domain: sdv.bbm.bp sdv phys

text:
Sarah Ameziane-Le Hir

Céline Raguénès-Nicol

Gilles Paboeuf

Aurélie Nicolas

Elisabeth Le Rumeur

Véronique Vié

Sarah Ameziane Le Hir

Cholesterol favors the anchorage of human dystrophin repeats 16 to 21 in membrane at physiological surface pressure

Keywords: Atomic force microscopy, Langmuir film, muscular dystrophy, protein lipid interaction, CRAC sequence, spectrin superfamily

come

Cholesterol favors the anchorage of human dystrophin repeats 16 to 21 in membrane at physiological surface pressure.

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SUPPLEMENTARY TABLE S1 Summary of values obtained after protein insertion in lipid monolayers and described in the text: surface pressure (∆π) and ellipsometric angles (δ∆) variations; apparent height of protein protrusions in AFM images.

SUPPLEMENTARY TABLE S2

Occurrences of CRAC sequences (L/V)-(X) 1-5 -Y-(X) 1-5 -(R/K) in the primary sequences of dystrophin (uniprot P11532), alpha erythroid spectrin (uniprot P02549), and beta erythroid spectrin (uniprot P11277) as determined by protein pattern find tool of Sequence Manipulation Suite (Stothard, 2000. Biotechniques 28:1102Biotechniques 28: -1104)). Amino acids of CRAC sequences are in blue, sometimes several patterns are overlapped.

Introduction

Dystrophin (DYS) is a long filamentous muscular protein of 427 kDa (about 175 nm long) [START_REF] Koenig | Complete cloning of the Duchenne muscular dystrophy (DMD) cDNA and preliminary genomic organization of the DMD gene in normal and affected individuals[END_REF][START_REF] Pons | Isolated dystrophin molecules as seen by electron microscopy[END_REF]. Most of this length is due to a long central rod domain composed of 24 spectrin-like repeats (R1 to R24) folded into a triple-α-helical bundle [START_REF] Muthu | The crystal structures of dystrophin and utrophin spectrin repeats: implications for domain boundaries[END_REF]. The biological function of dystrophin is to protect the membrane against stress during elongation/contraction muscular cycles. Its absence leads to the severe disease Duchenne muscular dystrophy (DMD), whereas the milder Becker muscular dystrophy (BMD) may be observed when truncated proteins are expressed [START_REF] Koenig | The molecular basis for Duchenne versus Becker muscular dystrophy: correlation of severity with type of deletion[END_REF]. Due to its high levels of STRs, dystrophin is a member of the spectrin-like superfamily [START_REF] Vishwanatha | Structural organization of the nine spectrin repeats of Kalirin[END_REF]. A relationship has been suggested between the structural organization of the repeat domains in these proteins and the membrane's protection against shearing stresses [START_REF] Muthu | The crystal structures of dystrophin and utrophin spectrin repeats: implications for domain boundaries[END_REF]. This function may result from a homogeneous protein distribution along the membrane caused by protein/lipid interactions. Indeed, spectrin is known to bind to the inner membrane leaflet due to interactions with phosphatidylserine [START_REF] Diakowski | Brain spectrin (fodrin) interacts with phospholipids as revealed by intrinsic fluorescence quenching and monolayer experiments[END_REF][START_REF] Macdonald | Temperature and ionic effects on the interaction of erythroid spectrin with phosphatidylserine membranes[END_REF][START_REF] Maksymiw | Electrostatic coupling of spectrin dimers to phosphatidylserine containing lipid lamellae[END_REF][START_REF] O'toole | Rapid diffusion of spectrin bound to a lipid surface[END_REF]. Moreover the presence of 10-20% cholesterol (CHOL) facilitates the insertion of spectrin into phosphatidylethanolamine/phosphatidylcholine monolayers that were preformed at low surface pressure, i.e. from 8 to 11 mN/m [START_REF] Diakowski | Cholesterol affects spectrin-phospholipid interactions in a manner different from changes resulting from alterations in membrane fluidity due to fatty acyl chain composition[END_REF]. Even during its liquid-expanded phase (LE), spectrin binding capacity depends on the fluidity of the monolayer. Our previous studies showed that fragments of the dystrophin rod domain interact with membrane system models through electrostatic and/or hydrophobic forces [START_REF] Legardinier | Mapping of the lipid-binding and stability properties of the central rod domain of human dystrophin[END_REF][START_REF] Vie | Specific anchoring modes of two distinct dystrophin rod sub-domains interacting in phospholipid Langmuir films studied by atomic force microscopy and PM-IRRAS[END_REF].

This study is focused on the interaction of the specific dystrophin fragment made of repeats R16 to R21 and a membrane model. This construct has several specific features that set it apart from other dystrophin fragments. For instance, it contains a hinge (H3) between the repeats R19 and R20 [START_REF] Koenig | Detailed analysis of the repeat domain of dystrophin reveals four potential hinge segments that may confer flexibility[END_REF][START_REF] Sahni | Exon edited dystrophin rods in the hinge 3 region[END_REF]. The presence of H3 in microdystrophin improves the capacity for muscle degeneration prevention, while deletion of this hinge leads to the BMD phenotype [START_REF] Banks | The polyproline site in hinge 2 influences the functional capacity of truncated dystrophins[END_REF][START_REF] Carsana | Analysis of dystrophin gene deletions indicates that the hinge III region of the protein correlates with disease severity[END_REF]. In addition, DYS R16-21 is coded by exons 42 to 53, which includes the hot-spot between exons 45 to 53 where about one third of BMD patients have mutations [START_REF] Nicolas | Assessment of the structural and functional impact of in-frame mutations of the DMD gene, using the tools included in the eDystrophin online database[END_REF].

The fragment also mediates specific protein interactions, such as the anchoring of neuronal NO Synthase (nNOS) to the sarcolemma through R16-R17 [START_REF] Lai | Dystrophins carrying spectrin-like repeats 16 and 17 anchor nNOS to the sarcolemma and enhance exercise performance in a mouse model of muscular dystrophy[END_REF]. The mislocalisation of nNOS in the membrane leads to a decrease of muscular strength in dystrophin-deficient mdx mice [START_REF] Li | Nitrosative stress elicited by nNOSmicro delocalization inhibits muscle force in dystrophin-null mice[END_REF]. R16-21 is located in the last part of the protein, where the cysteine-rich domain is known to bind to β-dystroglycan (β-DG) in a membrane protein complex [START_REF] Ervasti | Dystrophin, its interactions with other proteins, and implications for muscular dystrophy[END_REF] that is located in detergent-resistant membrane. Sarcolemmal CHOL depletion induces the weakening of force contractions through the redistribution of β-DG, and is accompanied by a reduction of β-DG/dystrophin interactions in favor of β-DG/caveolin-3 interactions [START_REF] Vega-Moreno | Cholesterol depletion uncouples beta-dystroglycans from discrete sarcolemmal domains, reducing the mechanical activity of skeletal muscle[END_REF]. All these studies showed that structural or lipid environmental modifications impact muscular activity.

Nevertheless, up to now little was known about the ability of DYS R16-21 to interact with lipid membranes.

Here, we address the question of the DYS R16-21 anchorage in mixed lipid films and more particularly of the effect of CHOL on the interaction. Since phosphatidylserine is specifically abundant in the inner leaflet of the sarcolemma membrane, the effect of the polar head charge was tested using a simplified mixture composed of anionic dioleoylphosphatidylserine (DOPS) and zwitterionic dioleoylphosphatidylcholine (DOPC) lipids. Because we chose unsaturated lipids, the monolayers are in the LE phase regardless of the surface pressure. A mix of dipalmitoylphosphatidylcholine (DPPC) and DOPC was used to mimic a detergent-resistant membrane as described [START_REF] Davis | Phase equilibria in DOPC/DPPC-d62/cholesterol mixtures[END_REF][START_REF] Schroeder | Cholesterol and sphingolipid enhance the Triton X-100 insolubility of glycosylphosphatidylinositolanchored proteins by promoting the formation of detergent-insoluble ordered membrane domains[END_REF]. Indeed DPPC has been reported to form liquid-condensed (LC) phase domains and also to create separated phases when mixed with DOPC [START_REF] Li | Cholesterol decreases the interfacial elasticity and detergent solubility of sphingomyelins[END_REF]. Cholesterol, also present in the inner leaflet up to 25% [START_REF] Gennis | Biomembranes : molecular structure and function[END_REF], is known to modulate membrane fluidity, and particularly at high surface pressure, it acts as a head group spacer because of its molecular orientation in the monolayer. Moreover, CHOL is located mainly in the condensed phase formed by DPPC [START_REF] Guyer | Phosphatidylcholine and cholesterol interactions in model membranes[END_REF][START_REF] Epand | Cholesterol-rich domains : Preface[END_REF]. Langmuir monolayers were used, which allowed for control of charge densities, of fluidity, and of lipid packing according to specific lipid compositions and surface pressures. Protein-lipid interactions and protein structure were determined by combining tensiometric and ellipsometric measurements with atomic force microscopy imaging. We evaluated the influence of 15%

CHOL on the interaction of DYS R16-21 with DOPC/DOPS and DOPC/DPPC lipid mixtures. Using two initial surface pressures, two different lipid packings were tested The higher of these (30 mN/m) is considered to be comparable to physiological membrane pressure [START_REF] Macdonald | The relationship and interactions between lipid bilayers, vesicles, and lipid monlayers at the air/water interface[END_REF][START_REF] Marsh | Lateral pressure in membranes[END_REF].

Our results show that DYS R16-21 interacts with the lipid monolayer no matter what mixture was used, while the surface pressure has a notable impact on the insertion. In addition, under high surface pressure CHOL was shown to stabilize the fragment at lipid interface, likely through the presence of two cholesterol recognition amino-acid consensus (CRAC) sequences on the protein surface [START_REF] Li | Peripheral-type benzodiazepine receptor function in cholesterol transport. Identification of a putative cholesterol recognition/interaction amino acid sequence and consensus pattern[END_REF]. The presence of CRAC sequences all along dystrophin allows specific protein/lipid interactions that result in homogeneous protein distribution along the membrane. This anchorage could be related to lateral force transmission during muscle elongation/contraction.

Experimental sections

Materials

All lipids were purchased from Avanti Polar Lipids (Alabama, US). The pGEX-4T-1 plasmid vector and GSTrap HP and Sephacryl S-100 columns were bought from GE Healthcare. BL21 bacteria and restriction enzymes were purchased from Ozyme (St.-Quentin-en-Yvelines, France). Purified bovine thrombin came from Stago (Asnières-sur-Seine, France).

Preparation of recombinant DYS R16-21 protein

The DYS R16-21 fragment begins at residue Ser 1994 and ends with Thr 2686, using the full-length human dystrophin annotation (NCBI NP_003997). In order to improve stability and as previously done for other dystrophin fragments [START_REF] Kahana | conformation and phasing of dystrophin structural repeats[END_REF][START_REF] Winder | Dystrophin and utrophin: the missing links![END_REF], we included a three-aminoacid extension in the N-terminus and a seven-amino-acid extension in the C-terminus. The protein was cloned in pGEX-4T-1 then expressed in a protease-deficient E. coli BL21 strain as a GST-tagged protein [START_REF] Legardinier | Sub-domains of the dystrophin rod domain display contrasting lipid-binding and stability properties[END_REF]. Cell lysis was obtained by sonication after lysozyme treatment.

DYS R16-21 was further purified using GST affinity chromatography. After thrombin proteolysis (50 UI for 2 hours at room temperature), the chromatography fractions were separated by salting-out with ammonium sulfate. The target protein was collected at 24% salt saturation and rediluted in TNE buffer (20 mM Tris, 150 mM NaCl, 0.1 mM EDTA, pH 7.5).

Purification was achieved with size-exclusion chromatography by passing it through a

Sephacryl S-100 column (L 2.6 cm x H 60 cm) at 1 mL/min using the same buffer. After purification, the protein was concentrated using a 30 kDa cut-off centrifugal concentrator.

Purity was assessed by 10% SDS-Polyacrylamide gel electrophoresis (SDS-PAGE), and concentration was determined using a bicinchoninic acid assay with a standard of serum albumin.

Circular dichroism spectra were acquired between 250 nm and 200 nm, with 1.5 µM of protein in TNE buffer at 20°C (J-815, Jobin-Yvon) using 0.2 cm path-length cells. The αhelical content was calculated from the mean molar residue ellipticity [θ] (mdeg.cm 2 .dmol -1 ) at 222 nm [START_REF] Greenfield | Computed circular dichroism spectra for the evaluation of protein conformation[END_REF].

Tensiometric and ellipsometric measurements

In order to investigate the interfacial properties of the protein with and without lipids, surface pressure and ellipsometric angles were measured in Teflon troughs (4, 8, or 60 mL) at 20°C. The Wilhelmy method was used to determine the surface pressure using a tensiometer (Nima Technology, Cambridge, UK). Ellipsometric measurements were performed with a conventional polarizer-analyzer null-ellipsometer setup. Briefly, a He-Ne laser (632.8 nm;

Melles Griot, Carlsbad, CA) with an angle of incidence of 52.18° (1° away from the Brewster's angle), was used as the light source. Reflecting surface properties were explored by recording the positions of the polarizer and analyzer having the minimal intensity of transmitted light. The analyzer angle was doubled to yield the value of the ellipsometric angle (Δ) [START_REF] Berge | Ellipsometry Study of 2D Crystallization of 1-Alcohol Monolayers at the Water Surface[END_REF][START_REF] Bottier | Structure and orientation of puroindolines into wheat galactolipid monolayers[END_REF][START_REF] Chieze | Fluid and condensed ApoA-I/phospholipid monolayers provide insights into ApoA-I membrane insertion[END_REF]. Both Δ and the surface pressure (π) were recorded as functions of time. The amphiphilic character was determined by reporting the surface pressure reached at this end of the absorption kinetics as a function of the protein's subphase concentrations (ranging from 0.0001 µM to 1 µM). For lipid/protein experiments, the lipid mixtures (DOPC/DOPS 1:1 molar ratio and DOPC/DPPC 1:1 molar ratio) were prepared in a 2:1 chloroform/methanol solution (v/v) at 0.5 mM, and where used, CHOL was added at 15% (molar ratio). The lipids were gently spread at the air/liquid interface of the TNE subphase at the desired surface pressure. These initial lipid surface pressures (π i ) ranged from 5 to 35 mN/m. At a final concentration of 0.03 µM, the protein was then injected into the subphase just beneath the lipid monolayer. The variations (δΔ and Δπ) induced by protein adsorption at the end of the absorption kinetics were reported as functions of time and π i . Each experiment was repeated up to four times and means ± standard deviation (SD) were provided for each series. The temperature varied between 19 and 21ºC.

Atomic force microscopy (AFM) observations

The interfacial film was transferred onto a freshly-cleaved mica plate using the Langmuir-Blodgett technique. The experiment was realized using a 102 cm² Langmuir trough equipped with two movable barriers and controlled by a computer (model 601 M, Nima Technology, Cambridge, UK). The transfer was performed at the end of the kinetics and at a constant surface pressure. The dipper speed rate was 1 mm/min. Transferred monolayers were imaged using a PicoPlus atomic force microscope (Agilent Technologies, Phoenix, AZ) equipped with a 10 μm scanner operating in contact mode. Images were acquired under ambient conditions using silicon nitride tips on integral cantilevers (ScienTec) with a spring constant of 0.06 N/m. For each measurement, the set point was adjusted before and during the scanning to minimize the force between the tip and the sample. In each case, images were obtained from at least two samples prepared on different days, with at least four macroscopically-separated areas in each sample. To determine the height of the objects, after processing the images with Gwyddion 2.26 software (http://gwyddion.net) we did a statistical analysis. Objects of interest were selected by threshold, and then areas and maximal heights were determined. The values of height versus area were plotted, resulting in a plateau. This plateau included the majority of the objects and was used to compare the different experimental conditions.

Three-dimensional molecular homology modeling and surface properties

Three-dimensional (3D) homology models were obtained using I-TASSER server [START_REF] Zhang | I-TASSER server for protein 3D structure prediction[END_REF]. As previously described [START_REF] Legrand | Computational study of the human dystrophin repeats: interaction properties and molecular dynamics[END_REF], the sequences of two successive repeats ("tandem repeats") with an overlap of a single repeat were submitted to I-TASSER (e.g. DYS R16-17, then DYS R17-18, and so on). The H3 sequence was submitted separately. Superposition of the shared repeat of two successive tandem repeats allowed the construction of an extended model which we further minimized by YASARA (e.g. the DYS R16-17 and DYS R17-18 models were used to construct a DYS R16-18 model). In this way, step by step, the global 3D structure was reconstructed. Surface hydrophobicity was calculated using PLATINUM [START_REF] Pyrkov | PLATINUM: a web tool for analysis of hydrophobic/hydrophilic organization of biomolecular complexes[END_REF] and surface electrostatic potentials in the presence of 50 mM NaCl were calculated using the Adaptive Poisson-Boltzmann Solver (APBS) software [START_REF] Baker | Electrostatics of nanosystems: application to microtubules and the ribosome[END_REF]. Visualization was done using VMD and PyMOL.

Results

Protein purification

SDS-PAGE of the recombinant human dystrophin fragment DYS R16-21 (figure 1A)

showed the protein at the expected molecular weight of 80 kDa. As expected with spectrinlike repeats [START_REF] Legardinier | Mapping of the lipid-binding and stability properties of the central rod domain of human dystrophin[END_REF][START_REF] Macdonald | Free energies of urea and of thermal unfolding show that two tandem repeats of spectrin are thermodynamically more stable than a single repeat[END_REF], the circular dichroism spectrum of DYS R16-21 displayed the typical predominant α-helix folding with two minima at 208 and 222 nm (figure 1B). Per Chen et al., the molar ellipticity of 222 nm allowed for determination of an α-helix content of 83% [START_REF] Chen | Circular dichroism of lipoprotein lipids[END_REF].

The ratio of the molar ellipticity at 222 and 208 nm (θ 222 /θ 208 ) was 1.05, indicating that the helices were interacting and folded in a coiled-coil [START_REF] Zhou | Synthetic model proteins. Positional effects of interchain hydrophobic interactions on stability of two-stranded alpha-helical coiledcoils[END_REF]. Therefore, it appears that the recombinant DYS R16-21 fragment was highly purified and correctly folded, conditions necessary for its further study.

Amphiphilic properties of DYS R16-21

For each DYS R16-21 subphase concentration between 0.0001 to 1 µM, surface pressure measurements were done and the final surface pressure versus subphase protein concentrations are reported in figure 2A. As the protein concentration increased, π increased until a plateau (π max ) of 21 mN/m was reached, indicating the amphiphilic character of DYS R16-21. From these adsorption kinetics results, the critical concentration required for interface saturation was determined to be 0.1 µM. For further experiments, a protein concentration of 0.03 µM (or 2.5 µg/mL) was chosen (indicated with an arrow in figure 2A)

to avoid a potential aggregation of proteins in the subphase or the formation of multilayers at the interface. Figure 2B shows a typical adsorption time for this protein concentration.

Whereas surface pressure is related to the protein's ability to cover the interface and to interact laterally, the ellipsometric angle depends on the amount of matter at the interface. As is usual, at the very beginning of the adsorption the ellipsometric angle increased quicker than the surface pressure. It reached an angle of 10° at equilibrium, a value corresponding to a monolayer as experienced in other proteins [START_REF] Desfougeres | Strong improvement of interfacial properties can result from slight structural modifications of proteins: the case of native and dryheated lysozyme[END_REF]. AFM imagery of this monolayer at the end of the adsorption kinetics (figure 2C) showed a homogeneous film with no aggregation.

DYS R16-21 interaction with lipid monolayers

The adsorption kinetics of DYS R16-21 was monitored at various initial lipid surface pressures (π i ). The variations of the surface pressure (Δπ) i.e. the difference between the initial lipid surface pressure and the final surface pressure were reported as a function of π i (figure 3). When π i increased, Δπ induced by adsorption of DYS R16-21 decreased. Indeed as the lipids compacted, the protein was less and less able to be inserted into the lipid monolayer. When there was no more variation between the initial and final surface pressures (Δπ=0 or intercept of the linear regression with the x axis as indicated with arrows in figure 3), the maximal insertion pressure (MIP) was reached [START_REF] Calvez | Parameters modulating the maximum insertion pressure of proteins and peptides in lipid monolayers[END_REF]. This was clearly dependent on the lipid mixture, with values of 26.5 mN/m for DOPC/DOPS and 33 mN/m for DOPC/DPPC.

For π i less than 20 mN/m, ∆π was higher with anionic than with zwiterrionic lipids, suggesting an attractive role for electrostatic interactions in the insertion of DYS R16-21. On the other hand, when π i was higher than 20 mN/m, ∆π was higher with zwitterionic lipids, suggesting presence of hydrophobic interactions during the protein's insertion process, while electrostatic interactions limit the insertion. Moreover, for the DOPC/DOPS monolayer at an initial surface pressure higher than π e (26 mN/m), the protein induced a surface pressure decrease (∆π = -1.6 mN/m), thus showing that attractive interactions were still taking place between DYS R16-21 and the compacted DOPC/DOPS monolayer.

To further analyze the organization of DYS R16-21 in lipid environments, we focused our work on lipid monolayers formed at two significant initial surface pressures: 20 mN/m, where the protein adsorption induced similar overpressures for all lipid compositions tested; and 30 mN/m, considered to be close to physiological membrane pressure [START_REF] Marsh | Lateral pressure in membranes[END_REF]. All reported values are recapitulated in table S1.

Cholesterol has few effects at a π i of 20 mN/m

Cholesterol is known to influence lipid packing therefore we studied the impact of 15 % CHOL on the DYS R16-21/lipid interactions. At π i of 20 mN/m (figure 4A), no changes were observed upon the addition of CHOL to the DOPC/DOPS film, while there was a slight increase of ∆π and δ∆ (1 mN/m and 0.4° respectively) in the DOPC/DPPC mixture. AFM images of pure lipid films were unmodified in the presence of CHOL. DOPC/DOPS ± CHOL (figure 4B) formed a homogeneous liquid-expanded (LE) phase. DOPC/DPPC ± CHOL images (figure 4E) showed the coexistence of an LE phase of DOPC and a liquid-condensed (LC) phase of DPPC, as previously observed [START_REF] Vié | Distribution of Ganglioside GM1 between Two-Component, Two-Phase Phosphatidylcholine Monolayers[END_REF]. The LC domains were several µm in diameter and their apparent height was 1.4 ± 0.1 nm.

Mixed protein/lipid films showed bright protrusions attributed to the protein's presence (figure 4C-H). In DOPC/DOPS ± CHOL films, these protrusions appeared mostly as wormlike curved filaments (figure 4C-D). CHOL induced both protein clustering and an increase in the apparent height of protrusions (2.1± 0.1 nm without CHOL versus 2.7± 0.1 nm with CHOL). These changes in DYS R16-21 organization occurred even though the lateral pressure and the amount of material at the interface were similar with or without CHOL (figure 4A).

In DOPC/DPPC films (figure 4E-H), the presence of DYS R16-21 drastically decreased the LC domain's diameter, which never exceeded 400 nm. Without CHOL (figure 4F-G), the proteins appeared as round objects or filaments with heights higher (2.0 ± 0.1 nm) than those of LC domains (1.4±0.1nm). The filaments were most often stuck to an LC phase (detailed view, figure 4G) whereas round objects were located in an LE phase. In the presence of CHOL (figure 4H), higher filaments (3.1 ± 0.2 nm) were located both in the LE phase and at the boundaries of LE/LC phases.

Cholesterol increases DYS R16-21 insertion at a π i of 30 mN/m

Ellipsometric and tensiometric measurements (figure 5A) showed that in any lipid condition, a π i of 30 mN/m induced the presence of less protein at the interface than with a π i of 20 mN/m. Injection of DYS R16-21 under the DOPC/DOPS/CHOL monolayer led to an increase in surface pressure (from -1.6 mN/m without CHOL to +2 mN/m with CHOL) and a decrease in the ellipsometric angle (from 0.6° without CHOL to 0.2° with CHOL), indicating modified protein/lipid interactions that favor the insertion of DYS R16-21. In the DOPC/DPPC films, the presence of CHOL had no significant effects on ellipsometric angle variation after injection of DYS R16-21, whereas the presence of CHOL induced an increase in ∆π (from 0.3 mN/m without CHOL to 2 mN/m).

The AFM images of DOPC/DOPS films with or without CHOL (figure 5B) were similar to those at 20 mN/m. The DOPC/DPPC/CHOL AFM images revealed that the presence of CHOL caused a slight decrease in height of LC domains (from 1.4 nm without CHOL to 1.2 nm) (figure 5E).

In the DOPC/DOPS/protein films (figure 5C), DYS R16-21 appeared as protrusions forming heterogeneous clusters higher (3.6 ± 0.4 nm) and larger than those previously observed at a π i of 20 mN/m. This is consistent with the variations in surface pressures and ellipsometric angles seen with the two surface pressures (figure 4A and5A). In the presence of CHOL (figure 5D), the proteins formed small protrusions that are smaller in length, width, and height (2.3 nm ±0.1) than those in the absence of CHOL or with a smaller π i . Finally, the protein organization was more homogeneous, with deeper insertion into the lipid monolayer and less material at the interface.

In DOPC/DPPC/protein films (figure 5F), the protein was present as scarce clusters in agreement with the low ∆π. Their heights (between 1 and 4 nm), were very heterogeneous.

By contrast, CHOL induced numerous clusters of DYS R16-21 (figure 5G-H) with heights of 2.6 ± 0.2 nm. The LC domain borders were much more irregular and angular than without CHOL. The protein clusters appear to be more elongated when in contact with LC domains than when embedded in LE domains (detailed view, figure 5H). Some objects also appeared to be embedded in LC domains, but since there was a depression surrounding these clusters it was obvious that there was no direct contact between the LC and the protein.

Computational analysis of DYS R16-21in presence of 50 mM NaCl

As expected for a soluble protein, the molecular surface of DYS R16-21 is mainly hydrophilic. Nevertheless one side of the fragment displays a higher presence of hydrophobic amino acids (figure 6). The electrostatic potential at the molecular surface shows a distribution of negative and positive charges along the molecule. Numerous positive patches were observed, mainly on repeats 16 and 17 of DYS R16-21, with the rest dispersed more widely along the molecule and embedded in the negative surface. A rotated view of the fragment showed a more contrasted surface with larger negative patches, separated by a few small positive zones that appear mostly on repeats 18 and 19. Interestingly, some hydrophobic patches colocalize or are very close to the cationic surface (for example on R16, R18 and R20). These may thus be anchorage points for interaction or insertion in anionic and zwitterionic lipid monolayers even under high initial surface pressures.

A search of the primary sequence of dystrophin revealed several occurrences of the [START_REF] Li | Peripheral-type benzodiazepine receptor function in cholesterol transport. Identification of a putative cholesterol recognition/interaction amino acid sequence and consensus pattern[END_REF]. Due to this sequence, a cholesterol binding cleft is created where Y and R/K interact directly with CHOL [START_REF] Jamin | Characterization of the cholesterol recognition amino acid consensus sequence of the peripheral-type benzodiazepine receptor[END_REF]. In DYS R16-21, the pattern was found in the C helix of R16 (V 2088 NKMYKDR 2098 ) and in the R21 B-C loop (L 2647 RDYSADDTR 2657 ) (figure 6). Most of the amino acids involved are exposed on the surface, and accessible enough to enable interaction with CHOL molecules.

CRAC pattern -(L/V)-(X) 1-5 -Y-(X) 1-5 -(R/K)-

Discussion

In this study, we examine the interaction of DYS R16-21 with lipid monolayers . This particular dystrophin fragment is characterized by the presence of hinge 3 (H3) and is frequently mutated in patients. DYS R16-21 displayed amphiphilic properties that were probably due to the presence of hydrophobic amino-acids all along the surface (figure 6).

Compared with previously-studied rod domain fragments, the adsorption of DYS R16-21 at the air/water interface was faster,the plateau was reached at a smaller protein concentration but has a similar value of 21 mN/m [START_REF] Vie | Specific anchoring modes of two distinct dystrophin rod sub-domains interacting in phospholipid Langmuir films studied by atomic force microscopy and PM-IRRAS[END_REF][START_REF] Sarkis | Spectrin-like repeats 11-15 of human dystrophin show adaptations to a lipidic environment[END_REF]. This indicates both that the interface is more rapidly covered by the protein, and that the molecules do indeed interact with each other.

This result could be related to the larger size of the protein as compared to other dystrophin fragments previously studied, and/or to the presence of H3, which may make the protein's structure more flexible [START_REF] Koenig | Detailed analysis of the repeat domain of dystrophin reveals four potential hinge segments that may confer flexibility[END_REF].

The monolayer's initial lipid surface pressure (π i ) is a major factor influencing the adsorption of amphiphilic proteins through electrostatic or hydrophobic forces because it determines lipid packing. When π i was below 20 mN/m, the protein was able to insert itself into both anionic and zwitterionic monolayers; however the maximal insertion pressure was significantly lower for the anionic mixture suggesting that attractive electrostatic interaction plays a major role in DYS R16-21-membrane interactions. The negative charge of the polar head attracts the protein toward the membrane, allowing further hydrophobic interactions to occur thanks to free interfacial space. At a π i of 20 mN/m, the protein insertion overpressures and ellipsometric angle variations seem to be equivalent for both lipid mixes. AFM observations show the protein as round or fibrous objects with regular heights (about one protein thick). The fibrous objects were present in the anionic mix, and interconnected filaments were only detected in contact with LC domains. These features imply that the molecules were stabilized at the interface by negative charges or LC domains and then adopted the same orientation at the lipid interface. This favors the existence of lateral protein/protein interactions in both lipid mixtures. At a π i over 20 mN/m, electrostatic interactions still occur but the space between lipid head groups becomes more and more limited, and the fragment insertion slows. Finally, when the π i of the DOPC/DOPS film was over 26 mN/m, interactions with lipid head groups caused a reorientation and relaxation of the films, as shown by a decrease in the final surface pressure, and the protein appears as irregular clusters. This phenomenon has been reported previously for the dystrophin R2 repeat [START_REF] Dewolf | Interaction of dystrophin fragments with model membranes[END_REF], DYS R20-24 [START_REF] Vie | Specific anchoring modes of two distinct dystrophin rod sub-domains interacting in phospholipid Langmuir films studied by atomic force microscopy and PM-IRRAS[END_REF] and in α-spectrin [START_REF] Diakowski | Brain spectrin (fodrin) interacts with phospholipids as revealed by intrinsic fluorescence quenching and monolayer experiments[END_REF]. In zwitterionic film, however, the insertion of DYS R16-21 occurs up to 31 mN/m. This is probably because the protein's orientation is not driven by electrostatic interactions and because its flexibility allows for penetration of the films even at higher surface pressures.

When comparing the anionic monolayer insertion of the previously-studied fragments DYS R1-3, DYS R11-15 and DYS R20-24 to that of the DYS R16-21 studied here, different behaviors are seen, thus demonstrating that these fragments are not interchangeable as had been stated [START_REF] Le Rumeur | A new twist to coiled coil[END_REF]. Indeed, at 30 mN/m, only DYS R11-15 inserts into the monolayer (∆π = +2 mN/m) [START_REF] Sarkis | Spectrin-like repeats 11-15 of human dystrophin show adaptations to a lipidic environment[END_REF], whereas both DYS R16-21 and R20-24 stayed at the head group level [START_REF] Vie | Specific anchoring modes of two distinct dystrophin rod sub-domains interacting in phospholipid Langmuir films studied by atomic force microscopy and PM-IRRAS[END_REF].

At 20 mN/m, a gradient seems to occur along the central domain, with greater insertion for fragments from the N-to C-terminal ends (∆π =7 mN/m for DYS R1-3, 6 mN/m for DYS R11-15, and 5 mN/m for DYS R16-21 and DYS R20-24). This intriguing observation must be further studied to decipher whether it has a physiological significance. It would be especially interesting to understand the relationship between the variation in sarcolemmal lateral pressure during muscle elongation/contraction, and the number or position of anchoring points along the protein. Interestingly, it has been reported that binding of spectrin to phosphatidylserine in the inner membrane increases the mechanical stability of red blood cell membranes [START_REF] Manno | Identification of a functional role for lipid asymmetry in biological membranes: Phosphatidylserine-skeletal protein interactions modulate membrane stability[END_REF].

We examined dystrophin/lipid/CHOL interactions because of cholesterol presence in the membrane and of its impact on the membrane's physical properties. Moreover we report the existence of a specific amino-acid sequence called CRAC (cholesterol recognition amino-acid consensus) and involved in CHOL binding [START_REF] Li | Peripheral-type benzodiazepine receptor function in cholesterol transport. Identification of a putative cholesterol recognition/interaction amino acid sequence and consensus pattern[END_REF][START_REF] Saxena | Membrane cholesterol stabilizes the human serotonin(1A) receptor[END_REF]. Like spectrin [START_REF] Ciana | Detergent-resistant membranes in human erythrocytes and their connection to the membrane-skeleton[END_REF][START_REF] Grzybek | Spectrin-phospholipid interactions. Existence of multiple kinds of binding sites?[END_REF], dystrophin has been shown to be associated in vivo to detergent-resistant membranes [START_REF] Doyle | Dystrophin Associates With Caveolae of Rat Cardiac Myocytes: Relationship to Dystroglycan[END_REF] and the interaction between β-DG and dystrophin is weakened by CHOL depletion [START_REF] Vega-Moreno | Cholesterol depletion uncouples beta-dystroglycans from discrete sarcolemmal domains, reducing the mechanical activity of skeletal muscle[END_REF]. The impact of CHOL on the structural and lateral organization of lipid layers is highly-documented [START_REF] Epand | Cholesterol-rich domains : Preface[END_REF][START_REF] Mcconnell | Condensed complexes of cholesterol and phospholipids[END_REF][START_REF] Silvius | Role of cholesterol in lipid raft formation: lessons from lipid model systems[END_REF], in particular its role as head group spacer and its usual localization in the condensed phase formed by DPPC [START_REF] Guyer | Phosphatidylcholine and cholesterol interactions in model membranes[END_REF]. In addition, several monolayer experiments have shown complex formation between CHOL and phosphatidylserine, an important component of the inner leaflet [START_REF] Radhakrishnan | Electric field effect on cholesterol-phospholipid complexes[END_REF]. Our study was performed at 15 % of CHOL since 10 to 20 % CHOL have been shown to increase spectrin insertion in phosphatidylethanolamine/phosphatidylcholine monolayers [START_REF] Diakowski | Cholesterol affects spectrin-phospholipid interactions in a manner different from changes resulting from alterations in membrane fluidity due to fatty acyl chain composition[END_REF] and also because this concentration is known to preserve lateral phase separation [START_REF] Silvius | Cholesterol modulation of lipid intermixing in phospholipid and glycosphingolipid mixtures. Evaluation using fluorescent lipid probes and brominated lipid quenchers[END_REF]. At the lower initial surface pressure (20 mN/m), macroscopic measurements in the lateral packing of the acyl chains. These hydrophobic mismatch defects result in lateral fragmentation of these domains, and this phenomenon is amplified by the presence of CHOL [START_REF] Killian | Hydrophobic mismatch between proteins and lipids in membranes[END_REF][START_REF] Phillips | Emerging roles for lipids in shaping membrane-protein function[END_REF]. At a higher initial surface pressure (π i =30 mN/m), CHOL amplifies protein insertion into lipid monolayers and favors an homogeneous organization of the protein in both LC and LE domains. Indeed, the sterol polar head group is small compared to choline or serine, while the hydrophobic part at this pressure is similar to that of a phosphatidylcholinesaturated chain [START_REF] Li | Cholesterol decreases the interfacial elasticity and detergent solubility of sphingomyelins[END_REF], thus CHOL plays the role of head group spacer. This promotes a deeper and more homogenous lipid insertion. The DOPS/DYS R16-21 electrostatic interactions are modified by CHOL. This could be linked either to PS/CHOL complex formation as suggested by Radhakrishnan et al [START_REF] Radhakrishnan | Electric field effect on cholesterol-phospholipid complexes[END_REF] or to specific DYS R16-21/CHOL interactions.

Bio-informatics analysis of the DYS R16-21 amino acid sequence revealed two occurrences of the CRAC sequence -(L/V)-(X) 1-5 -Y-(X) 1-5 -(R/K)- [START_REF] Li | Peripheral-type benzodiazepine receptor function in cholesterol transport. Identification of a putative cholesterol recognition/interaction amino acid sequence and consensus pattern[END_REF]. CRAC sequences are also present in spectrin and in other dystrophin fragments (Table S2). These patterns in DYS R16-21 are easily accessible for interaction at the protein surface. Not only has CRAC been identified in many proteins that interact with cholesterol, but it has been more widely suggested to be related to protein propensity for interaction with cholesterol-rich lipid domains [START_REF] Epand | Cholesterol interaction with proteins that partition into membrane domains: an overview[END_REF]. CRAC motifs are often reported in membrane proteins, but are compatible with peripheral proteins and was also reported in a secreted toxin [START_REF] Lai | Characterization of putative cholesterol recognition/interaction amino acid consensuslike motif of Campylobacter jejuni cytolethal distending toxin C[END_REF].

Conclusion

Our results highlight the fact that dystrophin fragments, and here especially DYS R16-21, can interact directly with monolayers at high surface pressure, at low fluidity, and in the presence of anionic lipids. We further show that the presence of dystrophin close to the membrane is probably not due only to a link with β-DG but also thanks to its own sub-part membrane-binding properties. It has now to be tested whether our in vitro results may be transposed in vivo. Does formation of direct interactions between dystrophin and cholesterolrich nano-domains stabilize the dystrophin/β-DG/sarcolemma in muscular cells? If so, cholesterol-rich domains in the inner sarcolemma layer could prevent ejection of dystrophin under surface pressure variations due to contraction/elongation cycles and contribute to sarcolemma resistance.

  FIGURE 2

  (π, ∆) were unmodified in presence of CHOL, but topographic images revealed an increase in the lateral size and height of protein protrusions, indicating changes in the molecular organization of DYS R16-21 within the lipid monolayers. CHOL favors lateral proteinprotein interactions, and since the lipids are in fluid phase, it increases steric packing in the hydrophobic part, limiting deep anchoring. In the DOPC/DPPC mixture, LC domain fragmentation is induced by the protein. DYS R16-21 insertion might induce local disorders

Figure 1 :

 1 Figure 1: DYS R16-21 sub-domain purification and secondary structure analysis. (left) 10 % SDS polyacrylamide gel of DYS R16-21 obtained after purification. Lane M is the protein ladder. (right) Circular dichroism measurements of 1.5 µM of DYS R16-21 in TNE buffer (pH 7.5) at 20°C.

Figure 2 :

 2 Figure 2: DYS R16-21 adsorption at the air-liquid interface. (A) Final surface pressure reached at the end of the adsorption kinetics versus DYS R16-21 subphase concentrations. Arrow indicates π reached for protein subphase concentration of 0.03 µM (B) Adsorption kinetics of DYS R16-21 at air-liquid interface for the subphase concentration of 0.03 µM, surface pressure (+), and ellipsometric angle (o). (C) AFM topographic image of transferred DYS R16-21 monolayer at the end of the kinetics (5 h). Surface pressure was 20 mN/m. Scan size 5µm x 5µm, z scale 10Å.

Figure 3 :

 3 Figure 3: Overpressure induced by adsorption of DYS R16-21 at air/lipid interface versus the initial lipid surface pressures of DOPC/DOPS (full line) and DOPC/DPPC (dotted line). Arrows indicate the values of maximal insertion pressures (MIP). The subphase concentration of DYS R16-21 was 0.03 µM.

Figure 4 :

 4 Figure 4: Characterization of DYS R16-21 at the initial lipid surface pressure of 20 mN/m. (A) Variations of the surface pressure (Δπ) and of the ellipsometric angle (δΔ) induced by DYS R16-21 adsorption on phospholipid monolayers (black) and phospholipid/CHOL monolayers (gray). (B-H) Typical AFM topographic images of monolayers as indicated on the panels. Scan size 5µm x 5µm for images B-F and H, and 2 µm x 2 µm for image G. Zrange 80 Å. When present, DYS R16-21 subphase concentration is 0.03 µM, CHOL 15 %. LE, liquid-expanded phase; LC, liquid-condensed phase.

Figure 5 :

 5 Figure 5: Characterization of DYS R16-21at an initial lipid surface pressure of 30 mN/m. (A) Variations of the surface pressure (Δπ) and of the ellipsometric angle (δΔ) induced by DYS R16-21 absorption on phospholipid (black) and phospholipids/CHOL monolayers (gray). (B-H) Typical AFM topographic images of monolayers as indicated on the panels. Scan size 5µm x 5µm for images B-G and 2 µm x 2 µm for image H. Z-range 80 Å. When present,

Figure 6 :

 6 Figure 6: Molecular modeling of DYS R16-21. The 3D structure of DYS R16-21 was created in I-TASSER. The hydrophobic potential at the molecular surface (top row) is colored by increasing hydrophobicity from green (polar) to yellow (hydrophobic). The electrostatic potential at the molecular surface (middle row) is colored by increasing levels from red (-) to blue (+). The N-terminus (R16) is located at the left of the molecules. Left and right images in the top rows are representations of a 180°-rotation along the molecule's long axis. Bottom row: 3D structure of DYS R16-21 in cartoon representation. Each dystrophin repeat is a different grey. Amino acids of the CRAC (cholesterol recognition/interaction amino acid consensus) patterns are red.

AUTHOR INFORMATION


Romain Guillaume

Caroline Thierry

Integration of ill-known requirements in production planning

come   L'archive ouverte pluridisciplinaire


M Rebouah

G Chagnon

Extension of classical viscoelastic models in large deformation to anisotropy and stress softening

Keywords: Viscoelasticity, Constitutive equation, Sphere unit model, Anisotropy, Stress-softening

In this paper, an extension of two classical viscoelastic models adapted in large deformation for incompressible rubber like materials or soft tissues is proposed. These models are built by using a three dimensional homogeneization by means of a sphere unit approach. Thus several comparisons between classical formulations and homogeneization on a sphere unit formulation is proposed. An adaptation of those models to describe anisotropy is proposed. Finally an extension of those models to take into account stress softening is described.

Introduction

Rubber like materials are remarkable materials, by means of their capacity to endure large deformations and cyclic loadings. Despite many publications, the prediction of their mechanical behavior is still an open issue. Since last decades, it is well known that soft tissues present a similar behavior to rubber like materials. The main difference is that most of the soft tissues present also an initial anisotropy. Both materials present many non linear phenomena as Mullins Effect [START_REF] Mullins | Softening of rubber by deformation[END_REF]; [START_REF] Beatty | A theory of stress-softening in incompressible isotropic materials[END_REF]; Diani et al. (2006);[START_REF] Vande Geest | The effects of aneurysm on the biaxial mechanical behavior of human abdominal aorta[END_REF]; [START_REF] Natali | Biomechanical behaviour of oesophageal tissues: Material and structural configuration, experimental data and constitutive analysis[END_REF]), permanent set [START_REF] Dorfmann | A constitutive model for the Mullins effect with permanent set in particulereinforced rubber[END_REF]; [START_REF] Peña | Mechanical characterization of the softening behavior of human vaginal tissue[END_REF]), hysteresis [START_REF] Lion | A constitutive model for carbon black filled rubber: Experimental investigations and mathematical representation[END_REF]; [START_REF] Peña | On the Mullins effect and hysteresis of fibered biological materials: A comparison between continuous and discontinuous damage models[END_REF]) and time dependent behavior [START_REF] Christensen | Theory of Viscoelasticity: An Introduction[END_REF]; [START_REF] Wineman | Mechanical response of polymers. An introduction[END_REF]; [START_REF] Holzapfel | A new viscoelastic constitutive model for continuous media at finite thermomechanical changes[END_REF]). Mullins effect and permanent set are classically taken into account for quasi static behaviors, where the influence of the strain rate is neglected. At the opposite, the time dependency is treated independently without taking into account the stress softening in many studies. Nevertheless for the most of rubber like materials the modeling depends mainly on the strain rate, and the response varies from purely rubber like material to glassy polymer [START_REF] Yi | Large deformation rate-dependent stress-strain behavior of polyurea and polyurethanes[END_REF]; [START_REF] Sarva | Stress strain behavior of a polyurea and a polyurethane from low to high strain rates[END_REF]). Soft tissues present also a strain rate dependency. Models were developed to take into account the viscoelasticity of soft tissues and the initial anisotropy due to their structure, see for example [START_REF] Fung | Biomechanics, Mechanical properties of living tissues[END_REF]; [START_REF] Miller | Constitutive modelling of brain tissue: Experiment and theory[END_REF]; [START_REF] Pioletti | Non-linear viscoelastic laws for soft biological tissues[END_REF]; [START_REF] Miller | Constitutive modeling of abdominal organs[END_REF]; [START_REF] Limbert | Three-dimensional finite element modelling of the human ACL: simulation of passive knee flexion with a stressed and stress-free ACL[END_REF]; [START_REF] Taylor | On modelling of anisotropic viscoelasticity for soft tissue simulation: Numerical solution and GPU execution[END_REF]).

The viscoelasticity in large deformations is classically described according to two different theories. The first theory is the one developed by [START_REF] Green | The mechanics of non-linear materials with memory: Part I[END_REF] and [START_REF] Coleman | Foundations of linear viscoelasticity[END_REF] by using the principle of Boltzmann superposition. Indeed [START_REF] Green | The mechanics of non-linear materials with memory: Part I[END_REF] introduced and supposed that relaxation functions depend only on the time and [START_REF] Coleman | Foundations of linear viscoelasticity[END_REF] considered that these functions depend on time and on the actual state of deformation. These models are expressed in term of convolution integrals and were readapted by [START_REF] Berstein | A study of stress relaxation with finite strain[END_REF] and [START_REF] Zapas | Correlation of large longitudinal deformations with different strain histories[END_REF] to finally be known as K-BKZ model for fluids. Later, [START_REF] Ogden | Large deformation isotropic elasticity -on the correlation of theory and experiment for incompressible rubber like solids[END_REF] and [START_REF] Christensen | A nonlinear theory of viscoelasticity for application to elastomers[END_REF] developed the extensions of [START_REF] Green | The mechanics of non-linear materials with memory: Part I[END_REF] at the first order for the solids. All these models are not numerically implemented except for fluid mechanics, i.e. the K-BKZ model. The second theory was developed by [START_REF] Green | A new approach for the theory of relaxing polymeric media[END_REF] by means of differential models with internal variables. The viscoelasticity is classically build up for uniaxial formulations with springs and dashpots and generalized to tensorial formulation. The two standard models used are those of Zenner and Kelvin-Voigt. The readaptation of the differential models, initially proposed by [START_REF] Green | A new approach for the theory of relaxing polymeric media[END_REF], was stimulated 40 years later by the work of [START_REF] Lubliner | A model of rubber viscoelasticity[END_REF]. This model is based on the decomposition of the deformation gradient into an elastic part and an inelastic one. Later several authors [START_REF] Simo | On a fully three-dimensional finite-strain viscoelastic damage model: formulation and computational aspects[END_REF][START_REF] Govindjee | Transition from micro-mechanics to computationally efficient phenomenology:carbon black filled rubbers incorporating Mullins' effect[END_REF][START_REF] Holzapfel | A new viscoelastic constitutive model for continuous media at finite thermomechanical changes[END_REF][START_REF] Lion | A constitutive model for carbon black filled rubber: Experimental investigations and mathematical representation[END_REF][START_REF] Kaliske | Constitutive approach to rate independent properties of filled elastomers[END_REF] proposed to separate the response of the material as the sum of an elastic and an inelastic stress, thus the internal variable is of "stress" type. [START_REF] Simo | On a fully three-dimensional finite-strain viscoelastic damage model: formulation and computational aspects[END_REF] model is equivalent to the K-BKZ model as proved by [START_REF] Reese | Material instabilities of an incompressible elastic cube under triaxial tension[END_REF], and was implemented in a finite element code. More recently, [START_REF] Bergstrom | Constitutive modeling of the large strain time dependant behavior of elastomers[END_REF] developed also a non linear evolution law for a stress internal variable, this model is today one of the most used. In the same way, [START_REF] Miehe | A micro-macro approach to rubber-like materials. Part II: The micro-sphere model of finite rubber viscoelasticity[END_REF] were the first to proposed a micro-spherical approach to take into account the time dependent viscous effects and the stress softening for initially isotropic materials. The model proposed by Diani et al. (2006) presents also a micro sphere unit representation to take into account the viscoelasticity and the stress softening includind the induced anisotropy by the Mullins effect. Other micro mechanics based model were developed in the last decades to take into accont non linear effects and proposed differents way for three dimensional homogeneization (Govindjee andSimo, 1991, 1992b;[START_REF] Rajagopal | A constitutive equation for nonlinear solids which undergo deformation inducedby microstructural changes[END_REF][START_REF] De Tommasi | A micromechanics-based model for the Mullins effect[END_REF], 2010) The objective of this paper is to compare a classical and a three dimensionnal homogeneization by a sphere unit formulation of two models recently studied by [START_REF] Petiteau | Large strain ratedependent response of elastomers at different strain rates: convolution integral vs. internal variable formulations[END_REF]. Thus, a Convolution Integral Model (CIM) and a Internal Variable Model (IVM) are used in this paper. The IVM used is the one proposed by [START_REF] Huber | Finite deformation viscoelasticity laws[END_REF], and CIM is an adaptation of [START_REF] Berstein | A study of stress relaxation with finite strain[END_REF] proposed by [START_REF] Petiteau | Large strain ratedependent response of elastomers at different strain rates: convolution integral vs. internal variable formulations[END_REF]. First, a presentation of the both classical models is made in the section 2. Then, the sphere unit formulation of these models is developed in section 3. A comparison between classical and sphere unit formulation is proposed in section 4 for different loadings. Then by means of the sphere unit formulation some extensions to turn the modeling anisotropic are proposed. The influence of several parameters are highlighted for different loadings states and for both models in section 5. Finally, stress softening is added in the viscoelastic spherical approach.

Presentation of the models

The two viscoelastic isotropic models used in this paper are those used by [START_REF] Petiteau | Large strain ratedependent response of elastomers at different strain rates: convolution integral vs. internal variable formulations[END_REF]. The first one is an IVM deduced from the theory of [START_REF] Green | The mechanics of non-linear materials with memory: Part I[END_REF] and the second is a CIM deduced from the theory of [START_REF] Green | A new approach for the theory of relaxing polymeric media[END_REF]. These two models are quickly recalled in this section, in the incompressible framework.

Internal Variable Model

The simplest IVM for viscoelasticity was derived by [START_REF] Huber | Finite deformation viscoelasticity laws[END_REF] and is represented by a Zenner model. This model is based on the decomposition of the deformation gradient F between an elastic part F e , which represents the deformation gradient of a spring and an inelastic part F i , which represents the deformation gradient of a dashpot as illustrated in Fig. 1. Thus, the deformation gradient F can be expressed as

F = F e F i
(1) Classically it is considered, that there is no rotation in F i and thus the rotation is only in the elastic part F e . This permits to keep unique the elastic-inelastic decomposition. Furthermore it is assumed that the material is incompressible, thus each deformation is defined as det F e = 1 and det F i = 1. The strain enery density associated to the rheological representation is defined by:

C 0 C 1 C 2 F i Fe F (
φ(F e , F i ) = φ 1 (F e ) + φ 2 (F i ) (2)
where φ 1 represents the strain energy density between the initial C 0 and the intermediary C 1 configuration and φ 2 the strain energy density between the intermediary C 1 and final C 2 configuration. According to the Second Principle of Thermodynamics

D int = σ : D -φ ≥ 0 (3)
where D int is the internal dissipation and D is the rate of deformation tensor. After some algebric manipulations described in [START_REF] Huber | Finite deformation viscoelasticity laws[END_REF] and [START_REF] Petiteau | Large strain ratedependent response of elastomers at different strain rates: convolution integral vs. internal variable formulations[END_REF], the second principle can be rewritten as

σ -2 ∂φ 1 ∂B B -2 ∂φ 2 ∂B e B e : D + 2F T ∂φ 2 ∂B e F e : D i ≥ 0 (4)
where D i is the inelastic strain rate tensor defined as D i = 1/2(L i + L T i ) and L i = Ḟi F -1 i the velocity gradient. B is the left Cauchy Green tensor (defined as B = FF T ) associated to the global deformation and B e is the left Cauchy Green tensor (defined as B e = F e F e T ) associated to the elastic deformation. According to the procedure described by [START_REF] Coleman | The thermodynamics of elastic materials with heat conduction and viscosity Arch[END_REF] for deformation processes, the incompressibility constraint and to [START_REF] Petiteau | Large strain ratedependent response of elastomers at different strain rates: convolution integral vs. internal variable formulations[END_REF] who defined a Neo Hookean strain energy [START_REF] Treloar | The elasticity of a network of long chain molecules (I and II)[END_REF] for each spring of the rheological model, it is deduced from Eq.( 4) that the Cauchy stress can be described as:

σ = -pI + 2C 1 B + 2C 2 B e (5)
and the dissipation

2F T ∂φ 2 ∂B e F e : D i ≥ 0 (6)
where C 1 the material parameter of the Neo Hookean model associated to the global deformation and C 2 is the material parameter of the Neo Hookean model of the elastic part. It is quickly recalled that a Neo Hookean model is defined as W = C 1 (I 1 -3), where I 1 is the first strain invariant.

It is deduced as in [START_REF] Petiteau | Large strain ratedependent response of elastomers at different strain rates: convolution integral vs. internal variable formulations[END_REF] that for uniaxial extension in the direction

→

x the first component of the stress tensor is:

σ = 2C 1 λ 2 - 1 λ + 2C 2 λ e 2 - 1 λ e ( 7
)
where σ is the stress for a tensile test along the direction →

x and λ and λ e the global and elastic stretches along the same direction. To determine the deformation of elastic and inelastic parts, i.e. the left Cauchy Green tensor of the elastic part, it is necessary to establish an evolution equation of B e , which is deduced as the simplest sufficient condition to satisfy Eq.(6). As described by [START_REF] Huber | Finite deformation viscoelasticity laws[END_REF] and [START_REF] Petiteau | Large strain ratedependent response of elastomers at different strain rates: convolution integral vs. internal variable formulations[END_REF], the evolution equation is

Ḃe = LB e + B e L T -4 C 2 η B e 2 - 1 3 I 1Be B e ( 8
)
where η is a material parameter. Particularly, in uniaxial extension the expression can be written by means of the stretch of the elastic part:

λe = λ e λ λ -4 C 2 3η (λ 3 e -1) (9)

Convolution Integral Model

The CIM model is an adaptation of a K-BKZ formulation [START_REF] Berstein | A study of stress relaxation with finite strain[END_REF]. By using the same model as [START_REF] Petiteau | Large strain ratedependent response of elastomers at different strain rates: convolution integral vs. internal variable formulations[END_REF] the stress endured by an incompressible material can be expressed as follow:

σ = -p(t)I + 2g 0 B(t) -2F(t) t -∞ g 1 exp - t -τ τ r ∂C -1 (τ ) ∂τ dτ F T (t) ( 10
)
where g 0 , g 1 are material parameters that represent the stiffness of the material and are expressed in MPa and τ r is a material parameter that represents a relaxation time in seconds. Adapted to uniaxial extension in direction →

x , the model can be defined as:

σ = 2g 0 λ 2 (t) - 1 λ(t) + t -∞ 2g 1 exp - t -τ τ r 1 λ(τ ) ∂λ(t) ∂τ 2 λ(t) 2 λ(τ ) 2 + λ(t) λ(τ ) dτ (11)
3. Homogeneization on the sphere unit

General definition

The formulations previously presented (Eq.( 5) and Eq.( 10)) are isotropic. To turn it into a possible anisotropic formulation it is proposed to adapt the model with a sphere unit formulation, i.e. a spatial repartition of n directions, where each direction is defined by an initial vector A (i) transformed in a vector a (i) after deformation by a (i) = FA (i) , where i is the number of the direction. It is often considered that each direction corresponds to a macromolecular chain and thus each chain can endure only tension-compression. To take into account the behavior of a material according to different directions, the fourth invariant

I (i)
4 is used. It is to note that the elongation and the fourth invariant are equivalent according to the definition:

I 4 (i) = A (i) .CA (i) = λ (i) 2 (12)
Where C = F T F is the right Cauchy Green strain tensor. The following sections describe how to turn the models described previously into spherical models.

Application of the sphere unit to the two viscoelastic models 3.2.1. IVM

For IVM formulations, the location of the rotation in the decomposition between the elastic and the inelastic parts (cf. Eq.( 5)) is still an open issue. Indeed it is often estimated that the rotation endured by the material occurs into the elastic part and not into the inelastic one. By means of this spherical formulation, the problem does not occur since each direction is only submitted to tension, that means that there is no hypothesis to formulate about the rotation in this approach. It is proposed to turn the IVM into a sphere unit formulation, thus each expression described previously has to be adapted. First, the evolution equation of the IVM is just adapted from Eq.( 8) to be defined along n directions, it writes for each direction i:

Ḃ(i) e = L (i) B e (i) + B e (i) L T (i) -4 C 2 η B e (i) 2 - 1 3 I 1Be B e (i) (13)
thus for uniaxial extension:

λ(i) e = λ e (i)
λ(i)

λ (i) -4 C 2 3η λ (i) e 3 -1 (14)
by means of this expression one can obtain the value of the stress in each direction:

σ (i) = 2C 1 λ (i) 2 - 1 λ (i) + 2C 2 λ (i) e 2 - 1 λ (i) e (15)

CIM

To adapt the formulation of the CIM, the previous equation Eq.( 11) must also be defined in n directions. Thus for the CIM, and due to the hypothesis of uniaxial extension the n directions can be described only by λ (i) (t). The stress for each direction can be deduced from the Eq.( 11):

σ (i) = 2g 0 λ (i) 2 (t) - 1 λ (i) (t) + t -∞ 2g 1 exp - t -τ τ r 1 λ (i) (τ ) ∂λ (i) (t) ∂τ 2 λ (i) 2 (t)
λ (i) 2 (τ ) + λ (i) (t) λ (i) (τ ) dτ (16)

Definition of the global stress

In this paper, it is proposed to use the spatial repartition of 42 directions proposed by [START_REF] Bazant | Efficient numerical integration on the surface of a sphere[END_REF]. This discretization is homogeneously spread in space and allow to describe an isotropic behavior. The choice of 42 directions is made because they well represent the space and limit also the anisotropy due to the discretization as explained by [START_REF] Gillibert | Anisotropy of direction-based constitutive models for rubber-like materials[END_REF]. To obtain the total stress endured by the material, it is necessary to sum the influence of each direction by applying a weight ω (i) . Thus, the stress induced by the material for both models is defined as:

σ = 42 i ω (i) σ (i) (a (i) n ⊗ a (i) n ) ( 17
)
where a

(i)
n is the normalized vector for each direction, i.e. a

(i) n = a (i)
a (i) and σ (i) is given by Eq.( 15) or Eq.( 16) according to the chosen model. The spatial repartition of 42 directions used and proposed by [START_REF] Bazant | Efficient numerical integration on the surface of a sphere[END_REF] is represented according to different plans in Fig.

Simulation of the sphere unit models

It is proposed in this section to compare the results for a same loading for classical and spherical formulations and for both models at several strain rates. The Fig. 3 (2012): C 1 = 1 MPa ; C 2 = 9 MPa and η = 1MPa.s. It is observed that the obtained behaviors are different for the two formulations, the results of the spherical formulation provide a lower response than the classical formulation for all the different strain rates studied. But the influence of the strain rate is similar for the two formulations. In the same way it is proposed to highlight the most requests directions for the spherical formulation. A weight stress tensor can be associated to each direction, i.e. w (i) σ (i) (a

(MPa) =0 s -1 =1 s -1 =10 s -1 =100 s -1 = s -1 (b) Cauchy stress (MPa) =0 s -1 =1 s -1 =10 s -1 =100 s -1 = s -1
(i) n ⊗ a (i)
n ), the contribution of the weight stress tensor along the tensile direction is illustrated in Fig. 4 for a strain rate of α = 100 s -1 . It is to note that each direction that belongs to a same circle (cf. Fig. 2) presents the same mechanical behavior. Moreover, it is observed that some directions do not have any influence: directions 2-3-8-9 which belong to the circle 1 as illustrated in Fig. 2. Indeed it is consistent that these directions do not present any influence since the directions of the circle 1 projected in the direction → x (which matches to the tensile direction) is equal to zero. The other directions present an increasing contribution for the circles 2, 3, 4 and for the direction 1 which corresponds to the tensile direction.

It is proposed to realize the same two cycles tensile test up to λ = 4 with the classical and spherical formulations of the CIM. The obtained results are presented in Fig. 5. The parameters used are identical as those used by [START_REF] Petiteau | Large strain ratedependent response of elastomers at different strain rates: convolution integral vs. internal variable formulations[END_REF]: g 0 = 1 MPa, g 1 = 9 MPa and τ r = 1 36 s. As previously, both formulations do not predict the same behavior, but the influence of the strain rate is similar. Indeed the spherical formulation induced a lower stress state than the classical formulation for the same strain state at different strain rates. The stress state for each direction can be observed in Fig. 6 for a strain rate of α = 100 s -1 and for the same mechanical parameter. As previously, the most request direction is the direction 1 since it is the direction that corresponds to the tensile direction → x .

Comparison of the two theories with a sphere unit formulation

As proved for the classical formulation by [START_REF] Petiteau | Large strain ratedependent response of elastomers at different strain rates: convolution integral vs. internal variable formulations[END_REF], it appears that the two models exhibit the same response, even if the strengthening is larger for CIM than for IVM. Furthermore, the hysteresis loop is larger for IVM than for CIM. These differences are more important at high 7 strain rates. With the sphere unit formulation it is observed that both models exhibit also similar responses, still with some differences, as illustrated in Fig. 3(b) and Fig. 5(b). Indeed for all strain rates the two models present a different strengthening (larger for CIM than IVM) and hysteresis (larger for IVM than CIM). The differences are still more important for high strain rates. It is to note that the results of the sphere unit formulation are similar to the classical formulation for both models with a scale factor at each strain rates. This sphere unit formulation provides similar results as those of [START_REF] Miehe | A micro-macro approach to rubber-like materials. Part II: The micro-sphere model of finite rubber viscoelasticity[END_REF]; Diani et al. (2006) for isotropic materials. It is to note that the sphere unit formulation is based on a Neo Hookean constitutive equation but could be represented by any phenomenological constitutive equation instead of micro mechanics model based on Langevin function as proposed by [START_REF] Miehe | A micro-macro approach to rubber-like materials. Part II: The micro-sphere model of finite rubber viscoelasticity[END_REF]; Diani et al. (2006). Anyway, the spherical formulation can be adapted to any strain energy density.

(MPa) =0 s -1 =1 s -1 =10 s -1 =100 s -1 = s -1 (b) Cauchy stress (MPa) =0 s -1 =1 s -1 =10 s -1 =100 s -1 = s -1

Comparison of the classical formulations and the sphere unit formulations for different mechanical loadings

In this part, it is proposed to compare the results obtained with a classical formulation and a spherical formulation for the IVM and the CIM according to different loadings states. For each formulation and model, it is proposed to simulate a two cycles test up to λ = 4 for uniaxial (UT), planar (PT) and equibiaxial (ET) tests. The stress equations for planar and equibiaxial tests are easily deduced from Eq.( 5) and Eq.( 8) for the IVM and from the Eq.( 10) for the CIM. The Fig. 7 illustrates the results obtained for the IVM and the CIM with the classical formulation. It is observed that the three mechanical tests provide approximately the same response, the first load and the beginning of the first unload. For the ET it is observed that the models cannot truly take into account the viscoelasticity since the tests present highly compressive stress during the unloading, these results are not consistent with the experimental observations. It is also expected to obtain a stress state more important for an equibiaxial test compare to uniaxial or planar state [START_REF] Treloar | The elasticity of a network of long chain molecules (I and II)[END_REF][START_REF] Marckmann | Comparison of hyperelastic models for rubber-like materials[END_REF][START_REF] Boyce | Constitutive models of rubber elasticity: a review[END_REF]. This phenomenon can be explained by the unidirectional modeling used for the classical models. The same cyclic tensile tests were performed with the sphere unit formulation for the IVM and CIM as illustrated in Fig. 8. It appears that the equibiaxial tests provide a stress state more important than the uniaxial or planar tensile tests for a same strain state with the same elementary hyperelastic models. Furthermore with the sphere unit formulation, it is observed that there is only a few compressive stress during the unloading. These results are more consistent than those obtained with the classical formulation for the both models. These results highlight that the sphere unit formulation is more adapted to describe an equibiaxial test than the classical formulation for this type of viscoelastic models for classical formulations. The uniaxial and planar tensile tests are also well described by the sphere unit formulation. In conclusion, these simulations prove that simple tensorial form models cannot be easily used to represent biaxial loading, and leads to inconsistent results. At the opposite the use of spherical approach to represent three dimensional loadings is efficient even if the three dimensional homogenization is based on simple tensorial form issue from uniaxial development.

Extension to anisotropic viscoelasticity

By means of the spatial repartition of the directions used previously, an isotropic viscoelastic formulation was written. It is proposed now to use the sphere unit formulation to obtain an anisotropic response by a parametric study of the different directions. That means that each direction is no longer represented by the same mechanical parameter. With the sphere unit formulation, the directions are homogeneously distributed in space, a projection of this distribution was presented in Fig. 2. This homogenous repartition of the directions induces a quasi isotropic behavior Figure 8: Stress-strain curves for uniaxial (UT), planar (PT) and equibiaxial (ET) tension for sphere unit formulation with the IVM and the CIM [START_REF] Gillibert | Anisotropy of direction-based constitutive models for rubber-like materials[END_REF]. Thus, the results obtained for a tensile test along the direction → x or → y are identical.

Variation of the viscoelastic parameter (η and τ r )

Presentation

It is proposed to induce a variation of the viscoelastic parameters η for the IVM and τ r for the CIM. The variation of the viscoelastic terms do not change the expressions of the stress described previously. Except that, it is considered that this parameter depends on the considered directions as illustrated in Fig. 9. To highlight the influence of an anisotropic formulation compared to an isotropic formulation, several numerical simulations are performed:

• First, a cyclic tensile test along the direction → x until a stretch of λ = 4 is performed for a sphere unit formulation where the viscoelastic term is constant in each direction: η(i) = 1 MPa.s and τ r (i) = 1 36 s ∀i, these tests are denoted (dir1isotropic).

• Then the same cyclic test along

→

x is performed for a variation of the viscoelastic term for each direction as described previously and illustrated in Fig. 9: η(i) = η 0 cos(θ i ) and τ r (i) = τ r0 cos(θ i ) where η 0 = 1 MPa.s and τ r0 = 1 36 s, these tests are denoted (dir1anisotropic).

• Finally the same cyclic tensile test, with the variation of the viscoelastic parameters (η(i) = η 0 cos(θ i ) and τ r (i) = τ r0 cos(θ i )), is performed along the direction → y , these tests are denoted (dir2anisotropic).

These simulations are performed for both models (IVM and CIM) and the same sets of mechanical parameters defined previously are used (cf. paragraph 3.2.4). All the results are simulated for a strain rate α = 100 s -1 .

Analysis

The results of the tests described previously for the IVM and the CIM are presented in Fig. 10. It is observed that the results are similar for the CIM (variation of τ r ), and for the IVM (variation of η). Indeed the stress strain curves obtained for (dir1isotropic) and (dir1anisotropic) are very similar, the predominant direction in the spherical approach is the closest one from the loading direction. As the viscoelastic parameters were not changed in direction 1 the mechanical behavior is not so modified. At the opposite, the stress strain curve (dir2anisotropic) is very lower than the stress strain curves corresponding to a tensile test along the direction →

x . Thus, it proves that with the spherical formulation the two models can easily represent a multidirectional anisotropic viscoelastic behavior. 1 andC 2 ) 5.2

Variation of the hyperelastic parameter of the inelastic part (g

.1. Presentation

It is proposed for this second study to vary a second material parameter. The C 2 (IVM) or g 1 (CIM) parameters, are not the same in all directions. Thus, the variation of the parameters is expressed in the same way as previously, that means that C 2 and g 1 are now depending on the orientation of the directions in space as illustrated in Fig. 11. As previously several tests are performed:

• First a tensile test along

→

x until a stretch of λ = 4 for a constant value of the parameters in each direction: C 2 (i) = 9 MPa and g 1 (i) = 9 MPa ∀i, these tests are denoted (dir1isotropic).

• Then the same tensile test in the direction

→

x with varying values of the parameters: C 2 = C 2-0 cos(θ i ) and g 1 = g 1-0 cos(θ i ) where C 2-0 = 9 MPa and g 1-0 = 9 MPa, these tests are denoted (dir1anisotropic). 13

• Finally, still the same tensile test along the direction → y with also varying values (dir2anisotropic) is performed.

Analysis

The results can be observed in Fig. 12, where g 1 and C 2 are varying and the other mechanical parameters are constant, the simulations are presented for a strain rate α = 100 s -1 . As previously it is observed that for a tensile test along the direction → x the results are quite similar for the both models, even if the anisotropic spherical formulation (dir1anisotropic) provides lower results as the isotropic spherical formulation (dir1isotropic). The tensile tests along the direction → y for the anisotropic spherical formulation (dir2anisotropic) present results lower than the two other stress strain curves for both models. Thus, by varying these other mechanical parameters it is also possible to obtain an anisotropic behavior for both models, and the behavior of the chain oriented in the loading direction is still predominant.

Conclusions

According to this study, it was observed that, by means of a sphere unit formulation, it is easy to obtain an anisotropic behavior. The anisotropy of a material was induced by means of the mechanical parameters. All the parameters do not induce the same anisotropic response. Thus, it is possible to turn a material anisotropic as wished. The results presented in this paragraph where performed at α = 100s -1 , but the conclusions are identical for other strain rates. Compared to others pre-existent models about viscoelasticity, it is to note that there is only a few proposed models in literature [START_REF] Miehe | A micro-macro approach to rubber-like materials. Part II: The micro-sphere model of finite rubber viscoelasticity[END_REF]Diani et al., 2006) to take into account a multidirectional anisotropic viscoelastic behavior. [START_REF] Miehe | A micro-macro approach to rubber-like materials. Part II: The micro-sphere model of finite rubber viscoelasticity[END_REF] proposed an adaptation of its initial model to take into account anisotropic behavior and Diani et al. (2006) proposed a model to take into account the induced anisotropy due to the stress softening. Most of the anisotropic models are developed for soft tissues but they only present two different directions, due mainly to the fiber reinforcement in the material [START_REF] Peña | An anisotropic visco-hyperelastic model for ligaments at finite strains. Formulation and computational aspects[END_REF]; [START_REF] Taylor | On modelling of anisotropic viscoelasticity for soft tissue simulation: Numerical solution and GPU execution[END_REF]. The model presented here allows to take into account the initial anisotropy of a material by the simple way of a sphere unit formulation and can thus represented a multidirectional anisotropy by means of a simple phenomenological model.

Viscoelasticity and Mullins effect

As explained previously the viscoelasticity is one of the non linear phenomenon associated to rubberlike materials and soft tissues. Nevertheless other phenomena as Mullins effect or permanent set are also current for this materials. It is proposed in this part to add the phenomenological stress softening model developed by [START_REF] Rebouah | Anisotropic Mullins stress softening of a deformed silicone holey plate[END_REF] at the CIM and IVM to take into account the Mullins effect in the previous sphere unit viscoelastic models. It is to note that any other constitutive equation for describing the Mullins effect can also be used, especially the micromechanics-based model as De [START_REF] De Tommasi | A micromechanics-based model for the Mullins effect[END_REF]; Diani et al. (2006); [START_REF] Miehe | A micro-macro approach to rubber-like materials. Part II: The micro-sphere model of finite rubber viscoelasticity[END_REF]. The main equations are recalled here where a repartition of n initial spatial directions A (i) is also used. The general form of the strain energy is:

W = W cc (I 1 ) + n i=1 ω (i) F (i) W (i) cf (I (i) 4 ). ( 18
)
where W cc represents the hyperelastic Neo Hookean model already used in the viscoelastic model, and F (i) is the Mullins effect evolution function similar as the one proposed by [START_REF] Zuñiga | Constitutive equations for amended non-gaussian network models of rubber elasticity[END_REF] and

W (i)
cf is the strain energy associated with each direction. The energy proposed by [START_REF] Kaliske | A formulation of elasticity and viscoelasticity for fibre reinforced material at small and finite strains[END_REF] is chosen:

W (i) cf = K 4 (I (i) 4 -1) 2 (19)
and the evolution function is:

F (i) = 1 -η m I 1max -I 1 I 1max -3 I (i) 4 max -I (i) 4 I (i) 4 max -1 I (i) 4 max I 4 max 4 (20)
where η m is a mechanical parameter that represents the intensity of the stress softening between the first and the second loadings, and I 1max , I 4 (i) max and I 4max are respectively the maximum values of I 1 , I 4 (i) for the direction i and I 4 (i) for all directions.

IVM

The Mullins effect model was added to the viscoelastic part and thus the rheological scheme is modified as illustrated in Fig. 13. The final expression of the IVM and Mullins effect model is expressed as:

σ = 42 i ω (i) 2C 1 λ (i) 2 - 1 λ (i) + 2C 2 λ (i) e 2 - 1 λ (i) e + 2λ (i) 2 F (i) ∂W cf ∂I (i) 4 (i) a (i) n ⊗ a (i) n (21)
It is proposed to simulate a tensile test defined by 5 cycles until λ = 2, then 5 cycles until λ = 3 and finally 5 cycles until λ = 4. A strain rate of α = 15 s -1 is imposed and the chosen parameter are: g 0 = 1 MPa, g 1 = 9 MPa, η=1 MPa.s, K=0.5 MPa and η m =4.5. The results are illustrated in (Mullins) are represented independently and finally the sum of these two behaviors is represented (total). It is observed that for each cycle the Mullins effect is well taken into account and can be very well treated independently of the viscous effects. The global behavior is finally composed of hyperelasticity, viscoelasticity and stress softening.

CIM

The Mullins effect part is added to the viscoelastic part of the CIM in the same way as described for the IVM. The evolution function, and the strain energy are the same. Thus, the following expression of the stress is obtained:

σ = 42 i ω (i) [2g 0 (λ (i) 2 (t) - 1 λ (i) (t) )] + t -∞ 2g 1 exp - t -τ τ r 1 λ (i) (τ ) ∂λ (i) (t)
∂τ 2 λ (i) 2 (t)

λ (i) 2 (τ ) + λ (i) (t) λ (i) (τ ) dτ +2λ (i) 2 F (i) ∂W cf ∂I (i) 4 (i) a (i) n ⊗ a (i) n (22)
It is also proposed to simulate the same test as the one described in paragraph 6.1. The results are obtained for a strain rate of α = 15 s -1 and the material parameters are: g 0 = 1 MPa, g 1 = 9 MPa, τ r = 1 36 s, K=0.5 MPa and η m =4.5. The results are illustrated in Fig. 15. As for the IVM it is observed that for cycles until a same strain state that the Mullins effect is well described and can be also very well treated independently as the viscous effects.

Conclusion

This paper has presented first an extension of [START_REF] Petiteau | Large strain ratedependent response of elastomers at different strain rates: convolution integral vs. internal variable formulations[END_REF] study by proposing an adaptation of classical viscoelastic model to spherical approaches to realize a three directional homogenization in an incompressible framework. This formulation can be adapted to rubber like materials and soft tissues. Two classical models of viscoelasticity (IVM and CIM) were written by means of a spherical formulation and compared to a classical formulation. It appears that, if a homogeneous repartition of directions in space is used (isotropic behavior), the sphere unit model allows to describe better different loadings state (uniaxial, planar and equibiaxial tension) than a classical formulation by correctly describing the equibiaxial extension. This permits to use these simple constitutive equations in a tridimensional framework and to be able to simulate any loading state even with simple constitutive equations. Then, the sphere unit formulation was used to induce an initial anisotropy by imposing different mechanical parameters in each direction. It was observed that an anisotropic behavior could be easily created in this way, simply by a parameter variation along the directions. Finally, a stress softening model was added to the spherical formulation to also take into account the Mullins effect.

Figure 1 :

 1 Figure 1: Representation of the different configurations by using a strain type internal variable (a) and the rheological representation of the components (b)

  2. The Fig.2(a) represents the directions in the plan ( → y , → z ). It appears that each direction belong to a circle. A representation of the different circles is illustrated in the plan ( → x , → z ) in Fig.2(b).

Figure 2 :

 2 Figure 2: Representation of the 42 direction of Bazant and Oh (1986) (a) in the plan (

  (a) highlights the results obtained for two cycles tensile tests up to λ = 4 along the direction → x for the IVM classical formulation (as presented in Petiteau et al. (2012)) and Fig.3(b) the results with the spherical formulation for the IVM. For both simulations the parameters used are identical as those used by Petiteau et al.

Figure 3 :

 3 Figure 3: Stress strain curve IVM for an isotropic formulation (a) and a sphere unit formulation with 42 directions (b)

Figure 4 :

 4 Figure 4: Stress strain curves for IVM for the 42 directions of the sphere unit model

Figure 5 :Figure 6 :

 56 Figure 5: Stress strain curve of the CIM model for an isotropic formulation (a) and a sphere unit formulation with 42 directions (b)

Figure 7 :

 7 Figure 7: Stress-strain curves for uniaxial (UT), planar (PT) and equibiaxial (ET) tension for isotropic formulation with the IVM and the CIM

Figure 9 :Figure 10 :

 910 Figure 9: Representation of the spatial distribution of Bazant and Oh directions in the plan (

Figure 11 :

 11 Figure 11: Representation of the spatial distribution of Bazant and Oh directions in the plan (

Figure 12 :

 12 Figure12: Variation of g1 and C2 for the both models IVM and CIM, the other parameters are still the same (where η and τ are constant) for a cyclic uniaxial tensile test.

Figure 13 :

 13 Figure 13: Representation of the viscoelastic and Mullins effect modeling

Fig. 14 .

 14 Fig.14. The behavior of the viscoelastic part (visco) and stress softening(Mullins) are represented independently and finally the sum of these two behaviors is represented (total). It is observed that for each cycle the Mullins effect is well taken into account and can be very well treated independently of the viscous effects. The global behavior is finally composed of hyperelasticity, viscoelasticity and stress softening.

Figure 14 :Figure 15 :

 1415 Figure 14: Simulation of 15 cycles with the IVM sphere unit model coupled with the Rebouah et al. (2013) stress softening model