E Ivanov

P Ienny

R Leger

A S Caro-Bretelle

R Kotsilkova

Strain localization in iPP/MWCNT nanocomposites using digital image correlation

Keywords: Polymer nanocomposites, polypropylene, multiwall carbon nanotubes, micro mechanism of deformation, digital image correlation, ultimate properties, striction

Polypropylene/multiwall carbon nanotubes nanocomposites with different rate of weight incorporation (0 to 1 %) were prepared by melt compounding and cast extrusion. The effect of maleic anhydride (5wt%) grafted on polypropylene is study through mechanical tests at different scales and morphological observations. In particular, the micromechanism of deformation was investigated through instrumented tensile experiments (at a macro and micro scale) using of a non-contact method known as Digital Image Correlation. The objective of this paper is first to characterize global behaviour (Young modulus, tensile strength, ultimate properties) and second to go further in local analysis. In particular optical instrumentation enables estimation of strain profile distribution onto the sample in a strictioned area. Statistical parameters extracted from these local profiles are promising tools to enhance mechanical properties in link with microstructural composition. Tensile tests confirm composites reinforcement at a low level of nanocomposites incorporation and local analysis enables quantitative measurements of adding maleic anhydride in formulations. The results reveal that addition of maleic anhydride delays strain localization in the strictioned area.

Introduction

Polymer nanocomposites filled up with carbon nanotubes (CNTs) gained intensive interest thanks to their unique and valuable mechanical, thermal, and electrical properties compared to the virgin polymers [START_REF] Du | The present status and key problems of carbon nanotube based polymer composites[END_REF][START_REF] Coleman | Mechanical reinforcement of polymers using carbon nanotubes[END_REF][START_REF] Moniruzzaman | Polymer nanocomposites containing carbon nanotubes[END_REF][START_REF] Lin | Functionalized carbon nanotubes for polymeric nanocomposites[END_REF][START_REF] Manchado | Thermal and mechanical properties of single-walled carbon nanotubes-polypropylene composites prepared by melt processing[END_REF][START_REF] Kashiwagi | Thermal and flammability properties of polypropylene/carbon nanotube nanocomposites[END_REF][START_REF] Zhang | Low percolation threshold in single-walled carbon nanotube/high density polyethylene composites prepared by melt processing technique[END_REF][START_REF] Xiao | Mechanical and rheological properties of carbon nanotubereinforced polyethylene composites[END_REF][START_REF] Meincke | Mechanical properties and electrical conductivity of carbon-nanotube filled polyamide-6 and its blends with acrylonitrile/butadiene/styrene[END_REF][10]. One of the CNTs advantages as a reinforcement filler is their large surface area that can induce a better adhesion with the polymer matrix, which is an important factor for an effective enhancement of the composite properties.

However, in order to take advantage of the unique properties of CNTs, the main challenge is to improve the dispersion and distribution of CNTs within the polymer matrix, CNTs having tendency to form agglomerates. This is due to the strong intermolecular Van der Waals interactions among the CNTs in combination with their high surface area and high aspect ratio. In the polymer composite, these agglomerates decrease the surface area, disturb the formation of the polymer network and are responsible for low mechanical properties. That is why the main task of processing is to dissolve such agglomerates as much as possible. Therefore, uniform dispersion of the CNT is required to realize the potentiality of the nanotubes as reinforcing fillers [START_REF] Park | Effect of carbon nanotube pre-treatment on dispersion and electrical properties of melt mixed multi-walled carbon nanotubes/ poly(methylmethacrylate) composites[END_REF][START_REF] Song | Influence of dispersion state of carbon nanotubes on physical properties of epoxy nanocomposites[END_REF][START_REF] Song | Modeling of effective elastic properties for polymer based carbon nanotube composites[END_REF][START_REF] Wang | A treatment method to give separated multi-walled carbon nanotubes with high purity, high crystallization and a large aspect ratio[END_REF][START_REF] Eitan | Surface modification of multiwalled carbon nanotubes: toward the tailoring of the interface in polymer composites[END_REF][START_REF] Kotsilkova | Thermosetting nanocomposites for engineering application[END_REF][START_REF] Kotsilkova | Isotactic polypropylene composites reinforced with multiwall carbon nanotubes. Part. 2. Thermal and mechanical properties related to the structure[END_REF].

The methods for preparation of polymer nanocomposites filled with CNTs are in situ polymerization, solution mixing and melt blending. Extrusion process is frequently used due to its industrial importance and ability to disperse CNTs in the matrix. From all polymer matrices, polypropylene (PP) is one of the widely used thermoplastic polymers for industry like automotive and food packaging for its low cost and well-balanced mechanical and physical properties as well. Other authors recently reported their findings on polymer chains grafted onto CNTs showing that they allow reducing the aggregation of nanotubes and improving the homogeneous dispersion in the matrix, and finally leading to the increase of the mechanical properties due to load transfer between matrix and nanotubes [START_REF] Blond | Enhancement of modulus, strength and toughness in poly(methyl methacrylate)-based composites by the incorporation of poly(methyl methacrylate)-functionalized nanotubes[END_REF][START_REF] Hwang | Efficient load transfer to polymer-grafted multiwalled carbon nanotubes in polymer composites[END_REF][START_REF] Shi | Enhancement of the mechanical performance of poly(vinyl chloride) using poly(n-butyl methacrylate)-grafted multiwalled carbon nanotubes[END_REF][START_REF] Wang | Enhancement of the mechanical properties of poly(styrene-co-acrylonitrile) with poly(methyl methacrylate)-grafted multiwalled carbon nanotubes[END_REF].

There are many studies in the literature on the macromechanical properties of the polymer nanocomposites filled with nanoparticles but there are very limited information about micro and nanomechanical properties and micromechanism of deformation [START_REF] Christmann | Mechanical behavior at large strain of polycarbonate nanocomposites during uniaxial tensile test[END_REF]. At nanoscale level the mechanics of the materials is different and interfacial interactions play an important role. One of the non-contact measurement techniques in the experimental mechanics is the Digital Image Correlation (DIC). The DIC method has been widely accepted and used as a powerful and flexible tool for the surface deformation measurement in the field of experimental solid mechanics. It directly provides full-field displacements and strains by comparing the digital images of the specimen surface in the un-deformed (chosen as the reference state) and deformed states, respectively. In principle, DIC is an optical metrology based on digital image processing and numerical computing [START_REF] Pan | Two-dimensional digital image correlation for in-plane displacement and strain measurement: a review[END_REF][24][25].

The main objective of the present work is to investigate the effect of multiwall carbon nanotube (MWCNT) concentration and MWCNT concentration at 5 wt% of maleic anhydride (MA) in isotactic polypropylene (iPP) on the micro mechanism of deformation of iPP/MWCNT nanocomposites based on tensile experiments and DIC. It is well known that the grafted maleic anhydride groups will form chemical or hydrogen bonding with the CNTs, improving the interfacial adhesion between CNTs and iPP matrix. This compatibilizer therefore acts as a coupling agent to enhance homogeneous dispersion percolated network structure of CNTs in the PP matrix [26].

Experimental study

Materials used and fabrication of nanocomposites

A masterbatch of 20 wt% MWCNT in PP was obtained in pellet form from Hyperion Catalysis International, USA. MWCNTs are commercially manufactured from high purity, low molecular weight hydrocarbons in a continuous, gas phase, catalyzed reaction. Typical outside diameter range of the tubes is from 10 to 15 nm, the lengths are between 1 and 10 microns, and their density is approximately 1.75 g/cm 3 . The masterbatch is produced by initially dispersing intertwined agglomerates of nanotubes into the polymer.

iPP homopolymer "Buplen"6231 with density ρ = 0.901 g/cm 3 and melt flow index MFI (230/2.16) of 12.2 g/10 min, supplied by Lukoil Neftochim Co., Bulgaria, was used as the matrix polymer. Maleic anhydride grafted polypropylene (MA-g-PP) Licomont AR 504, product of Clariant GmbH, Germany, was used as a chemical compatibiliser for improving the interaction between the polymer matrix and MWCNTs and the wetting of MWCNT particles as well.

Nanocomposites were produced by direct melt compounding in Brabender DSE 35/17D twin screw extruder according to a two steps process. First, the masterbatch was diluted to different CNT concentrations in the range of 0.05 to 1 wt%CNTs with neat iPP at melt temperature of 200 °C and a screw speed of 30 rpm. The extruded composites were cooled and pelletized. In order to improve the CNT dispersion, these compositions were extruded a second time. MA-g-PP (5% wt) was added into the nanocomposites during a second stage and pelletized. The compound is then calendered as sheets of various thicknesses (0.4 to 1 mm) at melt temperature of 230 °C and a screw speed of 30 rpm.

Characterization of composites

Microstructure

Optical microscope has been used to investigate the quality of the MWCNTs distribution and dispersion in the matrix. Strips of 1 mm thick have been cryo-fractured and observed. First of all neat and maleic iPP are characterized. Optical micrographs (Figs. 1) reveals large spherulites with diameter of 100µm and small spherulites of about 50µm for the neat iPP (Fig. 1a); addition of MA produces a non-homogeneous structure ranging from large spherulites of 200µm to a mixture of large and small spherulites from 10 to 100µm (Fig. 1b,c,d). The agglomeration of nanotubes in the nanocomposites has been also studied through this optical device (Figs. 2). CNTs aggregates are observed as dark spots with a broad particle size distribution (smaller to bigger as the CNTs concentration increases [40]). At these levels of incorporation, aggregates are well dispersed. After 0.5wt%CNTs, we observe a non-homogeneous structure with nondispersed aggregates smaller for the formulation with MA (Fig. 3). At a higher level of CNTs incorporation (Figs. 4) micrographs reveal non-homogeneous structure with non-dispersed aggregates of the masterbatch (smaller for the formulation with MA).

Standard tensile test instrumented by DIC

Tensile properties (uniaxial tensile test) were determined by using ZWICK TH010 universal testing machine equipped with a high resolution Charge-Coupled Device (CCD) camera (Redlake Megaplus II) with a telecentric lens of focal length of 55mm adapted for DIC and a load cell of 0.5 kN. Wedge grip were used for clamping the samples.

The camera is fixed on a Newport Motion Controller Model ESP301. Such equipment is very useful in case of large strains in order to follow the region of interest (ROI) at the center of the sample and remove the rigid body motion. The optical axis of the camera is perfectly perpendicular to the surface of the specimen. Acquisition of images (1920x1080 pixels, 256 grey levels) is performed by LabVIEW® software which allows the simultaneous acquisition of images and data provided by testing machine (such as load and crosshead displacement). According to the test speed of 10 mm/min, 2 images were recorded every second. The scale factor or magnification (k) which depends on the lens focal and the distance between camera and object was fixed to 32 µm per pixel. This scale factor constitutes a compromise between a necessary spatial resolution for the local strain measurements and the limitation of the perspective errors associated to the out of plane displacement [27].

As far as local analysis is concerned, expected samples were machined with a curved profile so that the neck initiation takes place within the observation field of the camera (Fig. 5a). This observed zone corresponds to the minimal section (4 mm) of the specimen. A radius curvature of 42.5 mm is used to keep the tensile load uniaxial and limit heterogeneities. The specimen thickness is between 0.4 mm and 1 mm.

A speckle pattern was applied before the tensile test. It consists, for the macroscopic samples in a first thin layer, of white paint and a sputtered layer of blue paint on the surface of samples. This creates the artificial image signature. The tensile tests were carrying out until breaking of the samples. Tensile direction is defined by x, and y is perpendicular to x (see Fig. 5).

Time, axial load and axial displacement were recorded by the tensile device software. The nominal stress (First Piola-Kirchhoff tensor) is defined by the following expression:

z y l l F S F P / / 0   (1)
where F is the axial recorded load and z y l l , are the initial useful section dimensions in the y and z directions. Nevertheless this definition neglects the cross-sectional area reduction occurring during the loading and the necking. Considering this variation, the true axial stress can be defined by: The image processing after the test is based on DIC computation. A two-dimensional digital extensometer is defined in the ROI (Fig. 5). The gauge length has to be small enough to accurately describe the strain gradients in the x direction, and large enough to evaluate the mean strain in the y direction. Each point of the virtual mesh corresponds to the centre of a pattern. This pattern, defined by its Correlation size (Cs=30pixels=0.96mm), is the representative area of the material point which is tracked (Fig. 5). Distance between mesh points is given by the Grid step parameter (Gs=3pixels=0.096mm). Deformation calculation is obtained by the first order derivation of the displacement field around the considered point. The dimension of the area where the in-plane gradient tensor is evaluated, is fixed by the number of steps parameter (Ns=4 in the x direction, Ns=20 in the y direction,) which qualifies the number of points to take into account in the horizontal and vertical directions. The bidimensional gauge dimensions (GLx and GLy) expressed in pixel and mm, used for different calculations are as follow: GLx=24 pixels=0.77mm; GLy=120pixels=4mm. It should be noted, that in the x direction, the gauge length estimated by the calculation is smaller than the correlation size (Cs). Also the effective gauge length should be slightly higher than the one estimated. Otherwise the strain gauge is chosen far from the sample boundaries to avoid edge effects (4mm<6mm). These correlation parameters have been optimized to establish reliable mechanical response of the material in the central zone where the necking takes place. It may be noted that in order to ensure good local analysis, the longitudinal gauge length is chosen around 1mm.

' ' / / z y true l l F S F    (2)
The DIC processing leads to an incremental displacement file. In-plane displacements fields u(x,y)=(u x (x,y), u y (x,y)) of a point M(x,y) were evaluated for each point of a virtual mesh in the ROI of the sample (Fig. 5) which included strain localization area. The calculation of the displacement gradient around a considered point allows the calculation of the two-dimensional displacement gradient tensor F (eq. 3) and thus the in-plane Green Lagrange strain tensor E is known for all points defining the ROI of Fig. 5 (eq. 4):

ROI      y) M(x, y), (x, y) (x, u 1 F (3)   ROI 2 1           y) M(x, y) (x, E y) (x, E y) (x, E y) (x, y) (x, y) (x, xy y x y, x y y x x - F F E T 1 (4)
It is therefore possible to define in-plane strain components of each point defining the ROI domain. Specific points M 1 , M 2 , M 3 and M 4 are located in the center of the ROI zone and will be useful for further investigations (Fig. 5). Strain cartography of the ROI domain leads to define the point where the maximum strain is reached. This point will be denoted M 5 and does not necessarily correspond to the point at the minimal section (M 1 ).

It is useful to define the in-plane stretches ratios in the x or y direction:

1 2 1 2 ' '       y) (x, y) (x, l y) (x, l y) (x, y) (x, l y) (x, l y) (x, y y x x y y x x E , E   (5) where y) (x, l y) (x, l y x , ( y) (x, l y) (x, l y x ' ' ,
) are the initial (current) sections dimensions of the square pattern in the x and y directions around M y) (x, . The out-of-plane stretch y) (x, z  is deduced from the in-plane principal stretches ratios in the longitudinal and transverse direction thanks to a transverse isotropy assumption [START_REF] Christmann | Mechanical behavior at large strain of polycarbonate nanocomposites during uniaxial tensile test[END_REF]. This last hypothesis has been validated by some authors for a similar material and in the same range of solicitation [28]. Under this assumption,

y) (x, S y) (x, y) (x, S y) (x, y) (x, S y z y 0 2 0     
and Eq. 2 can be rewritten as

y) (x, y) (x, S F y) (x, y true 2 0 /    . ( 6
)
The axial true strain x  and nominal strain N  are defined by

y) (x, l y) (x, l y) (x, l y) (x, y) (x, y) (x, l y) (x, l y) (x, l y) (x, dl y) (x, x x x N x l l x x x x x               ' ' ), ln( ln '    (7)

Results and discussion

Mechanical properties-global analysis from DIC

The elastic properties of the samples were first determined from the knowledge of both axial load and true local strain value measured by DIC analysis. For a low level of load, the reversible strain is analyzed all along the longitudinal direction. The Young modulus is therefore computed from eqs. 6 and 7:

ROI ) , ( ,    y x M y) (x, y) (x, E x true   (8)
To assess the reliability of Young Modulus identification, four evaluations have been made depending on their positions along the longitudinal axis (x direction) (points M 1 to M 4 in Fig. 5). For a given time, these points are at the same level of axial load but differ from the values of their current stretches and true stresses. x  is maximum at point M 1 , associated to the minimal section, and decreases until point M 4 . In fact the behaviour of these points is just shifted in time. As far as we are in the elastic range of strain (between 0.25% and 0.5% of axial strain), Young Modulus (E) is almost the same for all the studied points: 1540 ± 50 MPa (see Fig. 6 for the composite MWCNT (0.1wt%CNTs without MA). This analysis was repeated 3 times for all the formulations to ensure repeatability. All results are reported in Fig. 7.

The incorporation of nano charges even at 0.05 wt%CNTs induces a rise of the Young modulus until 0.5 wt%CNTs with and without MA. At 1 wt%CNTS the modulus almost reaches the initial value of pure iPP. The initial reinforcement is probably due to good spatial repartition of MWCNTs in the matrix (see section 2.2.1). The effects of CNTs on the crystallization behaviour of PP have been studied by some authors [30][31][32]. The CNTs served as nucleating agents to enhance crystallization and the crystallites strengthen the composite [29,33]. The MA conducts to a lower Young modulus which is incompatible with usual observations [29]: it seems that the spatial dispersion of nanotubes is not improved by the addition of MA (see section 2.2.1) and the better load transfer between polymer matrix and nanotubes produces by MA is not sufficient to improve Young modulus value.

Agglomeration of nanoparticles for higher wt%CNTs reverses this trend. Similar downturns of the mechanical properties have been observed for composites containing polymergrafted MWCNTs for the same level of incorporation [29]. Moreover, it seems that the percolation threshold is reached at almost the same level of CNTs incorporation with and without MA adding, but some more experiments should confirm this hypothesis.

Micromechanical models can be a good tool to understand this mechanical behaviour. As observed in SEM investigations, MWCNT are individual and randomly distributed under 0.1 wt%CNTs and agglomerate after that level. Nevertheless these agglomerates seem to be well distributed in the matrix. Mori Tanaka micromechanical model [34] is used to predict Young's modulus of reinforced iPP in both cases. For low level of incorporation, the effective elastic constants of the composite can be expressed as:

      -1 e e m f m A I A C - C C C f f f c c c     ) 1 ( (9)
where m C and f C are respectively matrix and fibre stiffness tensor, f c is the volume fraction of fibre,

    -1 m f m e C - C SC A 1 1   
where S is the classical Eshelby tensor (for fibrous reinforcement) and <.> is a weighted orientation averaging operator added to consider the arbitrary fibre orientation [35].

To implement this model, Matlab® [START_REF] Besson | Revue europpéenne des éléments[END_REF] software was used and appropriate elastic properties of matrix and fibres have been fixed. In [37], a longitudinal Young's modulus (denoted E L ) of 600 GPa has been chosen for the MWCNT fibre as a result of a literature review. As the nanotubes are transverse isotropic, four more parameters are necessary to evaluate f C : the transverse Young's modulus (equal to 6 GPa), the two Poisson's ratio (0.125 and 0.3) and the shearing modulus (2.5 GPa), The matrix is also supposed isotropic (Young's modulus of 1.017 GPa and Poisson's ratio of 0.3)). The rule of thumb conversion factor of 2:1 is used to convert weight fraction to volume fraction of fibres. In this model, strong assumption of perfect bonding and loading transfer is questionable [38]; for this reason, some authors reduce the effective occupied nanotube volume by 25% of the initial volume to match experimental data [37].



Mori Tanaka model well reproduced experimental data for the specimen without MA (see Table 1). At 0.5%CNTs of weight incorporation, the prediction overestimates the experiments. In fact at this level of incorporation, fibre agglomerates and the model is not applicable (see section 2).

The addition of MA leads to a new identification of parameter  (see Table 1). The smaller effective volume fraction is in accordance with the compatibilisation effect of MA: MA does not improve the dispersion as nanotubes are already well dispersed by the process but surrounds the NCTs by a phase softer than the matrix. Therefore the loading transfer decreases which leads to the following smaller equivalent volume of nanotubes.

For higher level of weight fraction, Eq. 9 is revisited to consider agglomeration of fibres. It becomes:

      -1 e e m f m A I A C - C C C  f f f c c c ) 1 ( ˆ    ( 10
)
where

    -1 m f m e C - C SC A 1 1    
and f c ˆis the volume fraction of agglomerates. In this equation f C is the stiffness matrix of agglomerate in which fibres are supposed to be arbitrarily oriented. Thus, the matrix embedded in the agglomerate is neglected. This matrix is therefore isotropic. The same assumption for the effective fibre volume is made and for the formulations with and without MA respectively. Results are summarized in Table 2. The equivalent weight fraction of agglomerates is higher for samples without MA. This result is in accordance with the SEM observation: add MA decreases volume fraction of agglomerates. In all cases the weight fraction of aggregates remains almost the same as the rate of incorporation of nanotubes increases. This analysis deals with a concept of equivalent distribution of nanofillers in the matrix but does not back up to the size distribution of aggregates or agglomerates.

Ultimate properties at the broken section of the sample, ie the strain at break and the tensile strength of the point M 5 (Fig. 5), are presented in Figs. 8 for all composites. These two parameters are deduced from the local stress-strain curves obtained by DIC and concern the axial true strain and the true maximum stress respectively. If the strain at break is increased with the addition of MA in the absence of nanofiller, this plasticizing effect does not seem to hold whatever the rate of incorporation of CNTs in the composite. Modulation of this observation is still required due to the high dispersion of results break. Much more tangible, the tensile at break follows the same pattern as the Young's modulus with the rate of incorporation of fillers. We first observed an increase of the stress at break with the rate of incorporation of nanofillers, and then a decrease probably due to the poor dispersion of fillers in composites. From this point of view no distinction can be made between composites with or without MA adding.

Mechanical properties-local analysis

Strain profiles distribution

This approach involves the analysis of the onset and growth of necking phenomenon in composite. The idea developed here is to understand how the incorporation and the dispersion of nanofillers affect the phenomenology of rupture of the composite. A first application of this methodology allowed understanding how the incorporation of silica within a polycarbonate could lead in a more homogeneous damage [START_REF] Christmann | Mechanical behavior at large strain of polycarbonate nanocomposites during uniaxial tensile test[END_REF]. Strain profiles distributions have been established for different states of stress for iPP and its composites along the longitudinal axis (x direction) of the tensile bar and through the necking area (ROI, Fig. 5). A typical result is presented in Fig. 9 for the composite iPP/0.05wt%CNTs. These strain profiles have been used to characterize the degree of flatness of each distribution. The shape of the strain profile depends on the formulation: the profile of iPP/MWCNT composites is higher and narrower than the one with MA. To enhance local behavior, we introduce a statistical coefficient called flatness coefficient which differentiates a sharp peak from a flat-topped in a non-Gaussian curve:

  MPa h i h i 25 max max / /    f C (11)
where i  is the peak width at half-height, max h the total height (see Fig. 9) and MPa 25 (.) is the value parameter for

MPa M N 25 ) ( 1  
. At this level of stress, all the tested specimens follow a homogeneous behavior far away from striction phenomenon. This flatness coefficient traduces the ability of the necking zone to spread over the tensile bar or to be localized in the machined minimal section. An important flatness coefficient indicates a large strain peak meaning a high dissipative energy by plasticity before fracture [START_REF] Christmann | Mechanical behavior at large strain of polycarbonate nanocomposites during uniaxial tensile test[END_REF]. This statistical coefficient is plotted (Figs. 10) as a function of axial strain for the point which presents the maximum strain (Point M 5 (1.8mm,0mm) in Fig. 9). For each formulation 3 samples have been tested. Presented results are an average of these 3 values which presents a very narrow dispersion regardless to the considered property (standard deviation is under 1% of the mechanical property).

The general evolution is the same for all the tested samples. The flatness coefficient shows a first drop corresponding to the onset of the necking associated to the growth of a strain peak and then, the coefficient stabilizes during the neck propagation.

Under 0.5wt% of CNT, the flatness coefficient is always higher when MA is added (see Figs. 10). Maleic anhydride polypropylene has a low surface energy and is expected to give a good compatibility between the CNT and the polymer by formation of stronger linkages in the interfaces and reducing CNTs surface tension. In particular, hydrogen bonding between maleic anhydride grafted PP and CNTs can be observed using infrared spectroscopy [39]. This better load transfer is in accordance with a better strain homogenization observed for all samples under 0.5wt% of CNT. The effect on the flatness coefficient of maleic anhydride is reversed beyond 0.5% of CNTs as illustrated for samples with 0.5wt%CNT (Fig. 10c) and 1wt%CNT (Fig. 10d). The reason is the particles agglomeration: they are responsible for a great heterogeneity of the strain profile, introducing sometimes a multi-peak strain curve. It is also important to note that the onset of necking does not occur necessarily to the minimal section when aggregates are present in the materials. To illustrate these points strain profiles are plotted for all formulations at same true stress equal to 36Mpa at the broken point (Figs. 11). As shown in Fig. 11a for nonmaleic samples, axial strain profile remains perfectly gaussian under 0.5wt%CNT and became heterogeneous for a higher level of nanofiller incorporations (see samples at 0.5wt%CNTs and 1wt%CNTs in Fig. 11b). The same kind of evolution is observed for the sample with MA (Figs. 11c-d). In Figure 11b each local peak should therefore correspond to an aggregate: in these cases, with incorporation rate higher than 0.5wt%CNT, characterization of local phenomenon via the statistical parameter defined by [START_REF] Park | Effect of carbon nanotube pre-treatment on dispersion and electrical properties of melt mixed multi-walled carbon nanotubes/ poly(methylmethacrylate) composites[END_REF] cannot be a good indicator of the observed phenomenology. As not expected, it is observed that the level of stabilization of the flatness coefficient is higher for higher rates of CNTs (Figs. 11). If this phenomenon is related to the preferential onset of fracture at the aggregate positions it would have been expected that presence of aggregates in composite prevents the necking from propagation. Reverse tendency is observed because several peaks appear from aggregate positions. So others statistical methods should then be established.

The presence of fillers in the PP matrix increases the flatness coefficient value and so helps the spreading out of the necking area (Figs. 12). On the other hand, for the same flatnesscoefficient value, which defines the shape of the peak, the maximum strain is greater for the composites containing MA. These results confirm the better properties for these last formulations due to a greater accommodation with plasticity.

Samples are loaded in tension, and their behavior before final fracture is analyzed. iPP with 0.05wt%CNTs presents a highly localized neck with a narrow peak. On the other hand, iPP with 0.05wt%CNTs and 5wt% MA presents a more diffuse necking that propagates in both sides of the tensile bar (Fig. 11c). This well corresponds to the strain profiles obtained respectively in Fig. 11b and Fig. 11d: a larger peak corresponds to a more important energy dissipated by plasticity.

Nominal and true stress as a function of maximum longitudinal strain

Let's now consider the central point M 1 in Fig. 5. For this point we are able to evaluate both true and nominal strain as a function of axial Green Lagrange component. As the nominal stress (Eq. 1) neglects the cross-sectional area reduction occurring during the necking, it reaches a maximum level and then remains constant or drops. Considering this variation, the true axial stress (Eq. 2) is computed. The evolution of stress-strain until the threshold around 40 MPa is not sensitive to the formulation (Fig. 13a). Moreover incorporation of 0.05 to 0.1 wt% of MWCNT leads to the maximal reinforcing effect compared with the pure iPP. Increasing of the content of MWCNT above 0.1 wt% leads to decreasing of the reinforcing effect and especially incorporation of 1 wt%CNTs conducts to mechanical properties lower than the pure iPP. This observation is in accordance with the agglomeration phenomenon already observed at this level of incorporation. The same trend is observed for samples with MA but the maximum strain is reduced in all cases: before the peak stress, the mechanical response appears to be independent of the CNT rates (Fig. 13b). This result corroborates the previous result obtained on the evolution of the Young Modulus with CNT content where a smaller variation is obtained with MA content. It seems that the addition of MA around CNT acts as an accommodation agent of the local stress. The stress-strain evolution for sample with higher rate of fillers (PP+1wt% CNTs+5wt%MA) is similar to the others except the fact that the fracture is obtained before reaching a maximum stress.

As a conclusion, the results confirm that the better mechanical properties are obtained in the range between 0.05 to 0.1 wt%CNTs in iPP without adding of MA compatibilizer with our methodology of fabrication. On the other hand, although weaker mechanical properties are obtained with the addition of MA (with the exception of the pure iPP), the use of MA as compatibilizer, homogenize the mechanical response of nanocomposites that remains the same until the threshold stress of 40 Mpa, regardless of the incorporation fraction of CNTs.

Conclusions

This work investigated the effects of incorporation of carbon nanotubes (CNTs) in polypropylene matrix (PP) on elastic properties and on the onset and growth of necking phenomenon in these composites.

As a first result, image correlation is diverted from its primary use and leads to the determination of some macroscopic properties as Young modulus, tensile strength and ultimate strain. This true strain value is measured locally by DIC analysis in the necking area. Between 0.05 and 0.5wt%CNTs most of these properties (excepted ultimate strain) are improved with the amount of CNTs. Nevertheless the major expected effect of iPP grafted maleic anhydride (better dispersion of nanotubes and an improvement in the load transfer between filler and matrix) is not interesting from a macroscopic point of view in our study: MWCNTs are already well dispersed in the iPP matrix as highlighted by the results obtained from strengthening these materials. The mechanical characterization has shown that both Young modulus and tensile strength of the nanocomposites increase by the addition of nanotubes until 0.5wt%CNTs. After that properties decrease with the addition of MWCNTs due to the presence of aggregates, found even when iPPg-MA is added.

The analysis of the onset and growth of necking phenomenon in composite is studied from strain profile distributions. Via a flatness coefficient, this approach provides a means of analysis the plastic dissipative potential before fracture. The local analysis highlights particular behaviour produced by iPP-g-MA. The enhancement of compatibility between CNT and polymer produces a strain homogeneization: striction in the central area of the sample is either the source of local breaking (neat iPP) or the origin of spreading through the sample (iPP-g-MA). Furthermore, these results reveal the benefits of anhydride maleic with regards to dissipative energy phenomena.

Finally this integrated local approach may be followed to introduce an energy analysis of the fracture of these heterogeneous materials by using numerical tool such as finite element modeling. Also instrumented tests in SEM should validate the phenomenology of deformation observed.

  section dimensions in the y and z directions.

Figure 5 .Figure 8 . 5 )Figure 13 -

 58513 Figure 5. Geometries of the hourglass specimens (quotations in mm, R radius of curvature), thickness between 0.4 to 1 mm

Table 1 : Experimental and fitted Young's modulus as a function of weight concentration

 1

	26

Table 2 : Weight fraction of aggregates as a function of weight concentration of CNT and MA.

 2

Figure 2 Optical micrographs of iPP/0.05wt%CNTs (a) and iPP/0.05wt%CNTs+5%MA-g-iPP composites (b)

Acknowledgements

This work was supported by grant COST Action MP0701 given to Dr. E. Ivanov for a research visit at Ecole des Mines d'Alès, Alès, France. The authors acknowledge the collaboration with Dr Sossio Cimmino, Dr. Clara Silvestre and Dr. Donatella Duraccio (ICTP-CNR, Pozzuoli, Italy) for the optical micrographs.


Moreno Marcellini
email: moreno.marcellini@yahoo.it

Cécile Noirjean
email: ce.noirjean@gmail.com

Dmytro Dedovets
email: dmytro.dedovets@saint-gobain.com

Juliette Maria
email: juliette.maria@saint-gobain.com

Sylvain Deville
email: sylvain.deville@saint-gobain.com

Cecile Noirjean

Time-Lapse, in Situ Imaging of Ice Crystal Growth Using Confocal Microscopy

come   L'archive ouverte pluridisciplinaire

Introduction

Understanding and controlling the growth of crystals, and in particular of ice crystals, is of interest in many domains, from geophysics to biology or food engineering. In materials science, the growth of ice crystals in colloidal suspensions is used in processing routes called ice templating or freeze casting, 1 where the unidirectional growth of the crystals and their successive removal by sublimation is used to template porosity in various materials. The microstructure, architecture, and properties of these materials is related, to some extent, to the morphology of the crystals grown during freezing. The dimensions and shape of ice crystals can be controlled, for example, by the cooling rate, the temperature gradient, the concentration and the composition of additives and particles, the pH condition, or the presence of externally applied magnetic or electric field. 2,3 The growth of the ice crystals may exhibit three different morphologies: dendritic, cellular (finger-like), and smoothed and faceted structures, [START_REF] Buckley | In Modern Biopolymer Science[END_REF] described by Gibbs-Curie-Wulff equilibrium. [START_REF] Li | [END_REF] Two of such examples are given by snowflakes or ice crystals in saline solutions. [START_REF] Untersteiner | The Geophysics of Sea Ice[END_REF] Several additives have been proposed to control the growth morphologies of ice crystals, with various success. These additives can be water-soluble solutes (sucrose, ethanol, glycerol) [START_REF] Fu | [END_REF] or polymers (PVA, PEG). 8 If these additives all affect, to some extent, the freezing point, growth kinetics, and morphologies of the ice crystals, the latter almost always adopt cellular or dendritic morphologies. A form of Gibbs-Curie-Wulff equilibrium could be achieved by using Ice-Binding Proteins in solution (also called Anti-Freeze Proteins AFPs).

Ice shaping is the results of their binding to ice. The first proteins to be discovered were the Anti-Freeze Glyco-Proteins (AFGP) in Antartic fishes, [9][10][11] followed by the report of Duman et al. 12 of a helical peptide in the serum of winter flounder. Both AFPs and AFGPs can control the freezing temperature, the crystal growth, and the size of ice crystals during freeze-thaw cycles 13 in the serum of fishes, 14 insects, 15 and plants 16 that experience sub-zero temperatures.

Zirconium acetate (ZrAc) and zirconium hydroxy-acetate (ZrHAc) are probably the only inorganic compounds that mimic the ice-faceting properties of AFPs. [17][18][19] The growth of ice crystals in presence of ZrAc probably occur under the Gibbs-Curie-Wulff equilibrium. ZrAc can thus be used as a simple model system to grow faceted crystals and measure their growth kinetics. ZrAc in water organizes in tetrameric stacks 20 which might mimic the repetitive motif of certain AFPs. 21,22 Although several methods have been proposed to follow the growth of ice crystals in situ, none of them can provide 3D, real-time observations without affecting the system. Optical microscopy is limited to 2D, surface observations. 23 Computed X-rays tomography can provide time-lapse, 3D observations, but the local absorption of synchrotron X-rays increases the temperature and affects the growth behaviour and morphology of the ice crystals. 24 Optical interferometry has been used to track the freezing front, but cannot provide informations about the morphology of ice crystals. 25 Transmission electron microscopy can be used to image the growth of crystals and the behaviour of particles repelled by the latter, but such experiments cannot yet be run routinely and the interaction of the electron beam with the sample is always questionable. [START_REF] Tai | Microscopy and microanalysis : the official journal of Microscopy Society of America[END_REF] Near infrared imaging spectroscopy was used to investigate the morphology of ice crystals in biological materials, but freezing was not done in situ, and the spatial resolution is still not sufficient to image small crystals. 27 Here we show how confocal laser scanning microscopy and image analysis can be used to investigate the growth of ice crystals as small as tens of micrometers in situ in ZrAc solutions, stained with Sulforhodamine B, in a confined thin slab. We will show that we can follow and reconstruct the ice crystals topography, estimate quantitatively the kinetic of the advancing ice front while freezing with a table-top instrument.

Results

Freezing process

We first verified that, at this concentration, Sulforhodamine B has no effect on the morphol- No regular structure is observable in the Sulforhodamine B trace. As expected, water, while freezing, rejects brine, which also contains the fluorophore accumulated at the grain boundaries. [START_REF] Untersteiner | The Geophysics of Sea Ice[END_REF]28,29 With the proper fluorophore, we can thus quantitatively and thermodynamically follow the growth of ice crystals, 30 with or without ZrAc in solution.

The freezing of #2(TMA-OH) and #2(NaOH) solutions was observed as a function of time in the x-y plane at full cooling velocity to refine the sample position, to check the extent of condensation, and to capture the longitudinal dendritic growth of the ice crystals.

Generally no freezing was observed until the temperature of the Peltier stage was below -15 • C. Due to the supercooled state of the solution, the ice initially bursts in small dendritic crystals which densely populate the slab, hindering any observation. These crystals will grow until the temperature matches the equilibrium melting temperature T m . This obliges us to carefully select the x-z plane to measure: to get an overall insight of the ice crystals geometries and growth kinetic we select a priori a plane where we expect that the Sulforhodamine B fluorescence will dominate over the transparency of the ice and we will observe a few large crystals. Because of the supercooling effects when cooling from room temperature, we opted for a seeding protocol. The solution is quickly frozen to trigger the nucleation of ice and the sample is then brought back to a temperature of -10 • C and thermalized for 10 minutes.

The ice re-crystallizes at this temperature, leaving just a few crystals. We can then use these seeds to grow larger crystals in controlled conditions.

The supercooled state can be explained by lack of nucleation points, the colligative depression of the freezing point caused by the electrolyte, and by the confined state of the solution (analogous to similar effect in very small capillaries 31 ). We believe that due to the confinement, the nucleating water must override a change in internal pressure. At the critical point of the water/ice phase transition, the ice faces a density change of ≈ -8%.

This change in density must be accommodated either by changing the volume (fixed internal pressure) or by changing the internal pressure (at fixed volume). We hypothesize that a change in internal pressure occurs, and the excess of pressure is accommodated by lifting the top glass slide, that is, by increasing the volume: an effect similar to frost weathering (ice can exert pressures in the order of thousands of bars 32 ). Due to this supercooled state and by observing small crystals after freezing burst, we speculate that in a free falling temperature experiment there are two stages for ice growth: 33 the rapid dendritic/cellular growth followed by an apparent regular Gibbs-Curie-Wulff equilibrium growth when the crystals tips reach the position where T = T m .

Imaging ice crystals

An example of what we can image both in the x-z and the x-y planes is shown in Fig. 2. The growth orientation on the x-y plane can be tilted with respect to the edge of the image (Fig. 2(b)). In the x-z plane, the ice crystals may thus be probed at a certain angle, which partially explain the elongated morphology of the crystals along the x direction (Fig. 2(a)). By collecting a series of similar images of the same x-z section as function of time (Fig. 2), the growth of ice crystals can be visualized as three-dimensional objects (Fig. 3). From the analysis on several crystals grown in controlled freezing upon seeding, we can obtain a mean value of the transverse growth velocities v t (table 1). For a cooling rate of ∆(T ) = -1 • C/min, the growth velocity in the TMA-OH solution is slightly lower than that in the NaOH solution. However, no differences are found for a cooling rate of ∆(T ) = -2.5 • C/min. The growth rate of the crystals in the NaOH solution is the same for the two cooling rates. We cannot at this point conclude whether the growth of the crystals is kinetic-or diffusion-limited. At this stage, the liquid water surrounding the crystals is heavily charged in solutes which slow down the freezing. The solutes diffusion is very limited because of the low temperature and the already high solute concentration.

To better understand how the morphology of the ice crystal changes during its growth, we developed a Python code which searches for the liquid-crystal interface. Each frame is firstly filtered by a Gaussian filter which smooth the image, that is then segmented. Isocontours are drawn and the largest closed one is selected as the ice crystal area. Filter and detection of contours are obtained from the Skimage [START_REF] Van Der Walt | the scikit-image contributors[END_REF] and Trackpy 35 packages. The code developed to track the contours and compute the growth velocities is available on Figshare for anyone to review, reuse, improve, or adapt to something else. [START_REF] Noirjean | Time lapse, in-situ imaging of ice crystal growth with confocal microscopy (Jupyter notebook)[END_REF] In Fig. 5 the evolving contours of the central crystal from Fig. 3 are shown together. The growth of the crystal follows several steps by changing its apparent geometry from quasi-triangular to hexagonal. The apparent geometry depends on the viewing angle, as exposed previously. In Fig. 6 we select 5 frames from Fig. 5 with different geometries.

From these contours we can measure the temporal evolution of the perimeter and area of the crystal as well as its growth velocity.

The results for 4 representative crystals grown from the #2(TMA-OH) solution cooled at ∆(T ) = -1 • C/min are shown in Fig. 7. The four crystals show similar growth dynamics.

The same conclusion can be drawn from the instantaneous velocity results (Fig. 8). The weighted average velocity v A is of 6.1 ± 0.4 µm 2 /s (computed here for one of the crystals). Nevertheless, we should stress that, because the cooling stage does not provide a good control of the temperature gradient, our measurements are somewhat position-dependent.

The same procedure was used to measure the longitudinal growth velocities v l in both solutions. The values obtained for the two solutions and two cooling rates are given in Table 2.

Table 2: Table of the average v l : ∆(T ) in • C/min, v l in µm/s. ∆(T ) #2(NaOH) #2(TMA-OH) -1 0.9 ± 0.4 0.9 ± 0.5 -2. 5 2.4 ± 0.5 2.4 ± 0.4

The values of v l only depends on the cooling rates: neither NaOH/HCl nor TMA-OH contents seem to play any role.

A typical 3D-topography of ice crystals grown from the #2(TMA-OH) solution (after final cooling at T = -21 • C) is shown in Fig. 9. Similar morphologies were obtained for all the solutions and cooling rates.

A more careful observation of single x-z images from the 3D-topography shows that single crystals can result from the effective merging of smaller ones protruding from the cold

Discussion

The freezing point of the solution is of particular importance. The best results, in term of growing hexagonal crystals, can be achieved if, after the rapid freezing due to supercooling, one is able to bring back the solution to a temperature close to the melting point and melt until a few small crystals remain. In this case only the small remaining crystals will grow and the solute concentration in the liquid phase will closely approximate the concentration in the unfrozen solution. A similar method is generally used, for example, to distinguish single ice crystals grown in solution with Green Fluorescent Protein (GFP) tagged AFP [START_REF] Pertaya | [END_REF]38 and to identify their positions with respect the ice faces. In the current experiments, we cannot tag the ZrAc with a fluorophore because the fluorophore has to be smaller than the tagged molecules to avoid or reduce the interference with the kinetic and dynamic processes of freezing. Moreover, small fluorophores (such as the pyrene-based ones 39 ) fluoresce under UV light that is unavailable for us with the current equipment, and tagging ZrAc with a fluorophore could limit the ability of ZrAc to self-assemble.

To better gauge and model the ice crystallization, the optical vertical resolution and the ability to isolate non-interacting crystals during the freezing should be optimized. Both conditions are inherent to the statistical analysis of velocities and morphologies. The ability to perform experiments with different liquid slab thickness should provide information on how the confinement affects the ice growth. If colloids are added to the initial solution of ZrAc, 18 the lateral ice growth is limited by the colloids engulfed between the crystals. Here, the lateral ice growth is only limited by the thickness of the slab and by the proximity of other ice crystals. Moreover, it seems that ice crystals do not prefer to grow in contact with the glass. We believe this is caused by the generation of brine which cannot escape because it accumulates between the ice and the two glass faces of the freezing chamber.

Phenomenologically, we can say that upon the rapid freezing and the subsequent thermalization (T = T m ), the mechanism of the crystal growth apparently follows the Gibbs-Curie-Wulff conditions while the crystal evolves along the thermal gradient. The ice crystals section expands monotonically, going through the shape change previously described, with a preference for the x-direction. This can be explained by the finite size of the liquid slab in the z axis, whereas the x-and y-direction seem infinite from the point of view of the ice crystal. It would thus be worthwhile to compute the surface free energy of the ice when in solution with and without ZrAc. Within our range of measurements and set-up, we cannot identify if the accretion of crystals is diffusion-limited or kinetic-limited. 40 To perform an optimal experiment it is strongly needed to control the sample temperature with great precision, of the order of less than 0.1 • C. We observed that the vertices of the ice crystals were the first to reshape when the temperatures became less stable. The tips, in particular, are the weakest features. This observation raises the following question: if we model the ice needles as a pyramid on the top of a hexagonal prism (which correspond to the morphologies reported in presence of ice shaping proteins), what will be the maximum ratio between the basal plane and the pyramid height at thermodynamic equilibrium? And thus, which are the equilibrium crystallographic planes of ice in the presence of such additives?

The observations obtained using the procedure described here could be used to validate or invalidate the different models proposed for this problem. ZrAc is easier to obtain and use than AFPs, it can thus be used as a model system to grow faceted ice crystals, difficult to achieve with other additives.

We hypothesize that, as proven, 20 ZrAc organizes in stacks, whose length is dependent on the concentration. Such stacks, through an adsorption mechanism similar to the one of antifreeze proteins, hinder the dendritic and cellular growth of ice crystals. The regular stacks intuitively resemble the repeating motif of several antifreeze proteins. The adsorption mechanism is mediated by the acetate anions. To follow where the ZrAc goes while water freezes, one could exchange one acetate with a small fluorophore, for example pyrene-based fluorophores, 39 whose size and weight are comparable with the ones of Zr tetramers (bearing in mind the limitations previously discussed on the length and weight of the fluorophore). In this way we may be able to track the dynamic of ZrAc clusters in proximity of the ice crystal and, by measuring the local fluorescence intensity be able to link the local density with the velocity of the ice front. 41 Similarly, we could identify how the ZrAc settles on the ice face, and if the mechanisms are indeed similar to that of AFPs. 42,43 Finally, by using a pHsensitive fluorophore in solution we could quantitatively and thermodynamically follow the freezing process to gain further insight on how the pH distribution is linked to Zr tetramers and stacks.

Experimental methods

Experimental set-up

All the measurements were carried out with a Leica TCS SP8 (Leica-Mycrosystems, Mannheim, Germany) confocal laser scanning microscope. Confocal laser scanning microscopy has been primarily developed to image cells, proteins or other biological machines tagged with a fluorophore. [START_REF] Pawley | Handbook of Biological Confocal Microscopy[END_REF] The technique is also of interest for studies in material science, [START_REF] Hovis | [END_REF] albeit rarely used so far.

The microscope is equipped with two external continuous laser sources, respectively at 488 nm (blue) and 552 nm (green) driven to the optical section by two independent optical fibres. The maximum powers on the optical plane are 10 mW for both of them. At these wavelengths the absorption coefficient for distilled water is ≈ 0.45 • 10 -3 cm -1 at 552 nm and ≈ 0.25 • 10 -3 cm -1 at 488 nm. 46,47 The minimal absorption and the small intensities of both laser beams imply that no energy is deposited along the optical path through the sample of few tens of µm, as opposed to X-rays tomography where the intense and collimated beam may locally melt the ice or change the temperature gradient. 24,48 To focus the laser light at the region of interest, the sample stage is equipped with 3-axes micrometric drive.

Sub-micrometer resolution for the z -axis (or vertical direction) is ensured by a galvanometric stage with a run of 200 µm.

The light-detection section includes a rotating head able to contain 6 objectives: two of The fluorescent light is collected in the 575-625 nm range. To improve the image contrast at the liquid-solid interface, the reflected light of the blue laser (λ = 488 nm) is integrated in the 485-490 nm range. With respect to the method and set-up proposed by Neils et al., 50 we image the ice-water interface through the optical axis.

The measured sample is prepared as following (Fig. 11): a 22 × 22 mm2 microscope cover slide (borosilicate D263M, VWR, thickness ≈170 µm) is attached on the copper plate by vacuum grease which improves adhesion and thermal conductivity.

Half of the sample slide is in contact with the cold side of the stage (copper plate), while the other half is suspended in air because the copper plate is elevated above the surrounding metal plate, avoiding any contact on this side. 12 µL of ZrAc solution are deposited onto the With this set-up we create a longitudinal temperature gradient through the sample. To limit condensation, the sample and objective are surrounded by a transparent protective shield. In general, the imaging is carried out close enough to the edge of the copper plate.

Condensation cannot however be avoided further on the cold side. We also regularly check for change in sample thickness: a sudden change may mean that the seal cracked during the freezing, in which case the sample is discarded.

By scanning the x-y plane (top-view) we can measure the longitudinal growth velocity (v l ) of the ice crystals on the optical plane and by scanning the x-z plane (section view)

we can measure the growth velocity of the edges (v t ) and their morphology. However, we cannot yet acquire time-lapse, three-dimensional images at a sufficient rate to perform the measurements in both directions at the same time. The data acquisition was thus performed in the following ways: we collected images

• of the x-y plane while cooling, to measure v l (freezing direction y)

• of the x-z plane as function of time during the cooling to measure v t

• at fixed temperature to obtain a 3D reconstruction of the crystals.

The gradient of temperature is thus along the y-direction, the thickness of the liquid slab is measured along the z-direction and the lateral size is measured along the x-direction.

Postprocessing and image analysis

The images are post-processed and analysed with Fiji. 52 The only digital manipulations done before the analysis are contrast enhancement and normalization in automatic mode. A cross-section of the frozen sample #2(NaOH) is shown in two colours (Fig. 13): magenta represents the fluorescent signal from Sulforhodamine B diluted in water, while cyan is the reflected signal from different interfaces (top and bottom cover slides, and water/ice interfaces). The dark hexagon-like regions are ice crystals.

Figure 13: Image of the sample section. In magenta emission from water, in cyan reflection from liquid-glass interface. The black hexagon-like shades are ice crystals.

A qualitative analysis was performed on the time and spatial x-z series, which we can use to reconstruct the topography of the ice crystals, using Fiji. 53 The time-series (x-z-t axes) can be used to image the growth of ice crystal, while the spatial-series (x-y-z axes) show the three-dimensional topography. Image segmentation was performed using the Fiji built-in tools.

Conclusions

We demonstrate here how laser scanning confocal microscopy can be used to investigate in situ the three-dimensional growth of ice crystals, using a simple set-up and a fluorophore.

The approach was done using a model system, an aqueous zirconium acetate (ZrAc) solution, that results in faceted growth of ice crystals. Upon freezing, in analogy with ice shaping proteins, ice crystals of predictable morphologies can develop in calibrated aqueous solutions.

Because, like almost any solute, the fluorophore (Sulforhodamine B) is expelled by the growing crystals, the ice crystals appear as black bodies immersed in a sea of fluorescence. We were able to follow qualitatively and quantitatively the kinetic of the growing crystals with solubilized ZrAc tetramers. This methodology can also be used to investigate, for example, the growth in presence of simple polymers such as PVA, new promising ampholitic polymers, 54,55 either tagged or untagged with fluorophores, such as it has been proved with ice shaping proteins. 38,56,57 As a minimal amount of energy is deposited along the optical path in the sample, confocal microscopy appears much more appropriate than X-rays computed tomography, known to induce artefacts in the growth morphology of ice crystals. 24

Figure 1 :

 1 Figure 1: 3D-topography of brine channels (with Sulforhodamine B) in ice. Visualisation of the trace of Sulforhodamine B cooled from room temperature to ≈ -22 • C at the rate of ∆(T ) = -1 • C/min. The fluorophore is expelled from the frozen water and concentrate in small, rounded pockets. Similar effect is found in freezing of salty water and the formation of brine pockets. The pockets are aligned along the freezing direction y. Thickness: 28 µm.

Figure 2 :

 2 Figure 2: Two dimensional view of ice crystals. a) x-z plane: section of #2(NaOH) frozen at a ramp rate of ∆(T ) = -1 • C/min. Temperature gradient along z. b) One of the possible x-y views of the same sample showing the dark shadows of the ice crystals. Temperature gradient along y. The c and a-axis direction are indicated in the lower left.

Figure 3 :

 3 Figure 3: Time-topography of ice crystals. Three-dimensional time-topography visualising the growth of ice crystals in solution #2(NaOH) as function of time at the fixed cooling rate of ∆(T ) = -1 • C/min. ZT, XT, and XZ views.

Figure 4 :

 4 Figure 4: Ice crystal evolution. Transversal evolution of an ice crystal in #2(NaOH) at temperature rate of ∆(T ) = -1 • C/min, without pre-seeding. a) at the reference time t = 0 b) after t = 336 s (121 frames). The white line represents the trajectory for computing the facet velocity.

Figure 5 :

 5 Figure 5: Evolution of the contour of the ice crystal with time. #2(NaOH) at ∆(T ) = -1 • C/min. The morphology of the crystal changes as function of time, from quasi-triangular to hexagonal.

Figure 6 :

 6 Figure 6: Frames contours. #2(NaOH) at ∆(T ) = -1 • C/min. Guides for the eyes through the different geometries.

Figure 7 :

 7 Figure 7: Growth dynamics. Area of individual crystals versus time. #2(TMA-OH) at ∆(T ) = -1 • C/min. The area was computed for the 4 crystals in the inset.

Figure 8 :

 8 Figure 8: Area velocity. #2(TMA-OH) at ∆(T ) = -1 • C/min post-seeded. Area velocities computed from data in Fig. 7.

Figure 9 :

 9 Figure 9: Topography of ice crystals 3D-topography of ice crystals grown from the #2(TMA-OH) solution cooled at ∆(T ) = -1 • C/min. The tips of the ice crystals look alike sharp chisels. Front view (top) and back view (bottom). Dimensions: 400 µm×200 µm×46 µm.

Figure 10 :

 10 Figure 10: Merging ice crystals. Selected sections from a series of 3D-topography of ice crystals grown in the #2(TMA-OH) solution cooled at ∆(T ) = -2.5 • C/min. Top to bottom: the three crystals, highlighted by dotted contours on top sub-figure, merged into one and through an intermediate step become a single crystal with an hexagonal geometry.

  the positions are occupied by a Leica HCX PL APO 40× dry objective and a Leica HCX PL APO CS 20× dry objective: working distances of both is 670 µm. This short length hampers the use of any encapsulating box filled with nitrogen to avoid the condensation of water on the top of the glass slide, such as in Ref.37. The collected light is detected by two photomultiplier detectors which are fed with photons whose wavelengths can be independently selected. The smallest optical window is 5 nm, and the two spectra cannot overlap. With this set-up one can independently capture two fluorophores. The digitalization centration at which ice faceting has been observed.18 The pHs of one (#2(NaOH)) of the batches were then adjusted to pH = 4.0 ± 0.1 by adding NaOH 25 mol/L solution and HCl solutions at 37% wt (Sigma Aldrich). The latter #2(TMA-OH) was carefully diluted with TMA-OH (tetramethylammonium hydroxide, Sigma Aldrich) at the concentration of 10% wt in water without overshooting pH ≈ 4. The objective was to titrate with two different base solutions to asses whether the ionic strength of the bases and the inorganic/organic nature of the bases influence the growth morphologies and/or kinetics. TMA-OH is a weak base (pK b =4.2 versus pK b =0.2 for NaOH). The final Zr concentrations for the TMA-OH solutions was estimated to 11.2 g/L.Local gelation of the solutions was observed while adjusting with the bases: ZrAc eventually gelates at pH > 4.4. One night of stirring was enough for the complete solubilization of the gelated part. Each solution was stained with 1 mM Sulforhodamine B (Fluotechnik, France), 60 µL/mL. The absorption/emission spectra of Sulforhodamine B are rather insensitive to pHs in the range of 3 and 10,49 and its absorption band is excited by the laser light at λ = 552 nm (absorption peak at λ max = 566 nm and emission peak at λ max = 584 nm in water).2

Figure 11 :

 11 Figure 11: Photograph of the sample set-up.Half of the sample slide is on the cold side of the Peltier cell and the other half is suspended in air because the copper plate is elevated above the surrounding metal plate, thus there is no contact on this side. A temperature gradient parallel to the cover slip as well as perpendicular to its surface is thus created and ice crystals grow from the cold side towards the hot side of the sample.

Figure 12 :

 12 Figure 12: Variations of the apparent measured thickness at various imaging resolutions. To measure the true depth, the optical path should be carefully calibrated. The two squared resolutions give values close to the expected ones.

Table 1 :

 1 Table of the average v

	∆(T ) #2(NaOH) #2(TMA-OH)
	-1	0.10 ± 0.03	0.07 ± 0.04
	-2.5 0.10 ± 0.05	0.10 ± 0.05

t : ∆(T ) in • C/min, v t in µm/s.

GNU Octave, version 4.0.1 (2016) John W. Eaton et al., www.octave.org

Data from: http://www.fluorophores.tugraz.at/substance/10

Acknowledgement

M.M. wish to aknowledge Dr. G. Discepoli (@gd_roger) for the fruitful discussion in Peltier control.

Funding

The research leading to these results has received funding from the European Research Council under the European Community's Seventh Framework Programme (FP7/2007-2013) Grant Agreement no. 278004 (project FreeCo).

of the integrated light is at 8-bits.

For the data collection the 20× objective was used to image the x-y plane and the x-z section for all the resolutions. For example, at a x-y hardware resolution of 1024 × 1024 px 2 (pixels 2 ), the area sampled is 775 × 775 µm 2 at the minimum optical magnification of M = 0.75× and frame rate of 0.388 frames/s with scanning frequency of 400 Hz, whereas for the x-z section a hardware resolution of 1024 × 256 px 2 gives a scanned section of 400 × 100 µm 2 at the minimum optical magnification of M = 1.45× and frame rate of 0.357 frames/s at a scanning frequency of 100 Hz. For the measurements in the x-z section, the optical plane is fixed and the sample is translated along the z-direction by the galvanometric stage. The microscope is controlled by the proprietary Leica LAS-AF software.

Sample preparation

A solution of zirconium acetate (in-house preparation of Saint-Gobain) at the initial concentration of 22.6 g/L (solution #1) of Zr (gravitometrically measured) 17 and original pH = 2.6

(measured by pre-calibrated Oregon Scientific pH-meter) was diluted in deionized water to obtain a solution with equivalent starting Zr concentrations of 13.3 g/L (pH = 3.2): two identical batches were prepared. This concentration was chosen because it is the lowest con-


Anne-Sophie S Caro-Bretelle

Patrick Ienny

Romain Léger

R Leger

Constitutive modelling of a SEBS cast-calender: large strain, compressibility and anisotropic damage induced by the process

Keywords: damage, digital image correlation, hyperelasticity

published or not. The documents may come

Constitutive modeling of a SEBS cast-calender: Large strain, compressibility and anisotropic damage induced by the process

Introduction

In this paper the effect of planar anisotropy induced by the forming process is modelled.

The two most important manufacturing methods with thermoplastic elastomers are extrusion and injection molding. Fabrication via injection molding is extremely rapid and highly economical. Both equipments and methods usually used for extrusion or injection molding of a conventional thermoplastic are generally suitable for TPEs. TPEs can also be processed by blow molding, thermoforming, and heat welding. In this paper, extrusion followed by injection molding or dip coating processes are used to produce thin films with a controlled thickness. Once the material is compounded, it is either shaped into sheets by flat path calender or by dip coating. In the first process polymers chains are uniaxially oriented, yielding pronounced structural anisotropy; on the contrary dip coating produces isotropic films.

The present study focuses on the mechanical behavior of a Styrene-Ethylene-co-Butylene-Styrene for which high deformability, compressibility and strain softening are relevant assumptions. This kind of polymer is widely used in industry because of its low cost, optical transparency, biocompatibility and flexibility. Its complex behavior is known and usually studied through filled materials [1][START_REF] Loew | Finite Elements in Analysis and[END_REF][3]. Nevertheless some authors modeled unfilled silicone rubber [4] but all of these studies assume that this material is isotropic.

A phenomenological model developed in the framework of continuum damage mechanics is proposed. Among numerous modeling [5,6], high deformability and compressibility are modeled via a Rivlin phenomenological approach [7]. Strain softening is assumed to be anisotropic and derives from Miehe's theory [8,9].

In order to identify parameters of the model, an experimental work including cyclic uniaxial tensile tests, biaxial and compression tests is performed.

The outline of this article is as follows. After the experimental section (2) mechanical tests results are discussed (3) to propose a phenomenological model suitable to reproduce experimental results (4). Throughout the article, various samples, cut in different directions in the plane of the sheet, are tested in order to study the influence of process on mechanical properties. It will be shown that the calender process induces an increase of both stress and damage in the calender direction in comparison with the transverse one. Parameter's optimization is presented in section (5) for samples cut in the calender and transverse directions and from uniaxial, biaxial and Oedometric tests. Finally, the model has been 3/30 validated for a sample cut in the diagonal direction. Lastly, in order to test the ability of the model to describe strain and stress of complex structure, a sample sheet containing two holes is uniaxially loaded. Finite elements simulation of this test is then performed and compared to experimental results.

Experimental

Materials and preparation

The SEBS triblock copolymer used in this study has been purchased at Plastic Technology Service LTD (reference SEBS Thermoflex 10H730). This system is characterized by an 11 shore A hardness, a low amount of polystyrene (PS) 5% by weight in its end block and by a ratio of butylene to ethylene less than one (30%wt butylene). This composition allows a very low crystallinity degree (less than 1%). Structures of these kinds of block polymers (or similar) has been widely studied especially when structure-property correlations are expected [10][11][12].. PS domains act as glassy phase with physical crosslinks while PEB are elastomeric blocks which give the rubbery behavior [13]. Depending on both the amount of PS and the ratio of butylene to ethylene the PS domains coalesce into cylindrical or lamellar rods packed in a two-dimensional hexagonal lattice [10], spherical domains embedded within the PEB matrix [11].

SEBS has been casted as films with a cast-extruder Polylab OS / Rheomex 19/25 of Thermofisher Company. This device is equipped with a 30cm wide flat-path to control the thickness from 0.2mm to 1.2mm and a cast-conveyor that helps cooling the film to control its crystallinity. The film is then rolled. Such process creates films with anisotropic properties which are mostly caused by preferred orientations of polymer chains developed during the severe plastic deformation such as cold rolling [11,12].

Other films have been prepared by solubilisation process. SEBS was first dissolved in xylene to prepare a 0.3 wt% solution. This solution was then deposited onto a flat support and let few hours under a hood at ambient temperature to evaporate the solvent. This sequence is repeated until obtaining the desired thickness of deposit. The film was annealed for 2 h in a vacuum oven at 140 °C.

Properties of casted films have been evaluated in the longitudinal direction (Calender Direction: CD), in the transverse direction (TD) or in the diagonal direction () of the roll.

The longitudinal direction corresponds to the extrusion direction; the transverse direction 4/30 corresponds to the direction parallel to the width of the extruded film, at right angles to the length of the roll and the diagonal direction is at 45° to the length of the roll (Fig. 1).

Fig. 1: Anisotropy induced by the process

Rectangular and discs shape specimens of a 0.6mm thickness were cut from films (see Fig. 2):

-Rectangular strips (height L=60mm, width l=30mm corresponding to the useful dimensions, see (a) in Fig. 2) were cut along the longitudinal, transverse and diagonal directions of the calender direction. Samples have been produced for both manufacturing process. An uniaxial load is applied to these samples.

-Rectangular strips (height L=10mm, width l=56mm corresponding to the useful dimensions, see (b) in Fig. 2); a uniaxial loading on these samples enables a biaxial solicitation in which transverse strain is prevented.

-Disc shape specimens (diameter d=20mm and height h=10mm, see (c) in Fig. 2): thin disks were stamped and stacked together to achieve a compression cylinder of total height h; oedometric compression tests were performed on these specimens to enhance the material compressibility.

-A rectangular shape specimen with two holes (see (d) in Fig. 2) was cut and loaded uniaxially to produce heterogeneous strains; a prior finite element simulation enabled the definition of this geometry to maximize shearing strain between the two holes of the specimen.

5/30

Quasi-static tests

Let (x,y,z) and (CD,TD,z) be two cartesian bases, attached to the device and to the material respectively (Fig. 3). x is the tensile direction, y is perpendicular to x and in the planar surface and z the out-of-plane direction. The -angle qualifies the material direction: ° corresponds to the CD direction, to the TD direction and to the diagonal direction). As usual, both basis are related by the relation: (x, y, z) t = P (CD, TD, z) t where

z z CD) TD - TD CD TD) TD CD CD P          )( sin( )( cos(  
and the superscript t denoted the transpose operator.

6/30

Mechanical loadings

Uniaxial tensile tests were conducted on a Zwick TH010 universal testing machine.

The crosshead speed was adapted to have the same longitudinal strain rate for the tensile and biaxial tests. The tensile test is conducted up to a defined nominal strain. The software used is TestXpert II ® and allows the recording of time, load (F) and stretch. In order to analyse the mechanical behavior, samples were subjected to several homogeneous loadings and one heterogeneous.

-Uniaxial tensile tests were carried out on rectangular strip specimens at room temperature (see Fig. 2). These tests were performed with crosshead speeds of 10mm/min and 100mm/min. During the test, nominal stress (first Piola-Kirchhoff)

x x   xx  σ
and deformation gradient tensor

y x z z y y x x F         xy z y x    
are assumed both to be homogeneous and to retain this form during the test even at high strain level. The axial stress and dip coated samples tested for both displacement speeds (10 mm/min and 100 mm/min). First of all the anisotropy induced by the process is clearly demonstrated: the response is stiffer in the direction of calendering. The comparison between the two processes shows a significant reduction of stiffness beyond 500% strain for the material processed by solubilisation. Otherwise and unexpectedly, the level of stress slightly increases with a decreasing strain rate: increasing strain rate should therefore induce a greater damage excepted for the dip coated samples for which the curves are superimposed. Nevertheless, the small effect of the time dependence behavior has led us to neglect this phenomenon for this study.  and yy  are unknown but can be deduced from theory and experiments (see section 3).

-A tensile test was also performed on a rectangular planar specimen containing 2 holes as shown in Fig. 2. The test was conducted with the same equipment: the load cell measure the axial force F and DIC gave the local in-plane stretches which is, in this case, heterogeneous.

-An oedometric device was also developed to evaluate the bulk modulus, necessary parameter to describe material compressibility. Samples (see Fig. 2) are subjected 9/30 to compressive load in a cylinder expected to be undeformable. Load and displacement cells enable the evaluation of bulk modulus.

Strain field measurements

During the mechanical tests, the deformations were measured by an optical extensometer including a high resolution Charge camera (Redlake Megaplus 2, 1920 by 1080 pixels, coded in 256 grey levels). A vertical translation table unit was used to follow the region of interest (ROI) of the tested specimen. The image acquisition is controlled by a LabVIEW ® software which allows the simultaneous record of images and data from the testing machine (such as load and crosshead displacement). The scale factor (k= 78 µm/pixel for homogeneous tests and k= 37µm/pixel for the heterogeneous test) is fixed to constitute a compromise between a necessary spatial resolution and the limitation of the perspective errors associated to the out of plane displacement. Different scale factors were used for both types of test on homogeneous and heterogeneous specimens [14,15]. To obtain a signature for the correlation treatment, some carbon powder is deposited on the observed surface of the specimen. Digital Image Correlation software CinEMA (produced by the Alès School of Mines) allows the determination of the two-dimensional displacement components with a subpixel accuracy: from two treated images, a correlation calculation is made at every meshing point (Fig. 6). Each point of the virtual mesh corresponds to the centre of a pattern.

This pattern, defined by its size denoted CS (for correlation size), is the representative area of the material point which is tracked. Distance between mesh points is given by the Gs (for Grid and the virtual mesh used for the homogeneous and heterogeneous tests, the cumulative error does not exceed 10 -4 and 10 -3 respectively to the level of strain considered [14,15] After evaluating of the two-dimensional fields of in-plane displacement a linear estimation of the displacement gradient realized on several points chosen around the studied point allows the calculation of the deformation gradient tensor.

Homogeneous tests

Cyclic behavior, material anisotropy

Uniaxial cyclic tensile tests are then performed on CD and TD specimens: they consist in achieving first three loading-unloading up to the axial stretch level of about  x1 =3. strain curves are shown in Fig. 7 for both samples. They behave as classical elastomer presenting a stress softening depending on the material direction. We observe a significant reduction on the stress at a given level of strain for loading sequences which follows the first loading, phenomenon known as the Mullins effect [16]. Moreover both specimens

show a relatively low level of hysteresis, which is already stabilized after the second cycle.

These observations are consistent with a low level of crystallinity supposed previously: this type of elastomer does not crystallize under loading [10]. Regarding to the low level of PS incorporation, the permanent set never exceeds x  =1.4 (40% of strain) and does not change beyond the first cycling. For the modeling, we can consider this value to be negligible with respect to the maximum elongation undergone by the material during the test (

x  >10). This result was also expected since this permanent set seems to be linked with the level of PS content [10]. It is possible to record volume strain E V up to large deformations, where E V is defined as:

.

1 E   z y x V    (1) 12/30
In the case of uniaxial tensile test performed in calender direction and under transverse isotropy assumption (see section 2.2.2), this volume strain can be deduced from surface measurements:

1 E 2   y x V  
. This volume variation increases first according to the growth of damage and decreases after 5 of stretch level (see "primary loading" in Fig. 8). This phenomenon has been already observed previously and can be attributed to a decrease of strain speed while the macro-molecular chains are aligned in the loading direction or the beginning of crystallization [17]. As commonly observed in the literature [18], this volume strain trend depends on mechanical cycling (see Fig. 8). While successive loadings lead to the same volume fraction evolution (as a function of longitudinal strain), the curves are gradually shifted to lower volume strains. It is important to note that, following an initial loading, the material is in compression state for a zero load level. This is probably due to a reorganization of the material microstructure under loading.

Biaxial test, material anisotropy

In the graph of Fig. 9, the evolution of axial stress is reported as function of the stretch ratios. It is observed the same behavior as for the uniaxial tensile test in the same proportion.

As expected, longitudinal stress for the biaxial test is greater than for uniaxial one at the same strain level. In addition to the already noted difference between the transverse and the 13/30 calendering directions, these results first highlight the strong influence of the type of mechanical loading on stress-stretch curves and the assumption of quasi-restrained test

( 1  y 
) which is thus validated.

Fig. 9: Axial stress versus stretches for CD and TD specimens for a biaxial loading

Oedometric test

To ensure reproducibility, the tests were performed three times. The samples (Fig. 2c) are stabilized under three loading-unloading cycles successively up to -4 kN, -3 kN, -2 kN, -1kN of the load. The height variation of the sample is deduced from the crosshead displacement reduced by the displacement corresponding to the machine and device stiffness.

The load -displacement curve (see Fig. 10) is fitted on the stabilized loading with a linear function (a is the director coefficient) between -3.5 kN and -2 kN, -2.5 kN and -1 kN, -2 kN and -0.5 kN, -1 kN and -0.5 kN. The bulk modulus is therefore deduced at each loading cycle and estimated by the expression

Phenomenological modeling of the mechanical behavior

Theoretical backgroung

As far as our analysis is phenomenological, the type of stress-strain curves is closed to the one of crosslinked rubber. A review of molecularly based model can be found in [19]. The studied SEBS exhibits a hyperelastic behavior: elastic (recoverable) but highly nonlinear. In particular, the constitutive behavior of such materials is usually derived from the strain energy potentials with the assumption that the material is isotropic. The Cauchy stress component can be expressed in the device frame as:

T W J F F    1 σ where J=det F. ( 2
)
In the phenomenological approach, this energy is defined through either strain invariants [5,7,[20][21][22] or stretch ratios [23] via a polynomial development. In this study, the 15/30 model of Rivlin [7] has been selected to describe this hyperelastic contribution. This energy enables a good approximation of the non linearity response at low and large strain because of the polynomial degree. The deviatoric part of the density energy is given by:

(3) where are the material parameters and , .

This model has been extended for compressible materials where an additive decomposition of hyperelastic potential in deviatoric and volumetric part is used:

(4)
The compressibility component is associated with volumetric expansion while the incompressibility component is associated with volume constant distorsion. Among the various forms present in the literature (see [24] for example) we choose:

(

) 5
where is the bulk modulus (see section 3).

From equation 3 to 5 the strain energy is therefore defined by:

W(F)= , ( 6
)
From equations 2 and 6 the nominal stress can be computed as follow: (7) where , Therefore, we assume that the hyperelastic part of the material behavior is isotropic unless global anisotropy is related to damage. Some authors have modeled anisotropic hyperelastic behavior using material directions decomposition [25] or additive principal invariants [26].

On the other end, sequenced tensile tests (Fig. 7) highlight a damageable behavior sensitive to the direction of calender associated to measured volume variation (Fig. 8). This softening behavior has been the subject of considerable study [27,28]. In the literature some authors related this phenomenon to failure/slip between matrix and inclusion, inclusions failure or disentanglement of matrix macromolecules [29]. Many phenomenological theories have been proposed in the literature. Some of them are based on the two phase microstructural models introduced by Mullins and Tobin [30] and developed by Johnson and Beatty [31]. Another class of models is developed within the framework of damage mechanics and 16/30 introduces a parameter of damage d which affected the strain energy density [8,9,32]. For example Miehe [8] proposed an isotropic modeling via two parameters which are related to the damage velocity () and to the level of damage (d ∞ ). It is common to define an effective stress from this damage parameter [33]. The stress softening is generally attributed to both breakage and shape of the PS domain [12]. Otherwise, the microstructural state is clearly not described with Miehe approach. Nevertheless this theory, which introduces a global damage parameter is simple and in accordance with our phenomenological observation. However, in order to consider micromechanical difference of conformation between CD and TD, we propose to introduce an anisotropic damage of SEBS, described through a tensorial expression in place of the single scalar parameter commonly used

Given the biaxial and uniaxial tests performed in the calendering direction (CD) and the perpendicular direction (TD), this anisotropic damage evolution in which damage coupling and plane-shear damage have been neglected: =

where are three parameters to be fitted. At the difference of Miehe but in accordance with [9] this experimental damage depends on the strain . This matricial expression can be expressed in the material frame device:

, ( 10
)
where P is the rotation matrix introduced in section 2.2.

In the TD direction (e.g. x corresponds to TD) the damage is reduced to Miehe's expression (no damage in the initial state) and in the CD direction (e.g. x corresponds to CD) the damage is affected by an initial value when it is not subjected to strain. This choice has been motivated by the material microstructural composition: due to the process, the macromolecular chains are preferably oriented in the calender direction and have been pre loaded in this direction, although in the transverse direction they are entangled. 17/30

The damage function was determined using experimental results (from sequenced tensile tests in CD and TD directions) [9]. We consider results of Fig. 11, where we reduced experimental data to the sequences of loading curves of cycles 3, 6 and 9. Primary loading curves correspond to simple uniaxial tensile test and are denoted by I 1 , I 2 I 3 for the three primary loading. are related to the three secondary loadings. From these data, we are able to determine discrete damage variables at stretches  x1 ,  x2 and  x3 by the expression:

TD. ( 11
)
This energetic approach does not permit the reconstruction of the reloading curves. In fact these discrete coefficients only traduce a ratio of area under the curve and no the shape of the curve. For more complete analysis see reference [9].

In the case of biaxial tensile test, the only non-zero stress components are and .

These conditions imply:

(14) (15)
For both experiments, the unknown out plane stretch is therefore defined in term of stretches and material parameters , .

Analytical definition of strain/stress component for the specimen with two holes is not possible since the test is highly heterogeneous. The mechanical modeling is thus based on the Finite Element Method and integrates the mechanical behavior introduced in part 4.1.

Numerical optimization

Assuming parameters ) known, theoretical stress / stretch curves were computed using procedure describe in section 4.

Results and discussion

Homogeneous study

After 10 iterations the minimization problem defined by ( 16) converges. The results of the identification process are listed in Table 3. To assess this identification procedure (accuracy and reliability), a sensitivity analysis has been made. The quality of this identification can be estimated via the ratio:

, 1..6, (18)
20/30

The lower is the more confidence we have in the identification process. Table 4 illustrates this issue. The parameter influencing the most (in eq. 16) is for the hyperelasticity and ( , ) for the damage. Therefore both of these parameters mainly control the stress distribution. results (see Fig. 13). For the biaxial test, in the calender direction, the discrepancies appear beyond 4 of axial stretch. Moreover the model mark the difference in behavior depending on the loading direction: no transitory phase is observed for the material tested in the calender direction due to the pre stretched undergone by the material during the process (Fig. 14) The evolution of continuum damage is shown in Fig. 15 (see eq. 9). For the diagonal sample, initial damage is lower than the CD one but the asymptotic value is the same. It is interesting to note that during the process the material is stretched in the CD direction and is damaged to a corresponding value of d=0.2. This damage corresponds to an equivalent dilatation =2 on the curve of the non pre-streched sample (see Fig. 15). On these curves, stars represent experimental discrete damage data at stretches Furthermore this modeling allows the numerical evaluation of the volume strain variation (see Fig. 16). From the knowledge of numerically, the equation (1) conducts to the numerical volume strain variation for biaxial tensile tests (CD, TD) and uniaxial one (TD).

We recall that, for the uniaxial tensile test performed in the calender direction (CD), the volume change is directly deduced from the measurement of in-plane stretch ratios associated with the assumption of transverse isotropy. It is clear that the biaxial test produces a greater volume strain variation than the uniaxial test (ratio of 10). In conclusion the model seems adequate to represent mechanical behavior of SEBS under homogeneous tests.

Heterogeneous study

To evaluate the capability of the model to describe a complete structure, a sample generating heterogeneous strain fields has been produced (see section 2.1). The numerical test was performed on ZeBuLON ( [START_REF] Besson | Revue européenne des éléments[END_REF]) with the constitutive model described in section 4.1

(already implemented in this software) and the fitted parameters given in Table 4 (see section The object of future works will concern the use of finite-element software to apply this inverse method to microstructural processes of deformation in a damageable area characterized by analytical and optical measurements (e.g. SEM, IR, or NMR).

LIST OF FIGURES

Fig. 2 :

 2 Fig. 2: Useful dimensions of specimens: for uniaxial (a), biaxial (b) compression loadings (c) and heterogeneous test (d) (quotations in mm, thickness 0.6mm)

Fig. 3 :

 3 Fig. 3: Material and device frames

  -of-plane measurement) provided stopping temporarily the test; the subscript' is relative to the deformed state. Reproducibility tests on a CD sample are shown in Fig.4where nominal stress is plotted versus stretch ratios. DIC was performed on tests 7/30 with or without stopping (DIC1 and DIC2 respectively); associated results confirm the reproducibility (see Fig.4in which DIC1x . These results validate DIC (stretches ratios deduced from DIC and measured by digital calliper correspond) and demonstrate transverse isotropy for the CD sample ( z y    ). This transverse isotropy assumption was validated in a previous article on polycarbonate nanocomposite specimen submitted to uniaxial tensile test[14]

Fig. 4 :

 4 Fig. 4: Uniaxial stress as function of measured stretch ratios (measured by DIC and calliper device)

xFig. 5 :

 5 Fig. 5: Uniaxial stress versus stretches ratios for SEBS manufactured by a calender process (CD, TD) and by dip coating process

  step) parameter. Deformation calculation is obtained by the first order derivation of the displacement field around the considered point. The dimension of the area where is evaluated the in-plane gradient tensor is fixed by the Ns (for Number of Steps) parameter which qualify the number of points to take into account in the vertical and horizontal directions, both directions being equivalent from this point of view. Of course, associated with the magnification, these variables characterize the bidimensional Gage Lenth (GL) expressed in pixel or mm: (GL = 2.N s .G s + C s ). The choice of these parameters should guarantee a good accuracy without losing local information and should reduce errors and the computation time (Table1

Fig. 6 :

 6 Fig. 6: Definition of the two-dimensional digital extensometer (CS: correlation size; GS: grid step

  7 and then successively up to about  x2 =5.3 and about  x3 =6.6 of the axial stretch. Nominal stress-11/30

Fig. 7 :

 7 Fig. 7: Stress-Stretch response during three cycles of loading-unloading at 3.7, 5.3 and 6.6 of stretch (for CD and TD samples)

Fig. 8 :

 8 Fig. 8: Volumetric variation as a function of mechanical cycling for a CD sample

.Fig. 10 :Table 2 :

 102 Fig. 10: Load-displacement curve in oedometric compression

Fig. 11 :

 11 Fig. 11: Construction of damage evolution: evaluation of its discrete components

2 . 1 . 30 ( 16 )Fig. 12 :

 21301612 Fig. 12: Identification and validation

  stress-stretch curves are shown in Figs. 13 and 14 for simple (ST) and biaxial tensile (BT) tests. Optimized parameters allowed the model to match the experimental data at least until 4.5 of axial stretch. This model is able to predict mechanical behavior: for the sample cut in the diagonal direction the fitted model gives satisfactory

Fig. 13 :Fig. 14 : 3 Fig. 15 :Fig. 16 :

 131431516 Fig. 13: Identification procedure based on uniaxial (ST) tests for CD and TD samples; Prediction for the sample cut in the diagonal direction (45°)

5. 1 )

 1 . The finite element geometry (deformed and undeformed) and the boundary conditions can be found in Figs. 17-18. It has been modelled via 382 linear triangular elements.Displacement (corresponding to the load F) is applied on the upper and lower sides of the sample, under plane stress assumption. The test was conducted on the instrumented testing machine under the conditions described in section 2.2.2. For each node shown on Fig.17, DIC gives access to the full in-plane strain components.

Fig. 17 : 30 Fig. 18 (Fig. 18 :

 17301818 Fig. 17: FE Mesh and line from which the data are extracted (a); photograph for DIC (b)

17

 17

30 Fig. 19 :Fig. 20 :

 301920 Fig. 19: Evolution of the shear Lagrangian strain component along the extended line between the two holes

Fig. 1 .Fig. 2 .Fig. 3 .Fig. 4 .Fig. 5 .Fig. 6 .Fig. 7 .Fig. 8 .Fig. 9 .Fig. 10 .Fig. 11 .Fig. 12 .Fig. 13 .Fig. 14 .Fig. 15 .

 123456789101112131415 Fig. 1.

Fig. 16 .

 16 Fig. 16.

Fig. 17

 17 Fig.17.

  Fig.17.

Fig. 18 .Fig. 19 . 30 Fig

 181930 Fig. 18.

Table 1 : Data parameter used by the DIC Process

 1 ).

	Specimen	Acquisition speed	Cs (pix/mm)	Ns	Gs (pix/mm)	GL (pix/mm)
	unnotched specimen	1 im/s	30/2.34	5	30/2.34	330/25.2
	Heterogeneous specimen	1 im/s	30/1.11	1	20/0.74	70/2.59
			10/30

Table 3 : Material parameters fitted from uniaxial and biaxial tests for sample in the calender and transverse directions

 3

	(MPa)	(MPa)	(MPa)
	0.0002	0.0641	0.0004	0.255	0.4020	1.4

Table 1 .Table 2 .Table 3 .Table 4 .

 1234 Data parameter used by the DIC ProcessVariation of the bulk modulus with the hydrostatic pressure Material parameters fitted from uniaxial and biaxial test for sample in the calender and transverse directions Sensitivity analysis

Acknowledgments

We thank the AREVA and PIERCAN for providing the SEBS samples and for fruitful scientific discussions.