defining k as the highest eigenvalue of the crystalline stiffness tensor Cc, Cref = ksup Æ I satisfies Eq. (10) [39]. This simple choice of Cref does not exactly define a Hashin– Strickman bound but a so-called upper-like Hashin–Strickman bound. For the lower bound, the original assumption leads us to consider the less rigid phase as the reference medium, in our case the amorphous phase, Cref = Cam. Note that the lower Hashin–Strikman bound is equivalent to the Mori– Tanaka model [40]. 3.2.2. Other models In order to get a better estimate of the elastic behavior of the equivalent homogeneous material, let us consider some other classic micromechanics models. All models presented in the following are detailed in Nemat-Nasser and Hori [41]. A self-consistent scheme is better suited for polycrystals or aggregate materials than for matrix containing inclusions [42]. Nevertheless, in order to provide a complete picture of micromechanical modeling applied to semicrystalline polymers, this model is also considered. The selfconsistent scheme is based on the assumption of inclusions surrounded by the homogeneous material. The Mori–Tanaka model [40] is suited for low volume fractions of inclusions, which is not the case in the present

study. The crystalline fraction is about 60%, which limits the use of this model. Nevertheless, since the Hashin– Strickman lower bound is equivalent to the Mori–Tanaka model, results are estimated for this model. Finally, when dealing with high volume fractions of the reinforcing phase, the differential scheme is appropriate [43]. In such a model, the crystalline fraction is added step-by-step using a dilute scheme until the required volume fraction is reached (see Ref. [12] for detailed equations). 3.3. Aggregate of composite layers 3.3.1. Layered two-phase inclusion behavior In this material representation (Fig. 4(b)), the length and width of the composite inclusion are considered very large in comparison to the thickness. This assumption allows us to consider a constant strain and stress field in each layer of the inclusion. The inclusion-averaged strain and stress tensor are given in terms of a volume average over the two phases: XI ¼ ð1
f c ÞXam þ f c Xc

for X ¼ r

and

X¼e

ð11Þ

where superscript I stands for the composite inclusion. At the interface, kinematic coupling compatibility of strain and stress continuity leads to I ecab ¼ eam ab ¼ eab

ða; bÞ 2 f1; 2g  f1; 2g

rci3

i 2 f1; 2; 3g

¼

ram i3

¼

rIi3

ð12Þ

in a direct basis (e1, e2, e3) for which direction e3 is the unit normal of the interface. Combining Eqs. (11) and (12), one obtains the elastic behavior of the composite inclusion [12,29]. Considering a set of randomly dispersed composite inclusions, several micromechanical models can be used to estimate the behavior of an aggregate of inclusions. 3.3.2. Bounds Upper and lower bounds are defined in the same way as Voigt and Reuss bounds [29]. For the Voigt (Reuss) bound, strains (stresses) are assumed identical in all inclusions. This assumption provides tighter bounds than the one obtained by applying Voigt and Reuss bounds to the amorphous and crystalline phases directly [29]. 3.3.3. Self-consistent scheme Considering the material as an aggregate of two-layered inclusions, the self-consistent scheme is theoretically well suited to estimate the elastic behavior of the homogeneous equivalent material [29,44]. Considering n inclusions randomly dispersed, the elastic stiffness tensor C of the homogeneous material is calculated by iteration, solving equations: n X 1 I C¼Cþ ðC
CÞ : Ak ðCÞ n k ð13Þ k¼1 Ak ðCÞ ¼ I þ Sesh C
1 ðCIk
CÞ

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where Sesh is the Eshelby tensor [35]. Tensors Ak must satisfy ÆAkæ = I, with angle brackets denoting the volume average over all considered orientations. In the case of non-spherical inclusions, this last equation is not necessarily satisfied and the self-consistent model is then generalized by adding the condition [45] Ak ¼ Ak  hAk i


1

ð14Þ

In the following, two models dedicated to polymer modeling are presented. 3.3.4. Hybrid scheme Lee et al. [7,8] first used a layered-composite inclusion representation, dealing with perfect plasticity. In this framework, two hybrid schemes have been defined based on a mix of Sachs [46] and Taylor [47] assumptions, the r-inclusion and the U-inclusion models. Both models have been recently used in the context of elastoviscoplasticity by Van Dommelen et al. [10]. The model formulation is recorded here in a simpler context of linear elasticity. 3.3.4.1. r-Inclusion scheme. Let us consider an aggregate of two-layered-phase composite inclusions submitted to a macroscopic strain field E. In order to compute the elastic behavior of the macrostructure, it is necessary, according to Eq. (7), to determine the macroscopic stress tensor R induced by the strain field E. Following Eq. (8), macroscopic strains and stresses may be defined as a volume average over all inclusions: E ¼ heI i and

R ¼ hrI i

ð15Þ

In the r-inclusion model, an unknown auxiliary stress field ^. Partial Sachs and Taylor conditions are is introduced as r applied at the local scale of each inclusion according to eIk ¼ Ek

for k 2 f3; 4; 5g

and

^k rIk ¼ r

for k 2 f1; 2; 6g ð16Þ

using Voigt notation in the local basis of the inclusion, e3 being the unit normal of the interface. Considering a system of n inclusions, 6 · n unknowns ðeI1 ; eI2 ; eI6 ; rI3 ; rI4 ; rI5 Þ; ^, remain from I 2 f1; ng and 6 global unknowns due to r Eq. (16). The composite inclusion elastic behavior (CI) being known, the equations eIj ¼ S Ijk  rIk ;

^j ¼ C Ijk  eIk ; j 2 f3; 4; 5g and r

j 2 f1; 2; 6g ð17Þ

written in the composite inclusion basis, provide 6 · n local equations. The global condition of macro-homogeneity E = ÆeIæ provides six global equations. Finally, one ends up with a (6 · n + 6) by (6 · n + 6) linear system to solve. The solution of this system makes calculations of R and C possible. 3.3.4.2. U-inclusion scheme. The aggregate of composite inclusions is now submitted to a macroscopic stress field R. To obtain the elastic behavior of the macrostructure,

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according to Eq. (7), it is necessary to determine the macroscopic strain tensor E induced by the stress field R. As for the r-inclusion scheme, an unknown auxiliary strain field ^e is introduced and the partial Sachs and Taylor conditions are still applied at the local scale of each inclusion according to eIk ¼ ^ek for k 2 f3; 4; 5g and rIk ¼ Rk for k 2 f1; 2; 6g ð18Þ From Eq. (18) remain 6 · n local unknowns ðeI1 ; eI2 ; eI6 ; rI3 ; rI4 ; rI5 Þ; I 2 f1; ng and six global unknowns due to ^e. As for the r-inclusion model, knowing the composite inclusion elastic behavior provides us with 6 · n local equations: ^eIj ¼ S Ijk  rIk ; j 2 f3; 4; 5g and

rIj ¼ C Ijk  eIk ; j 2 f1; 2; 6g ð19Þ

Finally, the global stress macro-homogeneity equation R = ÆrIæ defines six global equations. The resolution of a (6 · n + 6) by (6 · n + 6) linear system makes possible the calculation of E and C. Finally, let us end this theoretical section by noting firstly that it has not been possible to consider the differential schemes in the case of an aggregate of composite inclusions due to the requirement of a matrix in order to gradually add the reinforcing phase. Secondly, inclusion shape ratio parameters are required only for the self-consistent and differential schemes. In the following section, quantitative agreements between the predictions of the micromechanical models and the measured elastic constants for PP, PE, and semicrystalline PET are presented. 4. Results For both PP and PE there is a strong contrast between the mechanical behaviors of the two constitutive phases. PP morphology is relatively well estimated in Section 2.1.2, and none of the inputs required by micromechanical modeling is unknown. Considering PE, the crystallite dimensions, which provide inclusion shape ratios, remain unknown. However, Young’s modulus of the homogeneous material has been measured for a large range of crystallinity, which provides a good database to estimate the crystallite shape ratios. The PET was chosen for several reasons. Firstly, the elastic behavior of the amorphous phase is not questionable. Secondly, it is possible to control its crystallinity up to 41%. Finally, since the amorphous phase is in a glassy state, the contrast between both phases, in terms of elastic behavior, is largely reduced in comparison to PP and PE. 4.1. Comparison between models and PP experimental data Since all inputs required by the modeling are known for the PP, this material is chosen to compare several models. PP crystallizes in a narrow interval varying from 50% to

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60%. In this study, only two different macrocrystalline structures were obtained. Using the inclusions embedded in matrix representation (inclusion-matrix representation), theoretical estimates of Young’s modulus of PP are calculated for the lower and upper Hashin–Strickman bounds, for the differential scheme (DS) and the self-consistent model (SC). Dimensions of the ellipsoidal crystallite are taken as width/thickness = 100/12 and length/thickness = 1000/12, according to Section 2.1.2. Elastic behaviors of each constituent, crystallite and amorphous matrix, are defined in Section 2.2.

The results are plotted in Fig. 5 in terms of Young’s modulus vs. crystallinity rate. The Hashin–Strickman bounds are too wide to be useful. The SC scheme largely overestimates the experimental data, while the DS is in very good agreement with them. A comparison of experimental data with theoretical estimates of Young’s modulus using the composite inclusion aggregate representation is shown in Fig. 6. Again, bounds are too wide to be useful. The hybrid r-inclusion model appears to be very close to the Voigt upper bound, whereas the hybrid U-inclusion model underestimates the experimental data but remains relatively

12000 Experimental data Matrix-Incl: HS- bound Matrix-Incl: HS+ bound Matrix-Incl: SC - shape ratios W/T=10/1.2, L/T=100/1.2

Young's modulus (MPa)

9000

Matrix-Incl: DS - shape ratios W/T=10/1.2, L/T=100/1.2

6000

3000

0 0

10

20

30

40

50

60

70

80

Crystalline volume rate (%)

Fig. 5. Young’s modulus for PP vs. crystallinity: comparison between inclusion-matrix micromechanical model estimates and experimental data. The error bars are equal to twice the standard deviations.

4000 Experimental Data Composite-Incl: Voigt bound Composite-Incl: Reuss bound Composite-Incl: SC - shape ratios W/T=10/1.2, L/T=100/1.2

Young's modulus (MPa)

3000

Composite-Incl: S-Inclusion Composite-Incl: U-Inclusion

2000

1000

0 0

10

20

30

40

50

60

70

80

Crystalline volume rate (%)

Fig. 6. Young’s modulus for PP vs. crystallinity: comparison between composite inclusion aggregate micromechanical model estimates and experimental data. The error bars are equal to twice the standard deviations.

F. Be´doui et al. / Acta Materialia 54 (2006) 1513–1523

close to them. One of the interesting aspects of this model is the fact that no shape parameters are required. But the assumption of homogeneous strain and stress in each layer of the composite inclusion implies that shape ratios must be high. In the case of the PP, shape ratios are indeed high, and therefore this model assumption is compatible with the microstructure observations. By comparing Poisson’s ratio at 57% crystallinity, we obtained a correct value of 0.42 with the DS while the hybrid U-inclusion model provided an erroneous value of 0.38. So far, the inclusion-matrix DS model has proven to be better at estimating the elastic constant of the semicrystalline PP.

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45% and 67% of crystallinity, respectively, using the DS only. From the results obtained for both PP and PE, the inclusion-matrix DS seems to be the best model for estimating the elastic behavior of semicrystalline polymers while the amorphous phase is in the rubbery state. Moreover, the micromechanical modeling suggests that the elastic behavior of these materials does not depend on the spherulite macrocrystalline structure but on the crystallite shape ratios. In the following, our investigation continued considering a material of amorphous phase in the glassy state. The smaller difference between both phase behaviors may lead to other conclusions.

4.2. Comparison between models and PE experimental data 4.3. Comparison between models and PET experimental data In the case of PP and PET, different sizes of spherulites have been determined with crystallinity changes, but no significant variation of crystallite dimensions. Therefore, considering the inclusion-matrix DS, lamella dimensions have been fixed for all crystallinity rates. As shown in Fig. 7, a very good agreement between the model and experimental data has been obtained for shape ratios of width/thickness = 5 and length/thickness = 23. Results obtained using the U-inclusion model for a composite inclusion aggregate are also shown in Fig. 7. The discrepancy observed between the U-inclusion model and the experimental data may be understood by considering the shape ratio values obtained theoretically. From our simulation results, it appears that the crystallite shape ratios (width/thickness and length/thickness) of the PE seem much smaller than those of the PP. Therefore, the assumption of large shape ratios used in the U-inclusion model no longer holds for the PE. In terms of Poisson’s ratio, we have obtained very accurate values of 0.49 and 0.46 for

The PET microstructure has been precisely measured using X-ray diffraction. Crystallite shape ratios have been measured to be width/thickness = 2 and length/thickness = 2. The comparison between experimental data and Young’s modulus estimates obtained from several models is shown in Fig. 8. It appears that all models provide similar results. Considering the almost cubic shape of the crystallites, this last result is rather surprising. In order to understand it, the crystallite shape ratios were modified in the inclusion-matrix differential model. For a large shape ratio, as in PP (width/thickness = 100/12, length/thickness = 1000/12), Young’s modulus estimates do not vary significantly. It appears that in the case of PET, changing the crystallite shape ratio parameters has a small impact on the final results. This feature is due to the relatively small difference between the two phases in terms of elastic behavior. Then, in the case of the amorphous phase in the glassy state, a large number

2000 Experimental Data Matrix-Incl: DS - shape ratios W/T=5, L/T=23

Young's modulus (MPa)

1600

Composite-Incl: U-Inclusion

1200

800

400

0 0

10

20

30

40

50

60

70

80

Crystalline volume rate (%)

Fig. 7. Predicted and experimental Young’s modulus for PE vs. crystallinity. The error bars are equal to twice the standard deviations.

F. Be´doui et al. / Acta Materialia 54 (2006) 1513–1523

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Experimental Data

3700

Composite-Incl: SC - shape ratios W/T=L/T=2 Composite-Incl: U-Inclusion

Young's Modulus (MPa)

3500

Matrix-Incl: DS - shape ratios W/T=10, L/T=100 Matrix-Incl: DS - shape ratios W/T=L/T=2 3300 Matrix-Incl: SC - shape ratios W/T=L/T=2 3100

2900

2700

2500 0

10

20

30

40

50

Crystalline volume rate (%)

Fig. 8. Predicted and experimental Young’s modulus for PET vs. crystallinity. The error bars are equal to twice the standard deviations.

of models (the inclusion-matrix SC model, the inclusionmatrix DS, the composite inclusion U-inclusion model and the composite inclusion SC model) are equivalent. Nevertheless, even though the errors between experiment and theoretical estimates do not exceed 7%, the shapes of the theoretical curves differ from those of the experimental data. This last result shows a possible limitation in the ability of micromechanical models to represent the elastic constants of semicrystalline polymers. Actually, considering the results obtained for both PP and PE, one could have expected better results for PET. This might be due to some microstructure features not accounted for here. To answer this question, further work considering several semicrystalline polymers with glassy amorphous phases should be considered in the future. 5. Conclusion In this paper we have compared several micromechanical models in order to study the applicability of micromechanical modeling at the nanoscale of crystallites of semicrystalline polymers in the context of elasticity. For this purpose, three commonly used semicrystalline polymers have been considered: PP, PE, and PET. Comparisons between experiment and the models have shown that the micromechanical approach may be applicable for semicrystalline polymers. More precisely, a very good estimate of the Young’s modulus vs. crystallinity has been obtained using a DS and a matrix-inclusion representation for materials showing a large difference between both constitutive phases in terms of elastic behavior (PP and PE). In the same context, the hybrid U-inclusion model with a layered-composite aggregate representation has provided a relatively good estimate of Young’s modulus

for crystallites showing high shape ratios (PP), but poor results for moderate shape ratios (PE). These last results show the large impact of the crystallite shape ratios when considering materials whose amorphous phase is in the rubbery state. As for the hybrid U-inclusion model, poor estimates of Poisson’s ratios in all cases have been shown, which limits the general application of this model. It is important to note that these models have been applied without considering the particular spherulitic microstructure; this suggests that the mesostructure does not affect the mechanical behavior of these polymers when dealing with infinitesimal elastic deformations. Considering a semicrystalline polymer (PET), which shows a small difference in terms of elastic behavior between amorphous and crystalline phases, several models have been shown to provide similar results. In such a context, micromechanical models are only slightly sensitive to the crystallite shape ratio parameters. This explains why models with no crystallite shape ratio, like the U-inclusion model, still apply. The results obtained in the case of glassy amorphous phases are contrasted. In terms of absolute values, Young’s modulus is fairly well estimated by the models, but in terms of the shape of the curve, the convex nature of the experimental data curve is poorly represented. This last result may be a limitation for micromechanical modeling, which should be explored in the future with further comparisons. Acknowledgements The authors are grateful for the helpful discussion on theoretical aspects of homogenization with M. Pierre Gilormini. We very much thank M. Carlos N. Tome´ for sharing some of his numerical routines, which have significantly increased the accuracy of our results.

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