Perrine Freydier

Jérôme Martin

Béatrice Guerrier

Frédéric Doumenc

Pierre-Yves Jeannin

Cohe- Sion

Cohesion of natural cave sediments in the presence of thin liquid films

des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Cohesion of natural cave sediments in the presence of thin liquid films

Perrine Freydier 1 , Jérôme Martin 1 , Béatrice Guerrier 1 , Pierre-Yves Jeannin 2 , Frédéric Doumenc 1, 3 1 Lab. FAST, Université Paris-Sud, CNRS, 2 Swiss Institute for Speleology and Karstology, [START_REF] Clottes | Les cavernes de Niaux, art magdalénien de l'arriège[END_REF] Sorbonne Universités perrine.freydier@u-psud.fr, frederic.doumenc@upmc.fr

Experimental set-up and methods

Conclusion and continuation

Experimental results

Introduction : a multidisciplinary approach

The aim of this work is to study how to trigger a loss of sediment cohesion, that seems needed to form vermiculations, through experimental investigations coupled with field observations.  Vermiculations are small aggregates made of particles (clay, calcite, minerals, organic matter…) present at the surface of cave walls.

 Vermiculations are caused by the destabilization of the cohesive sediment layer.

 Reproduction of vermiculations in laboratory is awaited to help qualifying degradation factors of cave paintings.

 Hypothesis: Interactions of both hydrodynamics, heat & mass transfer and physico-chemical parameters can lead to mobilization, displacement and aggregation of particles forming vermiculations.

 These sediments, composed of fine particles, are yield stress fluids, with thixotropic properties.  Hypothesis: Decrease of the yield stress will lead to an easier mobilization of these sediments by external forces.  Fast decrease of the yield stress (τ c ) during the first 24h of soaking, followed by a continuous decrease.

IMA9 -2018

 Salted soaking solutions reduces the decrease of the yield stress.

 pCO 2 increases calcium carbonate dissolution and salt concentration in the soaking solution.

 Existence of a generic trend for this natural sediment and for all soaking conditions : τ c increases with the solid fraction (i.e. decreases with water content). 1 cm rough surface

In the humid environment of a cave, sediments on the walls may be subjected to:

-evaporation/condensation cycles.

-water infiltration coming from the surrounding karstic massif and containing salts.

-changes in surroundings pCO 2 (CO 2 partial pressure).

What impact on sediment cohesion ?

Context:

Methods:

1) Collection of natural sediments in cave.

3) Soaking in solutions varying soaking solution composition, soaking time, pCO 2 .

Study of scenarios involving a yield stress decrease of the sediment

Fig 2 :

 2 Fig 2: Degradation of wall paintings in the cave of Niaux, France 2,3 .

Fig 1 :

 1 Fig 1: Example of vermiculations. Up: cave wall, Dordogne, France. Down: ramified vermiculation 1 .

Fig 5 :Fig 8 :

 58 Fig 5: Scheme of the soaking experimental set up.

Fig 7 :Fig 9 :

 79 Fig 7: Yield stress (τ c ) of the sediment as a function of the solid massic fraction (W s ). In color, different soaking time with different soaking solutions composition (pure, CaCO 3 saturated, CaCl 2 concentrated). Black dots represent all other soaking experiments, without distinction of colors or symbols. These latter were performed with soaking time varying from 20 min to 2 months, and various solutions composition (pure, NaCl, KCl, CaCl 2 , MgCl 2, CaCO 3 ).

Fig 6 :paintings degradation 2 )

 62 Fig 6: Yield stress (τ c ) of the sediment as a function of soaking time in different soaking solutions composition.

Fig 4 :

 4 Fig 4: Shear rate as a function of time when applying different stresses at the sediment sample prepared at a solid fraction of 62.8%. Rheometer Anthon Paar MCR 50. In this example the measured yield stress τ c is 1200 Pa.

Acknowledgment : Financial support by French Ministry of culture and communication -DRAC Nouvelle-Aquitaine.

 We are developing an experimental set-up which reproduce the environment in cave including hydrodynamics aspects (flow rate, composition of solution runoff, temperature, evapo-condenstation, pCO 2 , substrate slope) (Fig. 8).

 We are looking for a more complex scenario leading to a drastic decrease of the yield stress in some specific critical environmental conditions (see Fig. 9).


domain: sde chim sdv

text:
Kassandra De Pao Mendonca

Perrine Chaurand

Andrea Campos

; Bernard Angeletti

Mauro Rovezzi

Ludovic Delage

Carole Borchiellini

André Le Bivic

Julien Issartel

Emmanuelle Renard
email: emmanuelle.renard@imbe.fr

Clément Levard
email: levard@cerege.fr

Hyperaccumulation of Vanadium in animals: two sponges compete with urochordates

Keywords: critical metal, bioaccumulation, Porifera, convergent evolution, reduced state, marine, cell types

Vanadium (V) is the second most abundant transition metal in the oceans. Nevertheless, its concentration in organisms is generally low despite its involvement in various physiological or enzymatic functions. Because exposure to high concentrations was shown to be harmful for various organisms including animals, the unexpected very high concentrations found in urochordates and annelids yielded additional studies to decipher their biological meaning.

Here we report cases of V hyperaccumulators in a distinct animal phylum (Porifera): the two homoscleromorph sponge species Oscarella lobularis and Oscarella tuberculata (up to 30 mg/g dw). These high concentrations overpass those reported previously in urochordates and annelids and are not found in the 5 other sponge species studied here.

In both Oscarella species, vanadium is mainly accumulated in the 100 m surface tissues, and in particular in mesohylar cells, as vanadyl (+4) before being partly reduced to V (+3) in the deeper tissues. Genomic surveys failed to find any previously described gene implicated in vanadium metabolism, suggesting that V hyperaccumulation emerged convergently in this lineage. This feature may be of interest for developing bioremediation strategies in marine ecosystems or bioinspired processes to recycle this critical metal.

Synopsis: This study reports extremely high concentrations of vanadium in two sponges. This unusual bioaccumulation of reduced V (+3) in mesohylar cells relies on unknown molecular mechanisms.

Introduction

Vanadium (V atomic number 23) is one of the most abundant elements (21 st ) on Earth and represents the second most abundant transition metal in oceans (Awan et al., 2021;Jeandel et al., 1987;Morford & Emerson, 2018;Schlesinger et al., 2017). It is naturally present in the terrestrial crust, river waters, and atmosphere (Huang et al., 2015;Schlesinger et al., 2017) and can show local enrichments, as for example in volcanic areas (Rehder, 2015). In nature, this redox sensitive element exists in three oxidation state forms: V (+3), V (+4) (i.e., vanadyl), and V (+5) (i.e., vanadate), and in a variety of coordination environments (from octahedral (Oh) to a five-coordinate (Py) or a tetrahedral geometry (Td)), depending on redox conditions and the pH of the surrounding environment (Wehrli & Stumm, 1989). In sea waters, the average V concentration is between 1 and 2 µg/L and Vanadium is often found as the vanadate form H 2 V 5+ O 4 - (Nielsen, 2020). Despite this abundance, Vanadium is considered as one of the 30 critical metals and materials listed by the European Union in 2020 (EU, 2020). V is used in a large variety of applications such as the manufacture of chemical catalysts, high-strength low-alloy steels for the aerospace and aviation industries as well as in nuclear reactors (Yan et al., 2021).

It is also involved in the production of pesticides, dyes employed in textiles and ceramics and in next-generation flow batteries (Moskalyk & Alfantazi, 2003;Treviño et al., 2019).

During the last decade, V has also been extensively studied for medical applications, especially in the formulation of treatments against diabetes and cancer (Treviño et al., 2019). The increasing use of V and of its subsequent release in the environment represent a threat for ecosystems (Moskalyk & Alfantazi, 2003;Schlesinger et al., 2017;Watt et al., 2018). For example, high quantities of V are released into the atmosphere via the combustion of fossil fuels (Schlesinger et al., 2017;Zoller et al., 1973) or from V-derived products stored in landfills (Larsson et al., 2015;Power et al., 2011). Finally, anthropic V is transported into the oceans via atmospheric fallout, rivers, and effluents; this means that all continental and marine ecosystems are concerned (Tulcan et al., 2021).

In contrast with other transition elements (Mo, W, Mn, Fe, Co, Ni, Cu and Zn), V does not seem to be an essential bioelement for most organisms (Gustafsson, 2019). Indeed, Vdependent biological functions or enzymes (vanadium nitrogenases and vanadatedependent haloperoxidases) have so far been reported only in a limited number of organisms (Rehder, 2015). This metal is thus often present in low amounts in most organisms with concentrations generally in the ng/g dry weight (dw) range, rarely up to µg/g dw (Fukushima et al., 2009). Exposure to relatively high concentrations of V (in the mg/L range) was shown to be toxic in most organisms, such as algae, land plants, and freshwater planktonic species (Wang & Liu, 1999, Schiffer & Liber, 2017, Gustafsson, 2019). This toxicity is mainly explained by the high similarity between vanadate and phosphate. Vanadate can thereby substitute for phosphate in enzymes such as phosphatases, kinases, phosphomutases,diesterases, ATPases and ribonucleases, thereby resulting in the inhibition of these enzymes (Huyer et al., 1997;Rehder, 2015;K. H. Thompson et al., 2009).

Considering this toxicity, the accumulation of very high amounts of V in a few species seems paradoxical. Among Metazoa (animals), the most well-known hyperaccumulators of V are urochordates Ascidiacea (sea squirts) and annelids of the Sabellidae family (fan worms) (E. D. Thompson et al., 2018). In fan worms, V concentrations range from 90 to 10,500 g/g dw depending on the considered species (Fattorini et al., 2010;Fattorini & Regoli, 2012;Ishii et al., 1994;Popham & D'Auria, 1982). In ascidians, where V body levels vary among taxa, Phlebobranchia suborder shows higher levels of V than that of the Stolidobranchia suborder. For example, in Ciona edwardsi, Ascidia ahodori, Phallusia mammillata and Ascidia gemmata, V concentrations in "blood" cells (named signet ring cells or vanadocytes) range from 1,500 to 21,000 µg/g dw which represents an enrichment factor from seawater of 10 4 (Michibata et al., 1987(Michibata et al., , 1991)). The proteins involved in the transport, storage, and metabolism of V in this ascidian lineage was thoroughly studied (Kawakami et al., 2009;Ueki & Michibata, 2011) but its biological role is still a matter of debate (Fattorini & Regoli, 2012;Ueki & Michibata, 2011).

The present study reports for the first time hyperaccumulation of V in two sponge species (Porifera) from the Mediterranean Sea. V distribution at the tissue and cell scales was determined and the oxidation state of V was investigated. Finally, we performed a gene survey to decipher whether some molecular mechanisms are shared with urochordates and annelids.

Materials and methods

Sampling

Seven sponge species (Porifera) from 3 different classes were sampled (Fig. 1A) in the Mediterranean Sea by divers: one Calcarea (Clathrina clathrus (Schmidt, 1864)), three Demospongiae (Agelas oroides (Schmidt, 1864), Aplysina cavernicola (Vacelet, 1959) and Axinella damicornis (Esper, 1794)) and three Homoscleromorpha (Corticium candelabrum (Schmidt, 1862), Oscarella tuberculata (Schmidt, 1868) and Oscarella lobularis (Schmidt, 1862)). All species (at least 3 individuals/species) as well as one sample of Ciona edwardsi (Urochordata, for comparison) were collected at the same site, Maïre island (43°12.760'N, 5°19.800'E) in the bay of Marseille (South littoral zone of France) where they can be found in sympatry (in contrast, Hexactinellida are absent at this sampling site). Concerning O. lobularis and O. tuberculata, additional samples were collected at other sites, depending on the distribution of the species: 6 other sites in the bay of Marseille and one site in the bay of Toulon. The Toulon sampling site was selected from a strategic point of view since Toulon is subjected to many industrial, military and port activities (Tessier et al., 2011). In addition, in order to evaluate environmental concentrations, surface sediments were manually collected at each of the eight sites (Supporting Information 1).

Vanadium concentrations in marine sponges and sediments

Samples were transported, cleaned and frozen exactly as described in detail in de Pao Mendonca et al. (2023). Then, 300 mg dw of each sponge or 50 mg dw of sediments were digested by acid treatment and mineralized with a microwave (see de Pao Mendonca et al., 2023, for a detailed protocol). After mineralization, the samples were diluted by a factor of 5-100 before analysis by Inductively Coupled Plasma Mass Spectrometry (ICP-MS Perkin Elmer, NexION® 300X). For each analysis series, technical replicates, procedural blank, and a standard sample (corn powder V463, INRAE) were analysed (Fehlauer et al., 2022) (Supporting Information 1). Results are expressed as µg/g of V dw and represent mean values obtained for each site.

Cellular localization of vanadium in Oscarellidae species

An adult sponge of each of the two sister species O. lobularis and O. tuberculata was collected at the Maïre site and then transported and cleaned as previously described. They were fixed using glutaraldehyde 2.5 % in a mixture of 0.4 M cacodylate buffer and seawater as described in Ereskovsky et al., 2013but without osmium (Ereskovsky et al., 2013). A cryofracture with liquid nitrogen was performed to prevent damaging internal structures. Finally, the samples were observed by Field Emission Gun Scanning Electron Microscopy (SEM, Zeiss GeminiSEM 500) coupled with an Energy Dispersive x-ray (EDAX SDD) detector, at 20 kV. 30 spots per cell type were analysed on the first 200 micrometres of the sponge tissue in order to determine the concentration of V for each cell type. Simulations performed with "Monte Carlo Simulation of electron trajectory in solids" (CASINO) V2.4 software, reveal that during EDX analysis of the marine sponges, the incident electron beam penetrates the area of ~ 1.5 µm depth for V (Supporting Information 2). The size of the electron beam interaction is therefore compatible with the size of the cells targeted in the sample (smaller cells are about 7µm long).

Oxidation state of vanadium measured in-situ in Oscarellidae species

V speciation in O. lobularis and O. tuberculata species sampled at the Maïre site was determined by X-ray absorption Spectroscopy (XAS) at the European Synchrotron Radiation Facility (ESRF, Grenoble, France). For comparison, the V speciation was also determined in the sample of Ciona edwardsi sampled at the same site. Prior to analysis, the organisms were transported and cleaned as previously described and stored at -20° C. For the analysis, samples were cut into 1 mm-thick cross sections in liquid nitrogen using a scalpel and then directly transferred to the XAS sample holder.

High Energy Resolution Fluorescence Detected X-ray Absorption Near Edge Structure (HERFD-XANES) spectra were collected at the V K-edge (5465 eV) on the FAME-UHD beamline (French CRG at the ESRF) (Levard et al., 2025). The incoming X-ray energy was monochromatized using a Si (220) cryogenically cooled and sagittally focusing double crystal monochromator. The monochromator was calibrated using a V metallic foil. The first maximum of the spectrum first derivative of the X-ray absorption spectrum was set at 5465 eV. The spectra were collected in step scanning mode with a beam size of approximatively 0.2 x 0.1 mm 2 . The sample temperature was kept at 10 K using a liquid Helium flow cryostat to prevent sample beam damage. The relatively small beam height (0.1 mm) is adapted to discriminate the average V speciation from surface tissue (100 µm thick) to internal tissue within the sponge cross-section. Samples were analysed in fluorescence mode using a crystal analyser spectrometer equipped with 10 Ge (331) spherically bent crystal analysers of 1 m radius of curvature aligned at the V Kα1 line (4953 eV, Bragg angle of 74.63°). The measured experimental broadening of the spectrometer was 1.6(1) eV. One spectrum represented an average of 4-6 scans for reference compounds, 2-7 scans for sponge tissues and 3 scans for ascidian.

A set of reference compounds in which V occurs in the three oxidations states (+3, +4 or +5) in coordination geometry from octahedral (Oh) to a five-coordinate (Py) or a tetrahedral geometry (Td) was prepared at 4000 ppm of V to avoid self-absorption issues.

Reference compounds were either diluted in boron nitride (BN) and pressed into pellets for solids or dispersed in ultrapure water for certain salts. It includes commercial solids

VCl 3 , V 2 O 4 , VOSO 4 , nH 2 O, vanadyl acetylacetonate C 10 H 14 O 5 V, V 2 O 5 , Na 3 VO 4 purchased
from Aldrich, a solution of Na 3 VO 4 dispersed in ultrapure water and a natural mineral Pascoite Ca 3 (V 10 O 28 ).17H 2 O. Oxidation state and symmetry of V in each of the reference compounds are detailed in Supporting Information 3 and 4. Finally, calibration, normalization, and data averaging were performed using Larch XAS Viewer 0.9.65 software.

Pre-edge analysis was carried out following a procedure inspired by Chaurand et al. (2007) and Levina et al. (2014). They demonstrated that the pre-edge features can be used for the determination of average speciation of V in complex biological or environmental matrices. They reported noticeable changes in both intensity and position of the pre-edge peak according to V oxidation state and coordination number, using a range of reference compounds. The pre-edge features were obtained by fitting a baselinecorrected pre-edge peak with a series of mixed Gaussian-Lorentzian curves (pseudo-Voigt functions) (Supporting Information 4). Care was taken in using the smallest possible number of components. The "pre-edge peak area, PEA" (sum of the integrated areas of each component normalized to V 2 O 5 pre-edge area) and the "pre-edge peak centroid energy, PECE" (area-weighted average of the position in energy of each component) were calculated.

Survey of genes involved in vanadium metabolism

Several genes involved in V transport, storage or metabolism in different metazoan species were searched for by blastP against the predicted proteome of Oscarella lobularis (genome draft (Belahbib et al., 2018)) using the local blast tool of Bioedit (Hall, 1999). The returned hit-blast with e-value < 10 -2 were then checked by a reciprocal best-hit approach against the NR database (NCBI), PDB and Swiss-Prot for domain predictions. The list of query sequences and of best hits obtained are provided in Supporting Information 5. The same strategy was adopted using, as queries, bacterial genes involved in V metabolism on the predicted proteome of Candidatus Rhodobacter lobularis, the dominant microorganism associated with this sponge species (Jourda et al., 2015). The list of query sequences and of best hits obtained are provided in Supporting Information 6. In order to refine the assignment of candidate sequences, phylogenetic analyses were performed.

Alignments (available upon request) were performed with MAFFT (Kuraku et al., 2013) followed by manual JALVIEW curation (Waterhouse et al., 2009). Phylogenetic analyses were performed using PHYML (WAG+G+I SH-LIKE model) (Guindon et al., 2010) and the trees were visualized with iTOL Newick tree (0.8-1 bootstrap cutoff) (Letunic & Bork, 2021).

Statistical analysis

In order to evaluate the statistical significance in V sponge body contents, a nonparametric Kruskal-Wallis test was used (significant p-value < 0.05) for comparisons between sites, and a non-parametric Mann-Whitney U test was performed (p-value <0.05) to compare V concentrations between species. To analyze the significant differences in V quantities between cell types, a Mann-Whitney U test was used with a pvalue <0.05. All statistical analyses and graphics were performed using R studio opensource edition package (https://www.rstudio.com/).

Results and Discussion

Vanadium hyperaccumulation in Porifera is a specificity of Oscarellidae Seven sponge species, members of three of the four classes of the phylum Porifera were sampled at the same site (Maïre island) where they live in sympatry and were analyzed by ICP-MS (Fig 1A; Supporting Information 1). Five of them, the three Demospongiae (A. cavernicola, A. oroides, A. damicornis), the Calcarea C. clathrus and the Homoscleromorpha Plakinidae C. candelabrum exhibit relatively low concentrations of V in their tissues ranging from 2.3 to 7.21µg/g dw (Fig. 1B). Slight variations (not statistically significant) are observed between these five species: C. candelabrum and A. cavernicola harbor the lowest concentrations (2.30 ± 0.43 µg/g and 3.06 ± 1.21 µg/g dw respectively). In contrast, C. clathrus contains the highest vanadium concentrations (7.21 ± 3.65 µg/g dw).

These values are lower than those found in the upper sediments collected at the same site (Maïre: 28.4µg/g dw, Supporting Information 1), suggesting that these species do not bioaccumulate V (i.e., exhibiting a bioconcentration factor (BCF)<1, Supporting Information 7). Unexpectedly, the two remaining species Oscarella tuberculata and O. lobularis, both members of Homoscleromorpha Oscarellidae, contain significantly higher concentrations of Vanadium -2,000 to 15,000 times higher-than any other sponge species studied here (Fig. 1B). These vanadium concentrations are also higher than the concentrations previously reported in 8 other demosponges ranging from 2 to 76.9µg/g dw in the Mediterranean Sea (Cebrian et al., 2003;Perez et al., 2005;Pérez et al., 2004).

The highest V concentrations are found in O. tuberculata with values ranging from 25,198 to 30,274 µg/g dw. In the sister species, O. lobularis concentrations range from 12,411 to 18,425 µg/g dw. To the best of our knowledge, such concentrations of V have never been reported before for marine sponge species. The BCF of these two species is higher than 1 (Supporting Information 7). It is worth noting that the concentrations measured in O. tuberculata exceed those of Ciona edwardsi, sampled at the same site (235 µg/g dw; Supporting Information 1), and those previously reported in urochordate species (Michibata et al., 1987(Michibata et al., , 1991)). The hyperaccumulation of V by the two Oscarellidae species reported here is therefore of great interest.

The hyperaccumulation of vanadium is not specific to the Maïre site populations

In order to test whether the hyperaccumulation of V was a specificity of Oscarellidae individuals growing at the Maïre site, we collected additional specimens of O. tuberculata and O. lobularis at 6 other sites in the bay of Marseille and 1 site near Toulon (Fig. 2A).

Even if statistically significant variations between sites exist, high V concentrations are observed in Oscarellidae whatever the site. For the sites where they are found in sympatry, O. tuberculata always shows higher V concentrations than O. lobularis, with values ranging from 25,198 to 33,479 µg/g dw (Fig. 2B). In addition, the adults collected at the previously mentioned Maïre site are, in fact, the specimens with the lowest concentrations in V with 15,204 ± 2,011 µg/g dw in O. lobularis (p value <0.05) and with 27,009 ± 2,833 µg/g dw in O. tuberculata. In contrast, the samples collected at the Fourmigues site, located near Toulon, show the highest concentrations in O. lobularis (28,212 ± 6,457 µg/g dw) followed by Riou site in the Marseille Bay (19,257 ± 9,508 µg/g dw).

Vanadium concentrations are higher in the mesohylar cells

A sponge's body plan is very different from that of the other V-hyperaccumulative animals belonging to urochordates and annelids. Even though sponges are sessile filter feeders like urochordates, they are devoid of organs, vascular system and blood cells. In sponges, filtration is performed via a completely different system (unrelated to a digestive tube) named the aquiferous system (Borchiellini et al., 2021) (Fig. 3A and Supporting Information 8). For this reason, it is necessary to determine in which part of the tissues this accumulation takes place and which cell types are involved.

In O. lobularis, the energy dispersive X-ray (EDX) mapping analysis shows a concentration gradient from the sponge surface down to the inner tissue (Fig. 3B): indeed, V (and P) concentrations are higher in the 100m surface tissues, in contrast to other elements such as C, S or K for which the distribution seems homogeneous (Supporting Information 9).

Figure 3: Localization of V in the tissues and cells of Oscarellidae species. (A) Backscattered SEM image of O. lobularis tissues (cryo section). ex: exopinacoderm, ec: excurrent canal (bordered by endopinacocytes), cc: choanocyte chambers. (B) V chemical distribution in O. lobularis from EDX analysis mapping (Kα emission line). (C, E) SEM images of the cellular organization in (C) O. lobularis and in (E) O. tuberculata. The only difference between O. lobularis and O. tuberculata is the presence of only one vacuolar cell type (type I vacuolar cell) in the mesohyl. (D, F) Vk quantities (counts) from microanalyses performed with electron beam pointed in the different cell types present in (D) O. lobularis tissues and in (F)

O. tuberculata (N= 30 for each cell type). In the boxplot, a different letter indicates a significative difference (p <0.05).

At higher magnification, the different cells can be clearly identified (Fig. 3C and3E): the outer epithelium called the exopinacoderm (ex) is composed of exopinacocytes (1). The choanocyte chambers (cc) are composed of choanocytes (2). The canals of the filtration system (ec) are lined by endopinacocytes (3). Between these epithelium layers, the mesohyl is a layer composed of vacuolar cells and bacteria (Supporting Information 8).

The separate analysis of chemical elements in the different cell types (4 for O. tuberculata, 5 for O. lobularis) (Fig. 3C and3E) shows that in both species the mesohylar cells (type I vacuolar cells and archeocytes (also named type II vacuolar cells)) have a significantly higher V concentration than exopinacocytes, endopinacocytes or choanocytes. Among mesohylar cells in O. lobularis, type I vacuolar cells display a higher V concentration than archeocytes with 444 ± 78 Vk counts. According to the statistical analysis, both choanocytes and endopinacocytes show significantly the lowest concentrations of vanadium (160 ± 13 Vk counts and 154 ± 16 Vk counts respectively) (Fig. 3D). Regarding O. tuberculata, type I vacuolar cells and choanocytes present the highest concentration of vanadium (386 ± 30 Vk counts and 376 ± 23 Vk counts respectively); while endopinacocytes present a lower vanadium concentration with 334 ± 21 Vk counts (Fig. 3F).

The fact that, in both species, the highest concentrations of vanadium are found in type I vacuolar cells, provides new insights into the largely unknown functions of this cell type, even though a storage/secretion property was already suggested, because of the presence of two to four large vacuoles (Gaino et al., 1986). Interestingly, in the V hyperaccumulator annelid P. ocellata, high levels of vanadium have been detected in the epidermal cells of the gill crowns within the apical vacuoles (Ishii et al., 1994). In ascidians, vanadium is also stored in the vacuoles of vanadocytes (Botte et al., 1979;Kalk, 1963).

Vanadium is mainly reduced to vanadyl when absorbed by the Oscarellidae species

HERFD-XANES spectra obtained for sponges (Oscarellidae species) and ascidian (Ciona edwardsi), collected at the Maïre site are shown in Supporting Information 3. Pre-edge fitting is detailed in the Supporting Information 4 and 10. The plot of pre-edge features comparing model compounds and samples (PEA vs. PECE, Fig 4 .B) reveals that V is incorporated by the Oscarellidae species mainly as vanadyl in both surface and internal tissues (V (+4)). Indeed, pre-edge features (PECE and PEA) are similar to those of reference compounds in which V occurs as V (+4) in octahedral geometry (Oh), especially VOSO 4 , nH 2 O (Supporting Information 10). Vanadyl is stable in moderately reducing environments, particularly when the organic matter concentration is high. It is highly reactive and forms a large number of strong complexes with organic ligands (Gustafsson, 2019).

Although dominated by the vanadyl speciation, differences were observed between surface and internal tissues. HERFD-XANES pre-edge of surface tissues (O. lobularis and O. tuberculata) exhibits a slight shift from V (+4) features toward higher PECE and larger PEA, characteristic of V (+5) (Fig. 4.B). This suggests that a minor fraction of V (+5) can coexist with V (+4) within the sponge surface tissue. In seawater, V is mainly present as the tetrahedral vanadate ion H n V(+5)O 4 (3-n)- (Abbasse et al., 2002;Gustafsson, 2019). The possible presence of small amounts of V (+5) at the sponge surface is probably due to the reduction of vanadate to vanadyl in the surface tissues.

On the other hand, the pre-edge features of inner tissues shift slightly toward those of V (+3) in octahedral geometry. This indicates that the average oxidation state of V within sponges changes as we go from the surface tissue (100 µm thick) to the internal tissue: Predominantly V (+4) at the surface being partly reduced to V (+3) in the inner tissue (Fig. 4.B). The case of Oscarellidae species differs from that of ascidians: In Oscarellidae, reduction to V (+3) in the internal tissues is apparently a minor process. Ascidians, assisted by Na+-dependent phosphate transporters, take up vanadate from seawater, which is then transferred to the cytoplasm and reduced to vanadyl. This reduction is done by vanabin, a vanadium-binding protein expressed in blood plasma in the presence of NADPH, glutathione, and glutathione reductase. After the reduction to vanadyl, V is further reduced to V (+3) by various, as yet unknown, mechanisms.

Finally, the HERFD-XANES spectrum of ascidian Ciona edwardsi exhibits pre-edge features similar to those of the reference compound, incorporating V (+3) in octahedral geometry (Fig. 4). This result is in accordance with the literature. V was shown to be accumulated in ascidians mainly as a V (+3)-SO 4 complex in a number of pioneer papers (Frank et al., 1998(Frank et al., , 2003(Frank et al., , 2008(Frank et al., , 2020;;Michibata, 1996;Ueki & Michibata, 2011). In these papers, V (+3) was identified as the dominating oxidation state in the blood cells, but vanadyl (+4) also contributes.

Gene surveys evidence a convergent emergence of vanadium hyperaccumulation in

Metazoa

A high-quality transcriptome and a draft genome are available for the species O. lobularis (Belahbib et al., 2018;Vernale et al., 2021), enabling us to search (Blast search) for genes previously reported in the literature to play a role in V transport, storage, or metabolism (in particular the genes of urochordates that were characterized in depth). None of the searched for metazoan genes that are specific to V metabolism (e.g., V-binding proteins from urochordates or V-dependant peroxidases from different protostomes, see Fig. 5 and Supporting Information 5) were found in O. lobularis. On the other hand, gene encoding for transketolase, glutathione S-transferase and heavy metal transporting ATPase are found (reported to be involved in the V metabolic pathway of urochordates). This result is not surprising, because these genes are highly conserved across Eukaryotes. Indeed, transketolase enzymes are involved in very important metabolic pathways such as the Calvin cycle, the pentose phosphate pathway, and the oxidative stress response; glutathione S-transferase (GST) and heavy metal transporting ATPase (HMTA) are usually active in the detoxification in many species (Lee, 2008;Takahashi et al., 2012;Ueki & Michibata, 2011;Xu et al., 2016). The absence of homologous vanadium specific genes argues for the independent emergence of V metabolism in the three distant metazoan lineages (Porifera Oscarellidae, Annelida Sabellidae and Urochordata Ascidiacea, Fig. 5). Altogether, with the different oxidation states in which V is stored, it means that the hyperaccumulation of V in Oscarellidae relies on different, yet to be discovered, metabolic pathways from the one involved in urochordates.

Possible involvement of Oscarella's bacterial symbiont and functional significance?

In addition, we searched for bacterial V-dependent enzymes in the predicted proteome of Candidatus Rhodobacter lobularis, the most abundant bacteria associated with O. lobularis tissues (Jourda et al., 2015). According to the Blast search (Supporting Information 6), three sequences seem to be related to the acid phosphatase/V-dependent haloperoxidase superfamily. While two of these candidates were clearly assigned to phosphatase and N-Acetylmuramic Acid alpha-1-Phosphate (MurNAc-alpha1-P) Uridylyltransferase (data not shown), the third sequence shares a best aminoacid conservation with the active site domain of the VHPOs C-terminal region compared to acid phosphatases. Phylogenetic analyses suggest that this sequence pertains to an orthology group of unknown function, distinct from both phosphatases and vanadium haloperoxidases (VHPOs) in the PAP2 superfamily (Hemrika et al., 1997) (Supporting Information 11). Interestingly, molecular analyses have shown a conserved pattern of amino acid residues for the interaction of vanadate or phosphate in the active site of these two enzyme classes (Littlechild et al., 2002), and biochemical experiments have demonstrated that two bacterial acid phosphatases are able to bind vanadate and to exhibit a reduced halide oxidation activity (Tanaka et al., 2002).

Further sequence and biochemical analyses would be required to evaluate the functions of this protein and its possible involvement in vanadium metabolism. Even though this finding may suggest that the V metabolic pathway present in O. lobularis may involve at some point its symbiotic Rhodobacter species, there is so far no evidence that the same or a closely related bacterial species is present in O. tuberculata (Gloeckner et al., 2013).

Whatever the molecular mechanisms involved are, it remains to decipher what the roles are: of this V hyperaccumulation and of a V-dependent metabolism in Oscarella species and their associated holobiont. Several hypotheses can be proposed in the light of roles described in other species: i) V can be used as final electron receptor for respiration by some bacteria (Antipov et al., 2000;Carpentier et al., 2005) in anaerobic conditions. This capacity may be useful if tissue oxygenation is heterogeneous in Oscarellidae, as it was shown in other sponges (Schläppy et al., 2010); ii) In vertebrates, V was shown to promote the antioxidative response by enhancing the expression of antioxidant compounds such as glutathione and GPx (Tripathi et al., 2018). Such implication in the defense against oxidative stress might partly explain such V burden in Oscarella tissues. However, V is also described as a pro-oxidant and the propensity of V to act as an antioxidant or pro-oxidant depends on numerous factors (for example, concentration and oxidation state);

iii) Microbial V-dependent haloperoxidases (VHP0s) were shown to play a major role in the synthesis and diversification of secondary metabolites (Baumgartner & McKinnie, 2021;Carter-Franklin & Butler, 2004;Ishikawa et al., 2022) that are known to be numerous and powerful in sponges (Mehbub et al., 2014); iv) In brown seaweeds, VHPOs were also shown to result in the production of volatile halocarbons (Fournier et al., 2014;Wever & Horst, 2013); but to date the profiling of odoriferous molecules secreted by sponges is only in its infancy, even though most sponge species (including the two considered species) emit a clearly perceptible unpleasant odour, which is even more intense when they are cut (Aresta et al., 2021;Christophersen et al., 1989;De Rosa et al., 2008); v) Antifouling, anti-predation or antimicrobial properties may also come from the toxicity of the V itself (Fattorini et al., 2010;Giangrande et al., 2017;E. D. Thompson et al., 2018;Ueki & Michibata, 2011); vi) Finally, the role of V to produce pigments, as mentioned in urochordates (Bandaranayake, 2006), may be of interest in the case of these colourful sponges (Renard et al., 2021).

Environmental perspectives of our finding

As well as a better understanding of the bio-geochemical cycle of V, our findings open up new perspectives:

• Firstly, in view of the toxicity of V for most organisms, the Oscarella species may provide new clues for developing bioremediation strategies for Mediterranean marine ecosystems contaminated by V, as was already proposed for plant-based remediation strategies for continental ecosystems (Chen et al., 2021).

• Secondly, the hyperaccumulation of V by Oscarella species implies very specific entry and storage routes that are of particular interest for developing selective processes for the extraction of V. Indeed, in the context of climate change, there is a need to shift from traditional pyro-and hydrometallurgy processes to more environmentally friendly extraction methods. In this regard, extraction processes using bioinspired or biosourced chelators for selective recovery of the targeted metal, may offer an interesting alternative for sustainable mining (extraction in water at ambient temperature and pressure, reduced number of steps and less waste products).

Although not yet developed for V, as far as we know, such strategies are currently being studied for the selective recovery of other critical metals, such as Rare Earth Elements (REEs) (Deblonde et al., 2020).

The development of bioinspired protocols requires a good knowledge of metal transfer routes. This knowledge can be obtained using omic approaches to determine molecular processes, including the nature of metabolites and proteins involved in the selective internalization of metals in living organisms. Because of the large number of studies and other resources available for Oscarella tuberculata and O. lobularis species, they would seem to be promising inspirational models.

Figure 1 :

 1 Figure 1: (A) Phylogenetic relationships between the four classes of Porifera and phylogenetic position of the sampled species. Hexactinellida (*) are found only in deep seas or marine caves and therefore cannot be found at the Maïre site. (B) Boxplot of V concentrations in sponge species sampled at the Maïre site (µg/g dw). The color code used for sponge classes is the same as in A. The stars show a significant difference between the considered sponge and O. lobularis (Mann-Whitney U test with p <0.05).

Figure 2 :

 2 Figure 2: (A) Location of the sampling sites (Southern of France; North-Western Mediterranean Sea). According to the geographic distribution of Oscarella lobularis populations, 7 sites in the bay of Marseille (Mejean, Frioul, Prado, Maïre, Perez cave, Caramassaigne, Riou) and 1 site out of the bay (Fourmigues) were sampled. The red dot represents Maïre Island (Fig. 1), (B) Boxplot of V concentrations in O. lobularis and O. tuberculata according to the different sampling sites. In the boxplot, a different letter indicates a significative difference (p <0.05).

Figure 4 :

 4 Figure 4: (A) Corrected pre-edge from HERFD-XANES spectra of Oscarellidae species (surface and inner tissues) and ascidian (Ciona edwardsi) collected at Maïre site (open circles). Solid lines show the fitting of the pre-edge with a series of mixed Gaussian-Lorentzian curves (sum of the 4 components in dashed lines). (B) Pre-edge features (PECE and PEA) extracted from (A). The pre-edge features are compared with those obtained for the reference compounds in which V occurs in the three oxidations states (+3, +4 or +5) in coordination geometry from octahedral (Oh) to a five-coordinate (Py) or a tetrahedral geometry (Td).

Figure 5 :

 5 Figure 5: Hyperaccumulation of V emerged at least 3 times during animal evolution. (A) Phylogenetic position of the three animal lineages to date reported to perform V hyperaccumulation in the metazoan tree: Urochordata Ascidiacea, Annelida Sabellidae and Porifera Oscarellidae are very distant lineages. (B) Summary of the results of gene surveys in the genome draft of O. lobularis. Genes encoding enzymes specific to V metabolism found in urochordates (purple) and spiralians (blue) are not found in O. lobularis.

Acknowledgment

The authors would like to thank the following divers who assisted in collecting sponge and sediment samples: Laurent Vanbostal and Dorian Guillemain of the OSU Pythéas ; Sandrine Chenesseau and Christian Marschal of the IMBE lab ; Sandrine Ruitton of the Institut Méditerranéen d'Océanologie (MIO), and Stephane Sartoretto of the Institut Français de Recherche pour l'Exploitation de la Mer (IFREMER). The authors would also like to thank the European Synchrotron Radiation Facility (ESRF) for providing synchrotron radiation facilities and the LA-ICP-MS platform elemental chemistry at Aix-Marseille. For their work preparing samples and providing efficient lab facilities, the authors want to thank Geneviève Dur (CEREGE), Sandrine Chenesseau (IMBE), Alexander Ereskovsky (IMBE) and Caroline Rocher (IMBE).

Finally, thanks to Thomas Smith, native English speaker, for proofreading the manuscript.

Funding sources

The K. De Pao Mendonca's PhD grant was funded by the CNRS under the 80|Prime program, which is acknowledged by the authors. The Eccorev research federation and the Labex DRIIHM, French program "Investissements d'Avenir" (ANR-1-LABX-0010), which is managed by the French ANR, jointly established this effort.


R Kadri

M Nait Abdelaziz
email: moussa.nait-abdelaziz@univ-lille.fr

B Fayolle

M Ben Hassine

G Ayoub

Y Nziakou

Micromechanical based model for predicting aged rubber fracture properties

Keywords: Aging, Rubber materials, Fracture properties, Micromechanical modeling, Damage Aging, Rubber materials, Fracture properties, Micromechanical modeling, Damage

Environmental aging induces a slow and irreversible alteration of the rubber material's macromolecular network. This alteration is triggered by two mechanisms which act at the microscale: crosslinking and chain scission. While crosslinking induces an early hardening of the material, chain scission leads to the occurrence of dangling chains responsible of the damage at the macromolecular scale. Consequently, the mechanical behavior as well as the fracture properties are affected. In this work, the effect of aging on the mechanical behavior up to fracture of elastomeric materials, and the evolution of their fracture properties are first experimentally investigated. Further, a modeling attempt using an approach based upon a micro-mechanical but physical description of the aging mechanisms is proposed to predict the mechanical and fracture properties evolution of aged elastomeric materials. The proposed micro-mechanical model incorporates the concepts of residual stretch associated with the crosslinking mechanism and a so-called "healthy" elastic active chain (EAC) density associated with chain scission mechanism. The validity of the proposed approach is assessed using a wide set of experimental data either generated by the authors or available in the literature.

Introduction

Rubber materials are used in a wide range of applications, prompted by their particularly high reversible deformations associated with low strength and their great damping and insulating properties. During operation, rubber materials are exposed to extreme mechanical and environmental perturbations, such as complex static and dynamic mechanical loadings, elevated temperatures, UV radiation, oxygen, and humidity. Therefore, understanding and modeling the operating conditions' influence on the mechanical and fracture properties is of high importance when designing structural rubber material components. Many experimental investigations showed the strong dependency of the mechanical and fracture properties of rubber materials on the loading condition history and on the chemical and physical aging exposure history. Those properties are directly influenced by the changes in the macromolecular network structure induced by a chemical processes that operate at the molecular scale [START_REF] Dunn | Changes in the stress-strain properties of natural rubber vulcanizates during ageing[END_REF][START_REF] Colclough | Oxidative aging of natural rubber vulcanizates. Part III. Crosslink scission in monosulfidic networks[END_REF]).

For the last few decades, many research investigations have focused on increasing our understanding of rubber materials' aging mechanisms and their effect on the mechanical properties. Different mechanistic and kinetic schemes of their thermooxidation process were proposed [START_REF] Bolland | Kinetics of olefin oxidation[END_REF][START_REF] Rincon-Rubio | A general solution of the closed-loop kinetic scheme for the thermal oxidation of polypropylene[END_REF][START_REF] Colin | Chemo-mechanical model for predicting the lifetime of epdm rubbers[END_REF][START_REF] Colin | A new analytical model for predicting the thermal oxidation kinetics of composite organic m atrices. Application to diamine crosslinked epoxy[END_REF]. A consensus among those studies describes the alteration of the macromolecular network structure by a three-step oxidation scheme: initiation, propagation, and termination. Initiation involves the creation of free radicals P °.

The free radicals P ° are extremely reactive with oxygen and consequently favor a chain reaction of chemical processes in the polymer, denoted as propagation. The termination consists in the transformation of the different radicals in the polymer into inactive products like gas release or the creation of new crosslinks or chain scissions. These two competing mechanisms (crosslinking and chain scissions) modify the macromolecular structure of the polymer and play a significant role in the degradation of the mechanical properties [START_REF] Fayolle | Review: degradation-induced embrittlement in semi-crystalline polymers having their amorphous phase in rubbery state[END_REF][START_REF] Planes | Evolution of EPDM networks aged by gamma irradiation -Consequences on the mechanical properties[END_REF][START_REF] Belbachir | Modelling of photodegradation effect on elasticviscoplastic behaviour of amorphous polylactic acid films[END_REF][START_REF] Gac | Description and modeling of polyurethane hydrolysis used as thermal insulation in oil offshore conditions[END_REF][START_REF] Howse | Experimental determination of the quantity and distribution of chemical crosslinks in unaged and aged natural rubber. II: A sulfur donor system[END_REF][START_REF] Bouaziz | Elastic Properties of Polychloroprene Rubbers in Tension and Compression during Ageing[END_REF].

The chemical composition, the macromolecular network structure, and the aging conditions of the rubber materials influence the relative predominance of the aforementioned aging mechanisms (Planes et al. 2010a).

With increasing aging time, a decrease of the fracture mechanical properties of rubber material whatever the degradation mechanisms is widely documented [START_REF] Celina | Accelerated aging and lifetime prediction: Review of non-Arrhenius behaviour due to two competing processes[END_REF][START_REF] Planes | Evolution of EPDM networks aged by gamma irradiation -Consequences on the mechanical properties[END_REF]Planes et al. , 2010b;;[START_REF] Gac | Description and modeling of polyurethane hydrolysis used as thermal insulation in oil offshore conditions[END_REF][START_REF] Pourmand | Deterioration of highly filled EPDM rubber by thermal ageing in air: Kinetics and non-destructive monitoring[END_REF][START_REF] Kashi | Mechanical, Thermal, and Morphological Behavior of Silicone Rubber during Accelerated Aging[END_REF][START_REF] Abdelaziz | New developments in fracture of rubbers: Predictive tools and influence of thermal aging[END_REF].

Over the last century, a significant number of models were proposed to capture the macro-scale hyperelastic response of rubber materials. Based on their theoretical frameworks, we can differentiate between phenomenological and physical-based models. The phenomenological models provide a macroscopic description of the hyperelastic stress-strain behavior of rubber materials that is not connected to the macromolecular structure [START_REF] Mooney | A theory of large elastic deformation[END_REF][START_REF] Yeoh | Characterization of Elastic Properties of Carbon-Black-Filled Rubber Vulcanizates[END_REF]). In physical-based models, we can distinguish between models describing the macromolecular network physics using Gaussian and non-Gaussian statistical approaches. The chains mechanics associated with the macromolecular network entropy decrease was described using a number of non-Gaussian-based theories . These non-Gaussian statistical theories account for the extensibility limit of randomly oriented molecular chains to capture the hyperelastic behavior of rubber materials using the probability distributions of [START_REF] Jernigan | Distribution functions for chain molecules[END_REF] and [START_REF] Kuhn | Relationships between elastic constants and stretching double refraction of highly elastic substances. Kolloid Z 101: Lake GJ[END_REF].

It is worth mentioning that failure associated with the changes in conformational entropy was rarely accounted for through those approaches, and mainly the mechanical behavior and microstructural parameters evolution of rubber materials subjected to multiaxial loading conditions were considered [START_REF] Vernerey | Statistical Damage Mechanics of Polymer Networks[END_REF][START_REF] Bahrololoumi | A multi-physics constitutive model to predict hydrolytic aging in quasi-static behaviour of thin cross-linked polymers[END_REF]). [START_REF] Volokh | Hyperelasticity with softening for modeling materials failure[END_REF][START_REF] Balakhovsky | Inflation and rupture of rubber membrane[END_REF] proposed modifying the free energy potential, paving the way to coupling the macroscopic fracture with the constitutive modeling. The "energy limiter" approach theory is based upon an evident axiom: since a material cannot absorb an unlimited amount of energy, its value must be bounded by a maximum, thus a new free energy potential using the properties of upper gamma function was developed to consider rubber fracture property. [START_REF] Dal | A micro-continuum-mechanical material model for failure of rubber-like materials: Application to ageing-induced fracturing[END_REF] pointed out the incompetency of phenomenological and statistical hyperelastic models to account for rubber materials fracture.

Accordingly, a modified entropic elasticity model describing the conformation capability of the chain and the energy of the interatomic bonds was proposed. The deformation and damage of a single chain coupled with advanced micro-macro transition [START_REF] Miehe | A micro-macro approach to rubber-like materials -Part I: The non-affine micro-sphere model of rubber elasticity[END_REF]) enable the description of the finite-strain macroresponse of rubber materials while using physical parameters.

Many other models describing the fracture of elastomeric materials can be found in the literature [START_REF] Mao | Rupture of polymers by chain scission[END_REF][START_REF] Tehrani | Effect of chain length distribution on mechanical behavior of polymeric networks[END_REF][START_REF] Lavoie | Modeling the mechanics of polymer chains with deformable and active bonds[END_REF][START_REF] Li | A variational phase-field model for brittle fracture in polydisperse elastomer networks[END_REF]. They take different approaches which will not be discussed here. Indeed, the purpose of this work is modeling the impact of aging on fracture properties of rubber. [START_REF] Belbachir | Modelling of photodegradation effect on elasticviscoplastic behaviour of amorphous polylactic acid films[END_REF]) studied the photodegradation effect on the mechanical properties of PLA and proposed a network alteration model by assuming that chain scission occurs mainly in the amorphous phase (rubbery phase). Later, [START_REF] Rodriguez | Effect of UV-aging on the mechanical and fracture behavior of low density polyethylene[END_REF]) and [START_REF] Ayoub | Modeling the visco-hyperelasticviscoplastic behavior of photodegraded semi-crystalline low-density polyethylene films[END_REF]) studied extensively the mechanism of photodegradation of semicrystalline polyethylene and developed a physically based composite-type model and introduced phenomenological damage evolutions of the macromolecular network to account for the alteration of the chemical and structural properties.

Macroscopic damage coupled with physical-based models introducing either a damage indicator or time-temperature equivalence approach were proposed to capture the effect of aging on the mechanical properties up to break [START_REF] Belbachir | Modelling of photodegradation effect on elasticviscoplastic behaviour of amorphous polylactic acid films[END_REF][START_REF] Abdelaziz | New developments in fracture of rubbers: Predictive tools and influence of thermal aging[END_REF][START_REF] Colin | Chemo-mechanical model for predicting the lifetime of epdm rubbers[END_REF][START_REF] Bahrololoumi | A multi-physics constitutive model to predict hydrolytic aging in quasi-static behaviour of thin cross-linked polymers[END_REF][START_REF] Kadri | A unified mechanical based approach to fracture properties estimates of rubbers subjected to aging[END_REF]. In those models the experimentally measured molar mass between crosslinks M c was established as a relevant indicator of damage induced by aging.

(Lake 2003) derived a useful relationship between the strain energy release rate (in the presence of a crack) and the molar mass M c , which has been successfully applied when the crosslinking aging mechanism predominates. However, when chain scission is the prevailing aging mechanism, those models predict an increase in the failure energy with increasing aging-induced degradation, which is in contradiction with several experimental observations reported in the literature [START_REF] Planes | Evolution of EPDM networks aged by gamma irradiation -Consequences on the mechanical properties[END_REF][START_REF] Gac | Description and modeling of polyurethane hydrolysis used as thermal insulation in oil offshore conditions[END_REF][START_REF] Kashi | Mechanical, Thermal, and Morphological Behavior of Silicone Rubber during Accelerated Aging[END_REF]. Indeed, chain scission mechanically increases the average value of M c , leading to an increase of the strain energy release rate.

Thus, the molar mass between crosslinks M c cannot be adopted as a generalized damage indicator independently from the aging mechanism. This issue will be addressed in this work.

The purpose of this work is to develop a physically based micro-mechanical approach capable of predicting the effect of aging on the mechanical and fracture behaviors of rubber materials regardless of the predominant degradation mechanisms. Physically based description of the aging mechanisms is proposed and coupled with the micromechanical model to accurately capture the experimental observations. The validity of the proposed approach is assessed using a wide set of experimental data either generated by the authors or available in the literature.

The paper is organized as follows: In section 2, the basic equations of the micromechanical model proposed by [START_REF] Dal | A micro-continuum-mechanical material model for failure of rubber-like materials: Application to ageing-induced fracturing[END_REF] are reminded. Furthermore, in section 3, the micro-mechanical model is modified to account for aging. When crosslinking is predominant, a residual stretch concept is introduced to capture the degradation of the fracture properties. In the case of chain scission prevalent mechanism, a healthy and degraded subnetworks scheme is proposed to explain the aging effects on the fracture. In each case, experimental data allows to check the capability of the modified model to capture the evolution of the fracture properties. This capability is widely discussed whatever the aging mechanism. Finally, concluding remarks and perspectives are then drawn.

Theoretical background

The rubbery microstructure can be conceptualized hierarchically starting from the mesoscale with the macromolecular network enclosing a large number of randomly oriented, variable-length, and elastically crosslinked active chains and at the micro scale with the Kuhn segments. In this section the main features of the physically based micro-mechanical approach for describing the mechanical and fracture properties of elastomeric materials are given. It reposed on the mathematical framework proposed by [START_REF] Dal | A micro-continuum-mechanical material model for failure of rubber-like materials: Application to ageing-induced fracturing[END_REF]. Then, the model is extended to account for the degradation due to aging.

The highly nonlinear hyperelastic macroscopic behavior of rubber materials is associated with the decrease of the macromolecular network entropy. The behavior of the macromolecular network is achieved by combining the chain length statistics with a non-affine micro-macro transition using the micro-sphere model [START_REF] Miehe | A micro-macro approach to rubber-like materials -Part I: The non-affine micro-sphere model of rubber elasticity[END_REF]. At the single-chain level, chains unfold with deformation, and they break when their limit of extensibility is reached. Therefore, the micromechanical behavior of a single polymer chain is achieved by accounting for the conformation capability of the chain and the energy of the interatomic bond [START_REF] Dal | A micro-continuum-mechanical material model for failure of rubber-like materials: Application to ageing-induced fracturing[END_REF].

Micromechanics of a single polymer chain

The single polymer chain in the network is represented by a large number N of rigid segments (Kuhn segments) of length l, the total length of the unfolded chain being L = N l (cf. Fig. 1). In this framework, the links between the rigid Kuhn segments (represented by springs in Fig. 1) are capable of deforming, and therefore a mechanical energy is stored and released (energy of the interatomic bonds), which is in contrast with the classical mathematical descriptions. These links represent the covalent bonds between the repeating units of the chain (348kJ/mol for C-C) and result in the polymer intrinsically exhibiting a high stiffness in the direction of the macromolecular chain.

Consequently, a simplified mechanical representation of the chain is therefore possible. The rheological model depicted in Fig. 2 represents each chain by a series of two springs: one capturing the energy storage resulting from the conformational change and the other the stored energy resulting from the interatomic displacement where u is the internal energy, T the absolute temperature, and s the entropy. An isothermal and isochoric deformation process induces a variation of the internal energy, resulting from the change in the interatomic distance, and of the entropy, resulting from the conformation evolution.

By correspondence with the rheological model described in Fig. 2, the internal energy corresponds to the energy stored by the interatomic bond and is noted ψ e , while conformation capability of the chain in the network corresponding to the free energy density of the chain is noted ψ η .

To describe the constitutive law associating the mechanical behavior of the chain conformations, we adopt in the modeling the well known form of the non-gaussian free energy density:

( ) ( ) ( )

1 1 η 1 ln sinh r rr r λ ψ NkT λ λ λ - - -  =+    L L L (2)
Where λ r corresponds to the relative stretch of the chain λ r = λ η .N

(-1/2)
, and

( )

1 r λ - L
is the inverse Langevin function, which can be evaluated through the Padé approximation:

( ) ( ) ( ) 2 1 2 3 1 r rr r - -  -    L (3)
The associated force (micro force of the entropic contribution) acting on a single chain by assuming Langevin statistics can be derived easily from eq.( 2). It comes:

( )

2 ηη η ηη 2 ηη d 3 d ψλ N λ f kTλ λ N λ  - ==   -  (4)
Reminding that we are interested in modeling and predicting the fracture properties of aged elastomeric materials, the internal energy and the stretch associated with the interatomic bond cannot be neglected in the vicinity of the locking stretch i.e.

the deformation limit. Accordingly, the constitutive equations describing the contribution of the interatomic bond to the total chain deformation is described using an expression derived from Morse potential [START_REF] Dal | A micro-continuum-mechanical material model for failure of rubber-like materials: Application to ageing-induced fracturing[END_REF]:

( ) ( ) e exp 2 2 exp 1 ψ e γ γ  = -- -+  (5) With: ( ) ( ) 2 1 e 2 11 γ α λ α J = -+ - (6)
Where e corresponds to the dissociation energy for a pair of atoms, J = det F is the Jacobian, F being the deformation gradient and α 1 and α 2 are parameters controlling the slope of the energy drop.

The micro force associated with the internal energy contribution ψ e acting on a single chain is expressed as follows:

( ) ( ) ( ) e e1 e 2 exp 2 exp ψ f α e γ γ λ  = = - -+ -  (7)
Fig. 3 illustrates the evolution of the internal energy potential ψ e and the associated micro-force f e for a single chain. (ψ e (λ e max ) = e/4), which will be denoted as the threshold bond energy.

Above the glass transition temperature T g , the mechanical response of rubbers is mainly associated with the change in entropy. This simplification is often assumed in classical hyperelastic modelling. However, the latter do not consider chains breaking, consequently the failure of the elastomer network. This can be explained by the deformation mechanism of the chain. Because of the weaker rotational stiffness of the Kuhn segments around the bond, the deformation process begins with the unfolding and orientation of the chain according to the stretch direction. This step mostly affects the entropic part of energy which is considered in the modeling across the Langevin spring. Once the chain is nearly unfolded, the extension of the interatomic bonds is initiated inducing the increase of internal energy ψ e . Its deformation is captured by the second spring. This is allowed by the high stiffness of this spring compared to that of the Langevin spring.

The total energy ψ is equal to the sum of the individual energies developed in each spring (ψ η and ψ e ), while the stress acting on each spring is the same.

Thus, through this micromechanical modeling, the physics of a chain behavior up to and beyond the fracture can be well captured. paving the way to modeling the physical effects of aging on the rubbers fracture properties.

Macromolecular network model

The macromolecular network behavior is defined at the macro-scale and related to the mechanics and thermodynamics of continuum media by describing for each chain the deformation mechanisms through the above described rheological model (Fig. 2), and transitioning to the macro scale using the micro-sphere model [START_REF] Miehe | A micro-macro approach to rubber-like materials -Part I: The non-affine micro-sphere model of rubber elasticity[END_REF]).

Rubber materials are generally assumed incompressible; therefore, the free energy function can be written in a volumetric-isochoric decoupled form:

( ) ( )

; =+ F Φ U J Φ g ( 8
)
Where g is the spatial metric,

- = 1 3
J FF corresponds to the isochoric part of the deformation gradient F that governs the isochoric stress.

Since for a single EAC, the free energy is given by eq.( 1), the free energy of a macromolecular network Φ is similarly decomposed and is expressed as follows:

( ) ( )

η η e Φ = n ψ λ + n ψ γ (9)
Where n corresponds to the density of EAC per unit reference in the macromolecular network.

In the same way, the Kirchhoff stress is obtained by adding the hydrostatic and the deviatoric contributions:

1 : =p -+ g ττ P ( 10
)
Where P is the fourth-order identity tensor.

The hydrostatic stress p is obtained through the derivative of the volumetric part U of the free energy ( ( )

p = J U J / J  )
where U is an incompressibility constraint that can take different forms [START_REF] Horgan | On the volumetric part of strain-energy functions used in the constitutive modeling of slightly compressible solid rubbers[END_REF] and is considered as a penalty function to approach the incompressible response.

The deviatoric stress τ is derived from the isochoric part of the free energy function Φ by using the intermediate derivative (i.e

1-q λλ   = g 2 / h with -2 q λ =  tt h ) (Miehe et al. 2004):
( )

; = 2 =    1-q Φ Φλ gF τ h g ( 11
)
Where λ is the macro stretch and is related to the micro-stretch λ of a single chain through a q-root averaging over the unit micro-sphere (λ q λ ), see [START_REF] Miehe | A micro-macro approach to rubber-like materials -Part I: The non-affine micro-sphere model of rubber elasticity[END_REF]).

 = Φ n f is the single elastic potential derivation and f is the micro force defined as

/ =   f ψλ .
Therefore, the isochoric contribution to macroscopic stress takes the following form:

η 1 - 1 - e ηe = + = qq λ λ n f f λ n f λ λλ        τ h h (12)
For more details about the mathematical developments of partial derivatives and calculation of stress see the appendix A. We can also conclude that the chain failure depends on the chain length and it occurs when the chain is fully stretched along the loading direction. At the molecular network level, the chain failure process depends on the statistical distribution of the chain length. This statistical distribution of chain length is accounted for through the non-Gaussian approach and the micro-macro transition carried out through the micro-sphere model.

Once these main features of the physically based model reminded, let us now pay attention to the consideration of the effects of aging.

Extension of the micromechanical model to account for aging

Environmental aging induces a slow and irreversible alteration of the rubber material's macromolecular network. This alteration is triggered by two mechanisms, crosslinking and chain scission, schematically illustrated in Fig. 6, which act simultaneously but one of them being generally predominant. hardening of the material is induced. This change in rigidity is generally well captured by assuming mass conservation [START_REF] Marckmann | A theory of network alteration for the Mullins effect[END_REF]Ayoub et al. 2011a[START_REF] Ayoub | Modeling the visco-hyperelasticviscoplastic behavior of photodegraded semi-crystalline low-density polyethylene films[END_REF], which is expressed as follows: 00 ( ) ( )

n t N t n N = (13)
N 0 and n 0 being the values at the reference state (as received material) of the average chain length and the EAC density, respectively. The mass conservation principle was widely used in the literature to theoretically predict the mechanical behavior of rubber during aging. However, can the use of this principle alone be sufficient to predict both the mechanical behavior and rupture of the material? In this subsection, we suggest to address this issue by using our own experimental data.

Fig. 7 shows the evolution of the uniaxial tensile behavior of an EPDM material vulcanized by a peroxide compound at different aging periods at 150°C. The experimental data show an increase in the stiffness along with a decrease of the strain at break with increasing exposure time. Knowing the experimental evolution of the EAC density n (its determination will be explained further), the segment number N is deduced using the mass conservation given by eq.( 13).

It is observed that the model only assuming mass conservation is able to capture the increasing stiffness with thermal aging. However, the model fails to predict the evolution of both stress and strain at break except for the reference state that is used to calibrate the parameters. Indeed, according to eq.( 9), the free energy increases with the EAC density n. Because the material crosslinks, n increases, leading to an increase of both the stiffness and the network free energy. This mechanically induces an increase of the stress at break, which is in contradiction with the experimental observations, as depicted in Fig. 7. That demonstrates the inability to predict fracture by only using the mass conservation principle. A first issue to be addressed will be therefore to build a novel concept able to model the fracture process at the micro-scale.

When chain scission (cf. Fig. 6.b) is the predominant mechanism during aging, the EAC density n decreases. The mass conservation principle implicitly lead to an increase of Kuhn segments number N, and an decrease of the EAC density because n(t).N(t) should remain constant. However, considering only mass conservation when chain scission is the predominant mechanism is a real issue because of the dangling chains which do not contribute neither to the stiffness nor to the fracture properties. How to estimate the real EAC density involved in the mechanical strength of the material? This is a second issue to be solved.

Moreover, the chain scission mechanism is accompanied by a decrease in EAC density n. Consequently, according to the mass conservation principle, the average chain length N increases, which theoretically provokes an increase of the strain at break (resulting from the dependency of λ lock on N). This is in contradiction with the experimental observations reported by many available investigations [START_REF] Planes | Evolution of EPDM networks aged by gamma irradiation -Consequences on the mechanical properties[END_REF](Planes et al. , 2010b;;[START_REF] Gac | Description and modeling of polyurethane hydrolysis used as thermal insulation in oil offshore conditions[END_REF][START_REF] Kashi | Mechanical, Thermal, and Morphological Behavior of Silicone Rubber during Accelerated Aging[END_REF]). Hence, mass conservation cannot be used to predict either the mechanical behavior or the fracture for this aging mechanism.

To partially conclude, the assumption of only mass conservation is not suitable to describe: a. The fracture properties of rubbers when crosslinking is the aging mechanism b. both the evolution of the mechanical behavior and the fracture properties when chain scission is the aging mechanism Since the mechanisms of crosslinking and chain scission are physically different, a unified theory that aims to capture the physical effects of both aging mechanisms on the behavior of rubber materials up to failure is difficult to conceive. Therefore, in this work we develop and propose approaches respectively suited to each mechanism, which allows to capture the aging effects on the behavior of rubber materials up to failure. Keeping in mind that few works deal with the problem in literature, and more particularly with the effect of chain scission mechanism on the evolution of failure properties, a novel theoretical framework for the modelling of ageing mechanisms and their impact on the mechanical and fracture properties is proposed and detailed in what follows.

Crosslinking mechanism

Residual stretch concept

Crosslinking mechanism lead to increase EAC density n in the network and simultaneously to the reduction of chain lengths N via the creation of new crosslinks. Understanding of what happens at the microscale being unavoidable to explain the macroscopic observations, we have attempted the described below reasoning which led us to propose a residual stretch theory.

When crosslinking operates, it creates new crosslinking nodes in the rubber network increasing over time the density of rubber network, as schematized in Fig. Higher the shortening of the chain, higher the residual stress. These residual stresses operate at the chain scale and lead to the stretching of the interatomic bonds. This residual micro-stretching reduces the energy capability of the bonds to absorb mechanical loadings and causes the alteration of fracture properties of rubber network during aging.

We can conclude that the increase of the crosslinking density induces residual stretches at the microscale. The network entropy decrease is accompanied by a progressive stretching of the network chains, as schematically represented in Fig. 8. Therefore, the remaining conformation capability of the chain in the network is reduced. With n and n 0 being the current and the initial EAC density, respectively. The parameter c is a macro-micro non-affinity parameter, which allows linking the degradation of the network at the macro scale to the micro-stretching at the chain scale (λ η res et λ e res ). For the as-received rubber material, λ res = 1; hence, no residual micro-stretch is applied.

Accordingly, a numerical implementation scheme of the model has been built and is detailed in Table 1. ( ) Fig. 10.a shows that the more the aging of the EPDM network progresses, the more the chain entropy decreases because of the decrease of the chain length. In the same time, chain shortening induces an increase of the bond residual micro-stretching λ e res . The consequence at the macroscale is shown in Fig. 10.b, the increase in λ e res involves a decrease of the macroscopic fracture strain which agrees with experimental observations. Therefore, the proposed micro-scale approach allows capturing the complex macromolecular network degradation resulting from chains crosslinking and its effects on the fracture properties.

1/ q m q i i i=1 λ = λ w        (average network stretch)
Precising that in the modeling, the value of energy dissociation of a pair of atoms (cf. eq.( 5)) is always the same during aging. Indeed, aging does not affect its value.

2022

). These results must be compared to that presented in Fig. 7 while assuming only the mass conservation principle. It is observed that the modified micromechanical model is able to capture the mechanical response and its alteration with thermal aging. In contrast with results presented in Fig. 7, the micromechanical model is successful in capturing the evolution of both stress and strain at break, which is clearly an improving and a highlights contribution of this work.

Experimental data and computational results a) Experimental data

In this work, both experimental data generated by the authors and extracted from the literature are used to discuss the validity of the proposed approach and to justify the suggested model modifications. For that purpose, two ethylene propylene diene monomer EPDM rubber were involved in the experimental investigation. The first one (denoted EPDM 01) is vulcanized by using a peroxide compound, and is widely used for thermal insulation, The second one (denoted EPDM02) is vulcanized using a sulfur compound (Nait [START_REF] Abdelaziz | New developments in fracture of rubbers: Predictive tools and influence of thermal aging[END_REF] and is used to manufacture electrical junctions. Sulfur vulcanization allows obtaining a high deformability of the rubber. The experimental data were generated by the authors and the EAC density evolution of the EPDM network was identified by using swelling tests 1 (Nait [START_REF] Abdelaziz | New developments in fracture of rubbers: Predictive tools and influence of thermal aging[END_REF][START_REF] Kadri | A unified mechanical based approach to fracture properties estimates of rubbers subjected to aging[END_REF] .

Thermo-oxidative aging was achieved by exposing the EPDM materials (noted EPDM01 and EPDM02) samples to variable aging periods at the following temperatures: 90, 110, 130 and 150°C in air-ventilated ovens for EPDM01 and 130, 150, 170°C for EPDM02. The tensile tests on EPDMs samples were carried out using an Instron device equipped with a 1kN load cell under a crosshead speed of 50mm/min and at ambient temperature. The tests were conducted according to ISO 37:2011 using H3 specimen geometry of 2mm thickness. To evaluate the material scattering, seven samples were tested for each aging condition.

b) Alteration kinetics

Based on the previous discussion, EAC density n(t) is considered as the best indicator for describing the degradation of the macroscopic network. The evolution of the EAC density n(t) is approximated by using an exponential law 2 (using least squares method) which parameters are summarized in appendix B.

Fig. 12 presents the evolution of the EAC density n of the studied materials at the reference temperature (150 and 170°C for EPDM01 and EPDM02 respectively).

The EAC density n monotonically increases with time, which is the marker of a predominant crosslinking mechanism during thermo-oxidative aging.

1 For EPDM rubber n(t) was determined experimentally using swelling tests, which consist of immersing a rubber sample in a suitable solvent. This kind of reversible experiment allows estimating the elastically active chains (EAC) concentration (Kraus 1963).

2 The evolution of the EAC density n(t) is approximated by using the following general exponential form:

n(t) = b 1 . exp(b 2 .t) + b 3 . exp(b 4 .t)
, where b i are fitting parameters.

c) Fracture properties prediction

The predictive capabilities of the model to capture the evolution of fracture properties are discussed in this section. Since the aging of EPDM rubbers is here predominantly controlled by crosslinking mechanism, only the parameter c of eq.( 14) needs to be additionnaly identified. This identification is conducted by using the experimental fracture strains at the reference temperature (150°C). The whole model parameters are summarized in table 2. different temperatures and periods. For that purpose, we used the time-temperature equivalence approach (the Arrhenius power law 3 ) to unify the material data of different aging temperatures (Nait [START_REF] Abdelaziz | New developments in fracture of rubbers: Predictive tools and influence of thermal aging[END_REF], allowing to build the master curves of stress and strain at break. The parameter that needs to be identified for the Arrhenius power law parameter is the activation energy, which was found equal to E a = 95 and 110kJ/mol for EPDM01 and EPDM02 respectively.

The evolution of the ultimate properties (strain and stress at break) of EPDM01 material as a function of reduced aging time at different aging temperatures is plotted in Fig. 13. The experimental data for different aging temperatures and periods are presented as dots, while the model is presented as a solid line. experimental fracture properties, in terms of either stresses or strains at break, and the model. The experimental data form a scatter around the model, proving the validity of our proposed approach. This ability is confirmed when analyzing the results for EPDM02 (Fig. 14) [START_REF] Mark | Dependence of the elastomeric properties of bimodal networks on the lengths and amounts of the short chains[END_REF] studied the effect of bimodal networks (i.e networks which are composed of chains statistically exhibiting two length average values) on the mechanical properties of rubber materials. It was reported that when long chains are predominant compared to small chains, the fracture is primarily controlled by the first ones. Paradoxically, [START_REF] Planes | Evolution of EPDM networks aged by gamma irradiation -Consequences on the mechanical properties[END_REF]) reported a decrease in strain at break in the case of chain scission which is contradictory, because the increase of the chain length theoretically induced by this mechanism should involve an increase of the strain at break which is proportional to N 1/2 . It was also reported in this work that a multimodal distribution of the chain length cannot explain these conflicting observations. The decrease of the strain at break experimentally observed was Two architecture network properties were identified to control the overall mechanical behavior: load bearing subnetwork and weakened zones comprising large free agglomerates which do not contribute to the stiffness.

Healthy and degraded subnetworks scheme

The creation of dangling chains resulting from chain scission mechanism induces a decrease of EAC density n. Consequently, according to these experimental observations, we assume that the subnetworks created during aging can be classified in two types: the first which contribute to the stiffness (load bearing network) and denoted "healthy" subnetwork, the second which can be considered as a "degraded" subnetwork which do not take part to the mechanical strength. In fact, even if the "degraded" subnetworks contains E , these latter cannot fully contribute neither to the mechanical stiffness nor to the fracture process because they are surrounded by dangling chains (cf. Fig. 15). Indeed, this heterogeneity in terms of subnetworks modifies the stress distribution, the load being essentially transmitted through the "healthy" zone. When breakage occurs, only the "healthy" part of the network is involved. Thus, because of these dangling chains, the mass conservation principle must be suited to account for the two kinds of subnetworks and can be rewritten as n 0 N 0 = n he N he +n de N de .

Where the subscript "he" and "de" corresponds to the "healthy" and "degraded" subnetworks respectively. In what follows we assume that only the healthy EAC density n he is involved in the fracture process. Moreover, during aging, the average chain length N he is not affected and keep the initial value N 0 . Indeed, there is no reason for the average chain length to evolve during aging, since the chain scission mechanism tends to make them inactive.

Assuming the above considerations, a "healthy" E density n he is therefore introduced as a damage indicator to capture the evolution of the fracture properties when chain scission is the predominant mechanism. This healthy EAC density n he is expressed in the following simplest form:

th he 0 th 0 n -n n = n n -n (15)
n 0 corresponds to the unaged material EAC density, n th corresponds to the threshold EAC density below which the material is supposed to be completely degraded (meaning that no load can be borne), and n is the overall EAC density determined experimentally.

Experimental data and computational results a) Experimental data egraded zone ealthy zone

To discuss the validity of the proposed approach and to justify the suggested model, experimental data extracted from the literature are used. For that purpose, two rubber materials are analyzed: (a) A polyether-based polyurethane PU: the experimental data were generated by [START_REF] Gac | Description and modeling of polyurethane hydrolysis used as thermal insulation in oil offshore conditions[END_REF], and (b) a silicone rubber SR: the experimental data were generated by [START_REF] Kashi | Mechanical, Thermal, and Morphological Behavior of Silicone Rubber during Accelerated Aging[END_REF]. Swelling tests results were not available for those materials. Therefore, the EAC density was deduced from tensile tests through the shear modulus value.

PU specimens were aged during 18 months at 70, 90, 100, 110, and 120°C in sea water renewed every day. The accelerated thermo-oxidative aging of SR was achieved by immersing the specimens in beakers filled with polyalkylene glycol (PAG) oil and placed in an oven at 195°C over 6 weeks. Aged PU material tests were performed using ISO 37-standard samples of 2mm thickness. The tests were conducted under a crosshead speed of 50mm/min. Before mechanical testing, samples were stored for 24h in sea water at room temperature (Le [START_REF] Gac | Description and modeling of polyurethane hydrolysis used as thermal insulation in oil offshore conditions[END_REF]).

The mechanical measurements on aged SR samples were achieved at ambient temperature using a crosshead speed of 200mm/min. The strain measurements were achieved by using a contactless extensometer. For each batch, three dumbbellshaped specimens were tested.

b) Alteration kinetics

Like for EPDM material, the evolution of EAC density n(t) was approximated by using an exponential law (least squares method). Those parameters for each studied material are summarized in appendix B.

The EAC density n was identified via the measure of the shear modulus μ by using the experimental tensile tests, n being deduced through the following relationship: μ kT. Where k is the Boltzmann constant, T the absolute temperature. For SR material, only one set of data at 195°C is available, hence, the model results are only calibration. The purpose of using SR data in the validation is to show the capability of the proposed approach to capture the evolution of the mechanical behavior of different rubber materials.

To assess the predictive capability of the model, a comparison with the experimental fracture properties of the studied materials aged at different temperatures and periods is achieved. For that purpose and like previously for EPDM material (section 3.1), we have built a master curve using the timetemperature equivalence approach (the Arrhenius power law) that unifies the data of PU material for different aging temperatures. The activation energy for this material was found equal to E a = 120 kJ/mol. Since only one aging temperature is available for the SR material, the time-temperature equivalence principle cannot be used.

The evolution of the ultimate properties in terms of stress at break as a function of aging time at different aging temperatures is plotted in Fig. 17. It is clearly shown that the modified micro-mechanical model introducing the concepts of the "healthy" E density allows capturing the experimental evolution. Moreover, either for crosslinking observed for EPDM material or chain scission in the case of PU material, the predictive capability was demonstrated since the parameters were identified at a reference temperature, the results for the other temperatures being model predictions.

Discussion

In this work, the behavior of aged rubber materials up to fracture is modeled using a coupling between the physically based description of the aging mechanisms and the micro-mechanical model based on the [START_REF] Dal | A micro-continuum-mechanical material model for failure of rubber-like materials: Application to ageing-induced fracturing[END_REF] approach. and "degraded" zones. ndeed, this heterogeneity at the microscale modifies the stress distribution, the load being transmitted essentially through the "healthy"

zone. Taking into account the above considerations, the noted "healthy" E density is therefore introduced as a damage parameter to capture the evolution of the fracture properties when chain scission is the predominant mechanism. The validity of the proposed approach is assessed using a wide set of experimental data generated by the authors and available in the literature. The comparison between the experimental and the computational results demonstrates the predictive capability of the model to capture the evolution of the mechanical and fracture properties of aged rubbers whatever the aging mechanism.

Rubber materials are highly viscous materials, and understanding the effect of aging on the viscosity is of high interest. The viscosity in rubbers is associated with the chain reptation mechanisms, which are associated with the thermal motion of chains and entanglements. Hence the chain length influences the rate of chain reptation, and therefore the degradation of the macromolecular network in theory affects the viscosity. The effect of aging on the viscosity needs to be considered by first understanding this effect through an extensive experimental investigation and second by including viscosity in the proposed model.

As a perspective, in the same way that proposed by [START_REF] Trapper | Modeling dynamic failure in rubber[END_REF] such a modeling can be implemented in a FE software to simulate the process of crack propagation due to aging. An attempt of its feasibility has been achieved

showing promising results (Fig. 18) Since we are assuming a series scheme, the stress in each spring is assumed to be equal to the total stress, and hence λ η and λ e are computed by using the following relation: The problem can be solved using any iterative scheme. In this work, a Newton-type procedure is used to solve the residual quantity η η e e =0 rf λ -f λ  .

Once the different micro-stretch contributions are determined, the isochoric kirchhoff stress  can be computed:

Fig. 1

 1 Fig. 1 Schematic representation of a single polymer chain with a magnified view representing interaction between Kuhn segments across bonds depicted by springs

Fig. 2

 2 Fig. 2 Rheological representation of the model representing each chain

Fig. 3

 3 Fig. 3 Evolution of the internal energy ψ e and corresponding micro-force f e of a single chain. The parameter values used for generating these results are the following: e=3.34 10-24 MPa, α 1 =9,α 2 =5α 1At the equilibrium (i.e. λ e =1) ψ e is null. On either side of this value, the energy

Fig. 4

 4 Fig. 4 shows the evolution of the different energies during stretching: total energy ψ, internal energy ψ e and the energy associated with the change of entropy ψ η as function of the total stretch λ.

Fig. 4

 4 Fig. 4 Evolution of the total strain energy density ψ, of the entropic. contribution ψη, and of the bond energy ψe of a single chain as a function of the total stretch λ. The parameter values used for generating these results are the following: n=4.5 10 20 cm -3 , N=10, e/k.T=3.44, α1=9, α2=5α1

Fig. 5

 5 Fig. 5.a illustrates the capability of the macromolecular network model to capture

Fig. 5 3 ,

 53 Fig.5(a) The true stress as a function of macroscopic stretch λ , (b) evolution of the sub-stretches λ η (solid line) and λ e (dashed line) as a function of the macroscopic stretch λ . The parameter values used for generating these results are the following: n=4.5 10 20 cm 3 , N=10, e/k.T=3.44, α1=9, α2=5α1

Fig. 5

 5 Fig. 5.b shows that the stretch associated to the bond energy λ e is negligible at the

Fig. 6

 6 Fig. 6 Schematic representation of (a) crosslinking and (b) random chain scission

Fig. 7

 7 Fig. 7Stress as a function of applied stretch for an EPDM aged at 150°C for different periods. The parameter values used for generating those results are the following: N=10, T= 296 K, e/kT= 3.44, q = 2, n0=4.5 10 20 (cm -3 ), α1=9, α2=5α1

  . 7 presents a comparison between the proposed micro-mechanical model predictions assuming only mass conservation and the experimental data. The unaged test data material was fitted to identify the material parameters n 0 and N 0 .

  8.a. The increase of the density network leads at the same time to a shortening of EAC length, schematically described in Fig. 8.b. To ensure the mechanical balance of the network during the shortening of chains induced by crosslinking, residual micro-stresses are involved at the chain scale, represented by arrows in Fig. 8.b.

Fig. 8 Fig. 9

 89 Fig. 8 Schematic representation of (a) the network densification and (b) chain stretching at the microscale induced by an increase of the crosslinking density

g

  Allow to identify values of λ η and λ e for each load refer to eq.(10)The effect of the residual stretch on the fracture behavior of the rubber can be explained through the effect of crosslinking on the macromolecular network. Indeed, the reduction of chains length induces a decrease in the network entropy and elongation at break. The relevance of the concept of residual stretch is demonstrated in Fig. 10 which represents the evolutions during aging of entropy and bond residual micro-stretching λ e res of an EPDM chain at 150°C (Fig. 10.a), as well as the influence of bond residual micro-stretch λ e res on the macroscopic failure strain (Fig. 10.b).

10

  Fig. 11 presents a comparison between the micromechanical model predictions and

Fig. 11

 11 Fig. 11 Comparison between the experimental stress vs. stretch and the modified micromechanical model predictions for an EPDM aged at 150°C at different aging periods

Fig. 12

 12 Fig. 12 Evolution of the EAC density n (mean values) during aging at reference temperature as a function of aging time (in days) for (a) EPDM01 and (b) EPDM02 materials (Nait Abdelaziz et al.2019;[START_REF] Kadri | A unified mechanical based approach to fracture properties estimates of rubbers subjected to aging[END_REF]

Fig. 13

 13 Fig. 13 (a) True strain and (b) true stress at break as a function of the reduced time for EPDM01 material

Fig. 14

 14 Fig. 14 (a) True strain and (b) true stress at break as a function of the reduced time for EPDM02 material

  the presence of sub-networks with different chain lengths.

Fig. 15

 15 Fig. 15 Schematic representation of the bimodal network induced by chain scission during aging (dangling chains in red)

Fig. 16

 16 Fig. 16 presents the evolution of the EAC density n of the studied materials. Only

Fig. 16

 16 Fig.16shows a decrease of EAC density regardless of the material, revealing a

Fig. 16 .

 16 Fig. 16. Evolution of the EAC density n during aging as a function of aging time (in days or weeks) for (a) PU (120°C) and (b) SR (195°C) materials c) Fracture properties prediction

  .

  3.

Fig. 17

 17 Fig. 17 True stress at break as a function of (a) the reduced time for PU and (b) the time (in weeks) for SR

  macromolecular network behavior is defined at the macro-scale and related to the mechanics and thermodynamics of continuum media by describing for each chain the deformation mechanisms and the effect of thermooxidative aging and transitioning to the macro scale using the micro-sphere model[START_REF] Miehe | A micro-macro approach to rubber-like materials -Part I: The non-affine micro-sphere model of rubber elasticity[END_REF]). The rheological representation of the model describes each chain by a series of two springs capturing the energy storage resulting from the deformation-induced conformational change and the stored energy resulting from the interatomic displacement. Environmental aging induces a slow and irreversible alteration of the rubber material macromolecular network. This alteration is triggered by two mechanisms, crosslinking and chain scission, which act simultaneously but with one generally being predominant[START_REF] Belbachir | Modelling of photodegradation effect on elasticviscoplastic behaviour of amorphous polylactic acid films[END_REF];[START_REF] Abdelaziz | New developments in fracture of rubbers: Predictive tools and influence of thermal aging[END_REF]). When crosslinking is the predominant mechanism, the EAC density increases, which induces a decrease both of the average chain length N and consequently of the maximum extensibility of the chains. Furthermore, it involves at micro-scale a pre-stretching of the chains, this residual stretch being aging dependent. This residual stretch is incorporated into the macromolecular model by adding a micro-stretch to the initial state before applying the external load. When chain scission is the predominant mechanism during aging, the EAC density decreases. In this work, we assume that the network becomes heterogeneous with the presence of sub-networks with different chain length denoted as "healthy"

Fig. 18

 18 Fig. 18 FEA simulation of the crack propagation in an aged elastomer

Table . 2

 . parameters values of the constitutive model

	Model parameter	EPDM01	EPDM02
	n 0 [cm	-3 ]	4.5 10 20	1.09 10	20
	N		10	16
	e [MPa]	1.8 10 -20	1.8 10 -20
	α 1		9	6
	q		2	3.5
	c		0.8	0.6

The predictive capability of the model is now assessed by comparing the model predictions and the experimental fracture properties of the studied materials aged at

Table . 3

 . Values of different parameters of the constitutive models.

	q		2	2
	n th [cm -3 ]	3.8 10 20	3.5 10 19
	(a)				(b)
	days				weeks
	Model parameter	PU	SR
	n 0 [cm	-3 ]	1.9 10 21	5.12 10	19
	N		120	345
	e [MPa]	1.25 10 -20	7.15 10 -19
	α 1		7	7

Appendix A

The kirchhoff stress is obtained by adding the hydrostatic and the deviatoric contributions:

P

Where P is the fourth-order identity tensor:

With δ being the Kronecker-delta symbol.

The hydrostatic stress p is obtained through the derivative of the volumetric part U of the free energy :

U is an incompressibility constraint that can take different forms [START_REF] Horgan | On the volumetric part of strain-energy functions used in the constitutive modeling of slightly compressible solid rubbers[END_REF] and is considered as a penalty function to approach the incompressible response.

The deviatoric stress τ is derived from the isochoric part of the free energy function Ψ by using the intermediate derivative (i.e et al. 2004):

( )

Where q is the non-affinity parameter between micro and macro stretch.

is the single elastic potential derivation and f is the micro force defined as

Appendix B