Towards the size estimation of a Representative Elementary Domain in semi-crystalline polymers

Accepted Manuscript

Towards the size estimation of a Representative Elementary Domain in semi-crystalline polymers Jose Teixeira-Pinto , Carole Nadot-Martin , Fabienne Touchard , Mikael ¨ Gueguen , Sylvie Castagnet PII: DOI: Reference:

S0167-6636(16)00004-1 10.1016/j.mechmat.2016.01.003 MECMAT 2536

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Mechanics of Materials

Received date: Revised date: Accepted date:

20 July 2015 8 December 2015 7 January 2016

Please cite this article as: Jose Teixeira-Pinto , Carole Nadot-Martin , Fabienne Touchard , Mikael ¨ Gueguen , Sylvie Castagnet , Towards the size estimation of a Representative Elementary Domain in semi-crystalline polymers, Mechanics of Materials (2016), doi: 10.1016/j.mechmat.2016.01.003

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RVE size has never been quantified for semi-crystalline polymers. Size of a Representative Elementary Domain for the macroscopic strain was determined. It was made possible by recent microstrain-fields obtained from in-situ DIC in a SEM. A statistical method was applied, associating the RED size with a wanted precision. For the tested precisions, RED size notably exceeds the size of one single spherulite.

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Towards the size estimation of a Representative Elementary Domain in semi-crystalline polymers

Jose TEIXEIRA-PINTO, Carole NADOT-MARTIN, Fabienne TOUCHARD, Mikaël GUEGUEN, Sylvie CASTAGNET

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Institut PPRIME (UPR 3346 CNRS – ISAE-ENSMA – Université de Poitiers), Département de Physique et Mécanique des Matériaux, 1 Avenue Clément Ader, BP 40109, 86961 Chasseneuil-Futuroscope cedex, FRANCE; [email protected], [email protected]

Abstract

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Like in any heterogeneous material, predicting the macroscopic mechanical behavior of semicrystalline polymers from scale transition approaches requires to estimate the size of a Representative Volume Element (RVE). However, no quantitative estimation has been reported so far in the literature for these materials. This study is a step forward, by estimating the minimal domain over which the microscopic strain must be spatially averaged to match the macroscopic strain, in a stretched linear low-density polyethylene. A statistical RVE determination method inspired from the work by Kanit et al. (2006) was applied to strain fields acquired at the spherulitic scale from in-situ Digital Image Correlation in a Scanning Electron Microscope. Results show that the size of a representative domain significantly exceeds the size of a single spherulite. Several spherulites are actually contained, depending on the wanted estimation accuracy and on the number of realizations one is ready to generate.

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Keywords: polyethylene, spherulite, microstrain-field, statistical method, RVE

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1. Background on the Representative Elementary Volume notion

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Like for any heterogeneous material, RVE is a crucial concept in semi-crystalline polymers, especially for micromechanical modelling which purpose is to estimate effective properties of multi-phase materials from the knowledge of the constituents’ properties and of microstructural morphology. Many definitions of an RVE exist in the literature for heterogeneous media. The pioneer definition was proposed by Hill (1963) with two major ideas (Salmi et al. (2012)): the statistical representativity of the microstructure and the independence of its apparent behavior with respect to details of boundary conditions applied to it, as long as their averages are uniform at larger scales. The principle of separation of length scales underlying the second idea of Hill, called “macrohomogenity”, has been formulated again by Hashin (1983): “the RVE should be large enough to contain sufficient information about the microstructure in order to be representative, however it should be much smaller than the macroscopic body”. Nevertheless, the independence on the types of boundary conditions reported in Hill’s definition, also advanced four decades later by Sab (1992), adds the 2

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supplementary idea of the existence of a unique effective behaviour. It is to be noted that Hill and Hashin definitions consider only a single realization of the heterogeneous medium and introduce conditions that allow to replace it by an equivalent effective homogeneous medium. Among theoretical works dealing with the conditions for the existence of such an effective behaviour, the one by Ostoja-Starzewski (1998) highlighted that the RVE is perfectly defined in two situations: “(i) a unit cell in a periodic microstructure, (ii) a volume containing a very large (mathematically infinite) set of micro-scale elements (e.g. grains), possessing statistically homogeneous and ergodic properties”. The above concepts plead rather for a large size of RVE in the case of random media. Drugan and Willis (1996) have proposed another definition which led to smaller RVE sizes. Their approach used the solution of the homogenization for an infinite medium, and did not consider statistical fluctuations of apparent properties over finite domains.

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For a practical determination of the RVE, there exist two main approaches. The first one is based on experimental observations of the microstructure with the aim to quantify rather a “morphological” RVE i.e. sufficiently large to be statistically representative of the material microstructure. This is done by combining morphological tools (covariance and covariogram of a random set (Jeulin, 2001; Zaoui, 2001) with stereological and image analysis techniques to quantify the geometrical dispersion of the medium. The second type of approaches focusing rather on the “mechanical” nature of the RVE is based on effective properties by means of analytical methods (Drugan and Willis, 1996 cited in the foregoing) or numerical analysis. Both types of approaches may be associated in the most recent contributions. In the case of numerical analysis, unit cell-type simulations (e.g. finite element simulations) and Monte-Carlo computations are combined. For a given property P, these computational methods are based on ensemble averages and variance of the apparent values of P over different (independent) realizations of volumes with increasing sizes. A first “deterministic” point of view defended that the RVE was estimated as soon as the average response over the different realizations was stabilized, and eventually the variance over the set of realizations, which tended to a minimal value (e.g. see Forest et al., 2000; Kanit et al., 2003; Michel et al., 1999; Vinogradov, 2001). This was also the position adopted in the experimental work by Ramirez et al. (2010) or Denay et al. (2012). Most of the time, the size of the “deterministic RVE” was very large and sometimes impossible to be reached, especially for high properties contrast between phases. This gave rise to the idea to work with apparent properties obtained on volume smaller than the “deterministic RVE”, as previously studied by Huet (1990) and Hazanov and Huet (1994). Kanit et al. (2003) therefore enlarged the RVE definition by introducing the relative error associated to the wanted effective property. Following such a statistical definition, the minimal RVE size is not unique any more but is associated with a given precision in the estimation of the effective property and a given number of realizations that one is ready to generate. Conversely, for a given volume size, it is possible to estimate the number of realizations necessary to reach the same level of accuracy. This approach has been first applied to a numerically generated two-phase artificial composite (Kanit et al., 2003) and to reconstructions of real materials from food industry (Kanit et al., 2006). Results showed clearly that RVE sizes for a given precision and a given number of realizations depend not only on the micro-geometry of the studied material (its morphology and volume fractions) but also on the investigated physical property, these sizes being different for effective elastic and thermal properties. Moreover, the RVE size for a geometrical property like the volume fraction was shown to differ from that found for a physical property. More recently, Dirrenberger et al. (2014) have proved

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Contrarily to the approach by Kanit et al. (2003), only tested for linear materials, the statistical procedure proposed by Gitman et al. (2007) dealt with non-linear mechanical behaviors. It was also based on the notion of accuracy but related to a confidence level and not to a relative error. Moreover, if the Kanit et al.’s method allowed to prescribe the number of realizations as an input, this number was fixed in Gitman et al.’s one. Considering that the size of a RVE was expected to increase with the non-linearity of the behavior (e.g. see Idiart et al. (2006)), this number was moreover too small for highly non-linear behaviors except for very large sizes. Therefore, Pelissou et al. (2009) combined the advantages of the two approaches to define a new statistical-numerical RVE determination technique able to guarantee, for a given RVE size and precision, a sufficient number of realizations and in the same time a good compromise between the RVE size and the total CPU time, this latter point being an improvement of the Kanit et al.’s methodology. The advanced technique was applied to both linear and non-linear properties.

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More recently and for random linear elastic composites, an interesting work due to Salmi et al. (2012) compared quantitatively two different points of view to define a RVE. The first one aimed to determine the minimum RVE size required to evaluate at some accuracy the effective properties of a heterogeneous material using computational approaches, without considerations on the conditions allowing describing the heterogeneous material with an equivalent homogeneous one. The RVE criterion proposed by Kanit et al. (2003) constituted an instance of such a “computational RVE”. The second viewpoint considered that the RVE size was reached when one was allowed to replace the heterogeneous material by an equivalent one in structural mechanics problems, without considering statistical expectations over possible realization of the material microstructure. Hill and Hashin RVE definitions, recalled at the beginning of the present introduction, belong to this second viewpoint defining rather an “equivalent medium-based RVE”. According to Salmi et al., (2012), the ratios between RVE sizes obtained with these two different viewpoints may be of the order five or more for similar materials and accuracy levels. This suggests that the question of a RVE size is not nowadays completely answered.

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Both analytical mean-field modeling and full-field simulations require the knowledge of a RVE of the material, for a relevant identification of the statistical parameters of the microstructure and/or for the definition of the computed volume. In semi-crystalline polymers, a very large majority of the proposed micromechanical models were mean-field analytical ones (Bedoui et al. 2006, Gueguen et al. 2010, Lee et al. 1993, Nikolov and Doghri 2000, Sedighuiamiri et al. 2010). The microstructure was generally defined as a collection of crystalline / amorphous lamellar stacks. Macroscopic isotropy of the material arose from the random orientation of a large number of these stacks. Non-trivial homogenization issues were addressed, especially related to the strong anisotropy of crystals. A rare example of full-field approach, with a microstructure schematization closer to the spherulitic one, was the work by Uchida et al. (2010), in which Finite Element Method was applied to RVE made of a few Voronoï cells, each of them aimed at modeling a spherulite, under periodic boundary conditions. A microstructural unit cell made of a bi-layer of crystalline and amorphous phase was prescribed at each Gauss integration point. Despite the above listed efforts to propose relevant micro-mechanical modeling for semi-crystalline materials, a quantitative estimation of RVE has never been reported in the literature. 4

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complex microstructure

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As depicted in the foregoing, the most current practical approach for the determination of mechanical RVE is based on mechanical properties averaging methods, from the microstructure scale up to the macroscopic one. Two major sets of difficulties arise in the case of spherulitic semicrystalline polymers, as summarized in Figure 1. The first one (sketched on the left part of Figure 1) deals with the high complexity level of the microstructure at several scales. Indeed, spherulites are made of a few nanometer-thick but long crystalline lamellae, branched and very often twisted. Crystalline lamellae may have variable thickness but this distribution cannot be captured since experimental techniques available to scan large volumes (X-ray scattering or calorimetry methods) only provide average values, moreover obtained from very simplified models (very often a onedimension lamellar stack). Poor data are available about other dimensions. At the micro scale, spherulites are variably isotropic entities, depending on the degree and/or perfection of the crystallization process that makes the lateral zones (i.e. the less crystallized zones on each side of the nucleus) more or less extended (see Figure 1). There is no evidence in the literature that intraspherulitic strain fields should be similar from one spherulite to another. For micro-mechanical modeling purpose, the microstructure parameters should be acquired in 3D, over large enough volumes. The electronic density contrast between crystals and the surrounding constrained amorphous phase is very weak. It makes experimental acquisition or numerical generation of the microstructure impossible today over large volumes and with high accuracy enough to be used in full-field simulations. Even if possible, meshing of the spherulites would be another challenge. Here are strong current limitations to full-field calculations that make necessary to introduce strong assumptions to schematize the real microstructure.

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micronic spherulites size distribution variable isotropy level tough cores

unknown 3D constitutive law of the amorphous component

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when confined between nanometric layers

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nanometric lamellar stacks variable size branching twisting defects

drastic simplification of the microstructure

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Figure 1: Overview of the main challenges to micro-mechanical modeling in spherulitic semicrystalline polymers The second difficulty (represented on the right part of Figure 1) is related to the local mechanical behavior of the amorphous phase. Amorphous segments connecting adjacent crystalline lamellae result highly confined and it is well admitted now that their mechanical behavior cannot be transferred from that of a bulk amorphous phase, even when the polymer is above the glass transition temperature. Its mechanical behavior cannot be tested separately because it cannot exist 5

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In this specific context, the present study proposes a first step towards RVE estimation. The aim is to estimate the size of a Representative Elementary Domain (RED) over which the spatial average of the microscopic strain would be representative for the macroscopic value within a stretched sample. Thus, the present viewpoint rather refers to considerations allowing to replace the heterogeneous medium by an equivalent homogeneous one. The purpose is thus to bring elements that could help to define an “equivalent medium-based RVE”. The proposed methodology is based on strain fields acquired from Digital Image Correlation (DIC) at the micro-scale during in-situ tensile tests in a Scanning Electron Microscope (SEM), Teixeira-Pinto et al. (2013). This technique has not been widely used so far in viscous and non-conductive materials like polymers. Its recent successful application to semi-crystalline polymers offers the opportunity to tackle the RVE issue in semi-crystalline polymers. Initially tested in a Polypropylene (Teixeira-Pinto et al. 2013), the technique is here applied to a Linear Low-Density PolyEthylene (LLDPE). PolyEthylene has been probably the most investigated semi-crystalline polymer in the literature for decades. Almost all micro-mechanical models for semicrystalline polymers are based on PE too (Bedoui et al. 2006, Lee et al. 1993, Nikolov and Doghri 2000, van Dommelen et al. 2003), probably due to the fine knowledge of crystalline stiffness and plasticity provided for decades in this polymer.

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The paper is outlined as follows. In the first section, strain-fields measurement from DIC method in the SEM is briefly reminded and applied to the studied LLDPE. Then, the results are exploited within the framework of a statistical approach for RED size estimation. Results regarding RED sizes obtained for different precisions in the estimation of the macroscopic axial strain are reported and discussed in a final part

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2. Strain-fields measurement at the spherulitic scale from in-situ DIC in the SEM

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2.1. Material

The studied LLDPE was obtained in pellets and then molded in 3-mm thick rectangular plates, out of which tensile samples were machined with the dumbbell geometry depicted in Figure 2(a) (gauge length 10 mm, width 2 mm). A picture of the microstructure, revealed by a JEOL 6100 SEM after permanganic etching, is displayed in Figure 3. The average spherulite diameter was about 9 µm. In the following analysis, the average surface of spherulites is estimated to 63 µm². Since the approach is based on in-situ DIC in the SEM, the average diameter of spherulites had to be significantly smaller than the measured field but had to exceed the DIC method resolution.

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Figure 2: (a) geometry of sample and (b) micro machine used for the in-situ tensile test in the SEM

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Figure 3: SEM micrograph of the LLDPE microstructure after permanganic etching

2.2. Experimental protocol and image processing

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For the first time in semi-crystalline polymers, strain fields at the micro-scale were obtained from insitu Digital Image Correlation (DIC) in the SEM. The method itself was deeply analyzed and validated after careful consideration of possible artifacts. It is briefly summarized below but one can refer to Teixeira Pinto et al. (2013) for more details and a first application to a PolyPropylene (PP). The experimental protocol was made of four steps, illustrated in Figure 4. First, the sample surface was prepared for SEM observation in a JEOL 6100. It was polished, exposed to permanganic etching in order to better reveal the spherulitic microstructure, and coated with a very thin gold layer (less than 20nm thick) to enable SEM observation. The second step was to take pictures of the microstructure in the SEM. As a third step, a very thin random grainy pattern (200 nm grain size) was laid on the etched surface of the sample. The use of this thin random grainy pattern allowed the measurement of the displacement fields at a very small scale (4 measurement points per spherulite). Samples were gold-coated again to enable SEM observation of the non-conductive random grainy pattern. The coating layer was optimized to avoid breakage in the explored deformation range. 7

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SEM observation sample micro-polishing permanganic etching gold layer

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During the fourth step, the sample was introduced again into the SEM chamber and fixed to the screw-driven micro-tensile machine represented in Figure 2(b). Grids on the sample were used to recover the position of the sample during step 2 (microstructure acquisition) and make possible any overlapping of the initial microstructure to the measured strain-field. This is not of first importance in the present study but was crucial for the analysis of intra-spherulitic strain fluctuations in Teixeira Pinto et al. (2013). The sample was then stretched with a crosshead speed of 0.02 mm/min, i.e. at an average conventional axial strain rate of 3.3E-5 s-1, slow enough to facilitate the DIC measurements during successive short interruptions every 30 seconds. Finally, the pixel size was 55nm; the spatial resolution for the displacement field measurement was equal to 2.2 µm.

tensile test in the SEM

microstructure acquisition

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Figure 4: Experimental protocol for in-situ strain-fields measurement in the SEM during tensile test.

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Images were processed in order to obtain the in-plane displacement fields by using the DIC software GRANU®. This software performed a parabolic interpolation of the correlation function in a 5² pixels² grid. The strain field was computed by derivative of the displacement field, using the derivative modulus of the ABAQUS® software. More precisely, the displacements measured from the DIC technique were associated with a 2D finite element (FE) mesh. Each node (measurement point) of the mesh corresponded to a displacement vector whose derivatives provide the strains at each point. Strain fields were derived from the measured displacement fields over a reduced region of the sample surface (so-called Domain Of Interest (DOI) in the following) which covered a surface of 1920 x 1280 pixels², i.e. 105.6 x 70.4µm². Parameters of the SEM (scan speed, working distance and stray electric field) were optimized to minimize spatial distorsion. The final sensitivity on displacement was  18nm. The drift distorsion was also evaluated. Final uncertainty on the strain was less than 7E-4. Axial and shear strain components were computed for three macroscopic axial strain levels macro: 3%, 4.6% and 10%. These macroscopic strain values were evaluated from DIC at the macroscopic scale at the same stress levels using the same micro-tensile machine and the same sample geometry than for the microscale measurements. Figure 5 represents only the axial strain fields in the tensile direction that will be exploited for the RED estimation in the next section. It can be mentioned however that the shear strain component remained expectedly low and negligible compared to the tensile one. Strain heterogeneities were observed in both fields, even at the smaller strain levels. These heterogeneities become more

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intense for increasing macroscopic axial strain, as attested for instance by the increasing gap between the least and most deformed regions in Figure 5.

Figure 5: Tensile strain fields in the DOI in LLDPE sample under tension, investigated by in-situ DIC in the SEM at (a) 3% (b) 4.6% and (b) 10% of macroscopic axial strain.

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3. Statistical method for the determination of a Representative Elementary Domain (RED) size

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From the axial strain fields derived above, the aim of this work is to estimate the size of a Representative Elementary Domain (RED) using a statistical approach. More precisely, it means to get the minimal size of domains over which the axial strain should be spatially averaged to get a given precision in the estimation of the macroscopic axial strain with a given number of realizations.

1 ( x ) ds S S

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S 

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The spatial average of the axial strain field xover a 2D domain (surface) S is classically defined as in Equation (1).

(1)

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Following the theory of samples, the absolute error e abs on the mean value of the random process

 S , obtained with N independent (different) realizations of domain S is deduced from the interval of

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confidence by Equation (2).

eabs 

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Hence, the relative error is given by Equation (3).

e 2D (S) erel  abs  S S N

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4D2 (S)

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In the present work, the target mean value is considered as known. It corresponds to the macroscopic axial strain noted macro in Section 2. The median value of 4.6% is here chosen to

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illustrate the advanced RED determination approach. Therefore, the function D  (S) is still to be

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determined. To this aim, different domain sizes S are considered. For each size S, a random partition (subdivision) of the largest measured strain field area (i.e. the DOI) is performed. Realizations can be located anywhere in the studied zone and partial or complete overlapping is allowed. For each size S, the number of realizations must be such that the relative precision on the mean is below a chosen relative precision (here 3%) estimated by usual sampling theory (Eq. (3)). The numbers given in Table 1, associated to the three domain sizes considered in this work, are checked to fulfil this condition. In addition, they are such that the product of the N by the surface of the domain does not exceed the total surface of the studied zone. Figure 6 illustrates the random partition of the studied zone for each of the considered domain sizes. Size of the domain S

Number of realizations N

Average number of spherulites contained

1045,44

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(µm²)

Table 1: Number of realizations N used for the considered domain sizes, and corresponding average number of spherulites contained.

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(a)

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(b)

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Figure 6: Location of the N realizations in the Domain Of Interest for a) S = 1045 µm2, b) S = 465 µm2 and c) S = 261 µm2.

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As inspired by Kanit et al. (2003) or Dirrenberger et al. (2014), the evolution of the variance as a

 B D (S)    S

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function of S is obtained by fitting the experimental values of D 2 (S) to the power law (Equation (5)): 

(5)

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2 

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This leads by identification to B = 0.022 µm2 and  = 1.207 for surfaces expressed in µm². The quality of the model may be seen in Figure 7.

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Figure 7: Fitting of D 2 (S) as a function of the domain size S.

Inserting Equation (5) into Equation (4), the number of realizations N necessary to estimate the macroscopic axial strain with a given relative error erel and a given domain size S becomes that given by Equation (6).

 2macro e 2rel

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

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In the same way, the minimal size of the RED, S precision, is given by Equation (7).

RED

1/ 

  4   B  2 2  e N  macro rel 

, for a given number N of realizations and a fixed

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S

RED

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N

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4. Results and discussion

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For a given target precision value, it must be underlined that the couple of values (N,S) are identical whatever obtained by prescribing S in Equation (6) or N in Equation (7).

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Figure 8 plots the size of the RED as a function of the required number of realizations to estimate the macroscopic axial strain with 1% and 2.5% accuracy. It is recalled that the macroscopic axial strain is 4.6%.

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The first obvious feature is that the RED size decreases with the number of realizations. Another way of expressing it is that the number of realizations decreases when the domain size increases. It means that the poor statistical information acquired over small domains may be compensated by the number of realizations. This relationship is clearly highly non-linear. Non-linearity increases with the target relative precision. For a given number of realizations, the RED surface increases with the target precision. As an example, a fixed number of 10 realizations leads to RED sizes containing about 55 and 12 spherulites, for a target precision of 1% and 2.5% respectively. The difference between areas associated with the two relative errors is clear for limited numbers of realizations, which is the most interesting framework to minimize experimental or calculation costs. In the present case, this difference clearly decreases when the number of realizations becomes large. Curves corresponding to higher accuracy would merge for a higher number of realizations. As illustrated in Figure 5, strain heterogeneities within the DOI increase with the macroscopic axial strain. An increase of the RED size is thus expected for higher macroscopic axial strains.

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Figure 8: Evolution of the statistical RED size as a function of the required number of realizations for an estimation precision of 1% and 2.5% at 4.6% of macroscopic axial strain.

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A common idea about mechanics of semi-crystalline polymers is that the spherulite would correspond to a RVE. To discuss this idea, the size of the RED was set equal to the average surface of one spherulite (63 µm²). Figure 9 displays the corresponding evolution of the number of realizations needed to estimate a macroscopic axial strain of 4.6%, as a function of the relative error on the estimation. Figure 9(b) is a magnification of Figure 9(a) for a better readability at low values. The shape of the curve is consistent with comments on Figure 8. Values show that several tens of realizations would be required to reach reasonable accurate estimations. As an example, 13 realizations are needed for a relative error of 10%. This value increases up to 1248 if the target error decreases down to 1%.

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To complete the discussion, it can be interesting to estimate the relative error resulting from considering a single realization over one spherulite only, as sometimes performed in the literature. For the present macroscopic axial strain of 4.6%, this error reaches the non-negligible value of 35%. Such values go against the idea that one single spherulite could be a Representative Volume Element for semi-crystalline polymers. Regarding the mechanical response, heterogeneity sources between spherulites arise from many factors among which size distribution, morphology of the final polyhedron, anisotropy, crystallization rate. As a comparison, studies on polycrystalline materials agreed on RVE including one or even several hundreds of grains (Diard et al. 2002, Mathieu 2006, Nygards 2003).

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Figure 9: Evolution of the number or realizations as a function of the relative error of the estimation for a macroscopic axial strain of 4.6% and for a surface domain S = 63µm² corresponding to the average surface of a spherulite in the LLDPE.

5. Conclusion

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In the same way, the question arises about a possible definition of the RVE size relatively to the average spherulite size. A similar study, briefly conducted in a PolyPropylene with larger spherulites (2826µm² on average), revealed that the average number of spherulites contained in the estimated RED was significantly different from the present LLDPE. The number of spherulites to be included into a RVE is thus expected not to be intrinsic. It would depend on the specific polymer, quality and degree of crystallization, anisotropy, tilt and branching of lamellae within the spherulite, etc… This is an important difference with other materials classically addressed in RVE estimations in the literature in which inclusions are more homogeneous.

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The aim of this study was to progress towards the quantitative estimation of a RVE size in semicrystalline polymers, which is lacking in the literature despite simultaneous effort to propose micromechanical modeling for this class of materials. Relevant estimation of a RVE size should involve statistical-numerical analysis dealing with mechanical property. It is not possible in semicrystalline polymers because the real microstructure and the local mechanical behavior of the amorphous phase cannot be experimentally accessed directly. Several strong assumptions, interrelated to the RVE notion, must be made at the same time and they cannot be separately evaluated. In this context, the present work did not actually estimate a RVE size, but focused on the strain. Following a statistical approach, it estimated the minimal size of a representative domain and the number of needed realizations over which the strain field should be averaged to estimate the macroscopic strain with a given precision. Strain fields obtained in a polyethylene at the sphrerulitic scale, by in-situ DIC correlation in a SEM during tensile test, were processed to this aim. It was logically confirmed that the size of domains over which the strain field should be spatially averaged increased for decreasing number of realizations. It was found out to increase too with the target precision of the macroscopic strain estimation. An important result is that for a limited number or realizations (less than ten), the minimal size of the treated domains significantly exceeded 14

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Acknowledgement

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Authors would like to gratefully acknowledge Dr. R. Séguéla (MATEIS laboratory, INSA Lyon, France) for kindly providing the material. CNRS and Région Poitou-Charentes are also gratefully acknowledged for granting J. Teixeira-Pinto’s PhD.

References

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Bédoui, F., Diani, J., Régnier, G., Seiler, W., 2006. Micromechanical modeling of isotropic elastic behaviour of semicrystalline polymers. Acta Materialia, 54, 1513-1523.

Denay, A.G., Castagnet, S., Roy, A., Alise, G., Coursault, C., Mellier, D., 2012. Size effect in glass-fiber reinforced and pure polyurethane foams in monotonic and creep compression. Journal of Cellular Plastics 48(3), 221-238.

M

Diard, O., Leclercq, S., Rousselier, G., Cailletaud, G., 2002. Distribution of normal stress at grain boundaries in multicrystals : application to an intergranular damage modelling.Comp. Mat. Sci., 25, 73–84.

ED

Dirrenberger, J., Forest, S., Jeulin, D., 2014. Towards gigantic RVE sizes for 3D stochastic fibrous networks. Int. J. Solids Struct. 51, 359–376.

PT

Drugan, W., Willis, J., 1996. A micromechanics-based non local constitutive equation and estimates of representative volume element size for elastic composites. J. Mech. Phys. Solids 44 (4), 497–524.

CE

Forest, S., Barbe, F., Cailletaud, G., 2000. Cosserat modelling of size effects in the mechanical behaviour of polycrystals and multi-phase materials. Int. J. Solids Struct. 37, 7105–7126.

AC

Kanit, T., Forest, S., Galliet, I., Mounoury, V., Jeulin, D., 2003. Determination of the size of the representative volume element for random composites: statistical and numerical approach. Int. J. Solids Struct. 40, 3647–3679. Kanit, T., N’Guyen, F., Forest, S., Jeulin, D., Reed, M., Singleton, S., 2006. Apparent and effective physical properties of heterogeneous materials: representativity of samples of two materials from food industry. Comput. Methods Appl. Mech. Eng. 195, 3960–3982. Gitman, I., Askes, H., Sluys, L., 2007. Representative volume: existence and size determination. Eng. Fracture Mech. 74, 2518–2534.

15

ACCEPTED MANUSCRIPT Gueguen, O., Ahzi, S. & Markradi, A. & Belloutar S., 2010. A new three-phase model to estimate the effective elastic properties of semi-crystalline polymers: Application to PET. Mechanics of Materials, 42, 1-10. Hashin, Z., 1983. Analysis of composite materials — a survey. J. Appl. Mech. 50, 481–505. Hazanov, S., Huet, C., 1994. Order relationships for boundaryconditions effect in heterogeneous bodies smaller than the representative volume. J. Mech. Phys. Solids 42, 1995–2011.

CR IP T

Hill, R., 1963. Elastic properties of reinforced solids: some theoretical principles. J. Mech. Phys. Solids 11, 357–372. Huet, C., 1990. Application of variational concepts to size effects in elastic heterogeneous bodies. J. Mech. Phys. Solids 38, 813–841.

AN US

Idiart, M., Moulinec, H., Ponte Castañeda, P., Suquet, P., 2006. Macroscopic behavior and field fluctuations in viscoplastic composites: second-order estimates versus full-field simulations. J. Mech. Phys. Solids 54 (5), 1029–1063. Jeulin, D., 2001. Caractérisation morphologique et modèles de structures aléatoires. In: Bornert, M., Bretheau, T., Gilormini, P. (Eds.), Homogénéisation enmécanique des matériaux, vol. 1. Hermès Science, pp. 95–132 (Chapter 4).

M

Lee, B.J., Parks, D.M. & Ahzi, S., 1993. Micromechanical modeling of large plastic deformation and texture evolution in semi-crystalline polymers,. Journal of the Mechanics Physics of Solids, 41, 16511687.

ED

Mathieu, J. 2006. Analyse et modélisation micromécanique du comportement et de la rupture fragile de l’acier 16MND5 : Prise en compte des hétérogénéités microstructurales., Thesis, École Nationale Supérieure d’Arts et Métiers.

PT

Michel, J., Moulinec, H., Suquet, P., 1999. Effective properties of composite materials with periodic microstructure: a computational approach. Comput. Methods Appl. Mech. Eng. 172, 109–143.

CE

Nikolov, S. & Doghri, I., 2000. A micro/macro constitutive model for the small-deformation behavior of polyethylene. Polymer, 41, 1883-1891.

AC

Nygards, M., 2003. Number of grains necessary to homogenize elastic materials with cubic symmetry. Mech. Mat. 35, 1049–1057. Ostoja-Starzewski, M., 1998. Random field models of heterogeneous materials. Int. J. Solids Struct. 35, 2429–2455. Pelissou, C., Baccou, J., Monerie, Y., Perales, F., 2009. Determination of the size of the representative volume element for random quasi-brittle composites. International Journal of Solids and Structures, 46, 2842-2855. Ramirez, C., Young, A., James, B., Aguilera, J.M., 2010. Determination of a Representative Volume Element Based on the Variability of Mechanical Properties with Sample size in Bread. Journal of Food Science, 75, 516-521. 16

ACCEPTED MANUSCRIPT Sab, K., 1992. On the homogenization and the simulation of random materials. Eur. J. Mech. Solids 11, 585–607. Salmi, M., Auslender, F., Bornert, M., Fogli, M., 2012. Various estimates of Representative Volume Element sizes based on a statistical analysis of the apparent behavior of random linear composites. C.R. Mécanique 340, 230-246.

CR IP T

Sedighuiamiri, A., Van Erp, T.B., Peters, G.W.M., Govaert, L.E. & van Dommelen, J.W.W., 2010. Micromechanical Modeling of the Elastic Properties of Semicrystalline Polymers: A three phase approach. Journal of Polymer Science: Part B: Polymer Physics Edition, 48, 2173-2184. Teixeira Pinto, J.M., Touchard, F., Castagnet, S., Nadot-Martin, C., Mellier, D., 2013. DIC strain measurements at the micro-scale in semi-crystalline polymers. Experimental Mechanics 53(8), 13111321.

AN US

van Dommelen, J.A.W., Parks, D.M., Boyce, M.C., Breklemans W.A.M. & Baaijens, F.P.T., 2003. Micromechanical modeling of the elasto-viscoplastic behaviour of semi-cristalline polymers. Journal of Mechanics and Physics of Polymers, 51, 519-541. Vinogradov, O., 2001. On a representative volume in the micromechanics of particulate composites. Mech. Compos. Mater. 37, 245–250.

AC

CE

PT

ED

M

Zaoui, A., 2001. Changement d’échelle: motivation et méthodologie. In: Bornert, M., Bretheau, T., Gilormini, P. (Eds.), Homogénéisation en mécanique des matériaux, vol. 1. Hermès Science, pp. 19–39 (Chapter 1).

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