David Costantini
email: david.costantini@mnhn.fr

Pierre Blévin

Dorte Herzke

Børge Moe

Geir Wing Gabrielsen

Jan Ove Bustnes

Olivier Chastel

Higher plasma oxidative damage and lower plasma antioxidant defences in an Arctic seabird exposed to perfluoroalkyl carboxylic acids

Perfluoroalkyl and polyfluoroalkyl substances (PFASs) may cause detrimental effects on physiological function and reproduction of Arctic animals. However, there is a paucity of information on the link between PFASs and oxidative stress, which can have potential detrimental effects on key fitness traits, such as cellular homeostasis or reproduction. We have examined the correlations between multiple blood-based markers of oxidative status and several perfluoroalkyl carboxylic acids (i.e., with 8 or more carbons) in male Arctic black-legged kittiwakes (Rissa tridactyla) during the pre-laying period. Higher protein oxidative damage was found in those birds having higher concentrations of perfluorododecanoic acid (PFDoA), perfluorotridecanoic acid (PFTriA) and perfluorotetradecanoic acid (PFTeA). Lower plasmatic non-enzymatic micro-molecular antioxidants was found in those birds having higher concentrations of perfluoroundecanoic acid (PFUnA), PFDoA and PFTeA. Effect size estimates showed that the significant correlations between PFASs and oxidative status markers were intermediate to strong. The non-enzymatic antioxidant capacity (including antioxidants of protein origin) was significantly lower in those birds having higher plasma concentration of linear perfluorooctanesulfonic acid (PFOSlin) only. In contrast, the activity of the antioxidant enzyme glutathione peroxidase in erythrocytes was not associated with any PFAS compounds. Our results suggest that increased oxidative stress might be one consequence of long-chain PFAS contamination. Given the correlative nature of our study, experimental work will be needed to demonstrate whether PFASs cause toxic effects on free-living vertebrates through increased oxidative stress.

INTRODUCTION

Ecotoxicological studies have so far extensively directed their attention toward legacy persistent organic pollutants (POPs) like organochlorine pesticides (OCPs) and polychlorobiphenyls (PCBs). 1 In contrast, less attention has been given to the environmental toxicity of other organic contaminants in the Arctic, such as chlorinated paraffins, phthalates, siloxanes or the perfluoroalkyl and polyfluoroalkyl substances (PFASs). [START_REF]Chemicals of emerging Arctic concern. Summary for Policy-makers[END_REF] Among these, PFASs remain comparatively much less investigated. [START_REF] Dewitt | Toxicological effects of perfluoroalkyl and polyfluoroalkyl substances[END_REF] PFASs are synthetically manufactured chemicals, produced since the 1950s, that are widely used for numerous industrial and commercial purposes as water repellents and surfactants (e.g., impregnation agents for carpets, papers and textiles, fire-fighting foam, non-stick coating and waterproof clothing). [START_REF] Jensen | Emerging endocrine disrupters: perfluoroalkylated substances[END_REF][START_REF] Kissa | Fluorinated Surfactants and Repellents[END_REF] Chemically and thermally stable, PFASs are highly persistent in the environment and have been detected globally in both wildlife and humans. [START_REF] Lau | Perfluoroalkyl acids: a review of monitoring and toxicological findings[END_REF][START_REF] Key | Fluorinated organics in the biosphere[END_REF][START_REF] Muir | Trends of legacy and new persistent organic pollutants in the circumpolar arctic: overview, conclusions, and recommendations[END_REF] Because of oceanic currents and atmospheric long-range transport, PFASs and their precursors and breakdown products can reach high latitudes, such as the Arctic Ocean. 9- 11 Once deposited in the Arctic marine ecosystem, PFASs bio-accumulate in living organisms and bio-magnify along the food webs. [START_REF] Fang | Trophic magnification and isomer fractionation of perfluoroalkyl substances in the food web of Taihu Lake, China[END_REF][START_REF] Haukås | Bioaccumulation of per-and polyfluorinated alkyl substances (PFAS) in selected species from the Barents Sea food web[END_REF][START_REF] Kannan | Perfluorinated compounds in aquatic organisms at various trophic levels in a Great Lakes food chain[END_REF][START_REF] Kelly | Perfluoroalkyl contaminants in an Arctic marine food web: trophic magnification and wildlife exposure[END_REF][START_REF] Tomy | Fluorinated organic compounds in an eastern Arctic marine food web[END_REF] Importantly, (i) PFASs have long half-lives, which facilitates their bio-accumulation through the food webs depending on the species and PFAS congener 8 ; (ii) the long and odd carbon-chain-length PFASs appear to be more bio-accumulative and toxic than the short and even-chain length PFASs in wildlife. [START_REF] Berntsen | Time-dependent effects of perfluorinated compounds on viability in cerebellar granule neurons: dependence on carbon chain length and functional group attached[END_REF][START_REF] Conder | Are PFCAs bioaccumulative? A critical review and comparison with regulatory criteria and persistent lipophilic compounds[END_REF][START_REF] Martin | Identification of long-chain perfluorinated acids in biota from the Canadian Arctic[END_REF][START_REF] Verreault | Perfluorinated alkyl substances in plasma, liver, brain, and eggs of glaucous gulls (Larus hyperboreus) from the Norwegian arctic[END_REF] It is worthwhile to note that while PFASs have been produced for over 50 years, it is only since late 1990s that their occurrence in the environment has come under scientific scrutiny. PFASs have raised recent concerns about their potential physiological disrupting properties and negative impacts on reproductive fitness in wildlife (multiple species in [START_REF] Bossi | Preliminary screening of perfluoroctane sulfonate (PFOS) and other fluorochemicals in fish, birds and marine mammals from Greenland and the Faroe Islands[END_REF] ; lesser black-backed gull in [START_REF] Bustnes | Perfluorinated and chlorinated pollutants as predictors of demographic parameters in an endangered seabird[END_REF] ; northern fulmar in [START_REF] Braune | Biomarker responses associated with halogenated organic contaminants in northern fulmars (Fulmarus glacialis) breeding in the Canadian Arctic[END_REF] ; zebrafish in [START_REF] Liu | The thyroid-disrupting effects of long-term perfluorononanoate exposure on zebrafish (Danio rerio)[END_REF] ; tree swallow in [START_REF] Custer | Exposure and effects of perfluoroalkyl compounds on tree swallows nesting at Lake Johanna in east central Minnesota[END_REF] ; black-legged kittiwake and northern fulmar in [START_REF] Nøst | Halogenated organic contaminants and their correlations with circulating thyroid hormones in developing Arctic seabirds[END_REF] ; black-legged kittiwake in [START_REF] Tartu | Endocrine and fitness correlates of long-chain perfluorinated carboxylates exposure in Arctic breeding black-legged kittiwakes[END_REF][START_REF] Tartu | The stress of being contaminated? Adrenocortical function and reproduction in relation to persistent organic pollutants in female black legged kittiwakes[END_REF] ; black-legged kittiwake in [START_REF] Blévin | Perfluorinated substances and telomeres in an Arctic seabird: Cross-sectional and longitudinal approaches[END_REF][START_REF] Blévin | Contaminants and energy expenditure in an Arctic seabird: Organochlorine pesticides and perfluoroalkyl substances are associated with metabolic rate in a contrasted manner[END_REF] ; glaucous gull in [START_REF] Melnes | Dissimilar effects of organohalogenated compounds on thyroid hormones in glaucous gulls[END_REF] ; common eider, black guillemot, black-legged kittiwake, glaucous gull, arctic skua and great skua in [START_REF] Haarr | DNA damage in Arctic seabirds: baseline, sensitivity to a genotoxic stressor, and association with organohalogen contaminants[END_REF] ).

Increased molecular oxidative damage and disruption of antioxidant defences are suspected as important mechanisms through which PFASs could be detrimental for cell function and, possibly, for organism health. Experimental evidence on laboratory models found that PFASs may increase production of reactive oxygen species (ROS), increase molecular oxidative damage and up-or down-regulate antioxidant defences. [START_REF] Liu | The thyroid-disrupting effects of long-term perfluorononanoate exposure on zebrafish (Danio rerio)[END_REF][START_REF] Yao | Genotoxic risk and oxidative DNA damage in HepG2 cells exposed to perfluorooctanoic acid[END_REF][START_REF] Eriksen | Genotoxic potential of the perfluorinated chemicals PFOA, PFOS, PFBS[END_REF] Further work found that the perfluoroundecanoic and perfluorododecanoic acids (PFUnA and PFDoA, respectively) are equally potent inducers of stress response genes relative to perfluorooctane sulfonic acid (PFOS) and perfluorononanoic acid (PFNA) and that the effect of carbon-chain-length was more important than the functional group in determining oxidative stress. 35 There is thus good reason to expect that long-chain PFASs might cause dysregulation of the oxidative homeostasis, leading to accumulation of oxidative damage to key biomolecules like proteins or nucleic acids. However, the effect of PFAS exposure on oxidative stress is almost unknown for wildlife. To the best of our knowledge, only two studies have addressed this topic in free-living animals. 36 did not find any significant relationship between plasma PFAS concentrations and the activity of the antioxidant enzyme superoxide dismutase in plasma of white-tailed eagle nestlings, while [START_REF] Haarr | DNA damage in Arctic seabirds: baseline, sensitivity to a genotoxic stressor, and association with organohalogen contaminants[END_REF] did not find any significant relationship between PFASs and amount of DNA damage in lymphocytes.

Long-lived species, like many polar seabirds that occupy high trophic levels, are exposed to a greater risk of accumulation and sensitivity to high concentrations of contaminants. In Svalbard (European Arctic), a number of studies showed that blacklegged kittiwakes (Rissa tridactyla, hereafter "kittiwake") are chronically exposed to a complex cocktail of organic contaminants and trace elements, which are known to correlate with physiological metrics, impaired individual fitness and population dynamics. [START_REF] Tartu | Endocrine and fitness correlates of long-chain perfluorinated carboxylates exposure in Arctic breeding black-legged kittiwakes[END_REF][START_REF] Tartu | The stress of being contaminated? Adrenocortical function and reproduction in relation to persistent organic pollutants in female black legged kittiwakes[END_REF][START_REF] Blévin | Perfluorinated substances and telomeres in an Arctic seabird: Cross-sectional and longitudinal approaches[END_REF][START_REF] Blévin | Contaminants and energy expenditure in an Arctic seabird: Organochlorine pesticides and perfluoroalkyl substances are associated with metabolic rate in a contrasted manner[END_REF][37][38][39] It is, however, unknown whether exposure of kittiwakes to PFASs is associated with markers of oxidative damage and antioxidant protection.

In this study, we have examined the correlations between blood-based markers of oxidative status and several PFAS compounds in male kittiwakes during the pre-laying period, while controlling for a number of potential confounding factors that might affect the oxidative status independently from PFASs (i.e., body condition, body size, both time and day of blood sampling, hormonal status; reviewed in 40 ). As with the hormonal status, we measured plasma levels of testosterone, baseline corticosterone and luteinizing hormone because prior work found large individual variation among kittiwakes [START_REF] Tartu | Endocrine and fitness correlates of long-chain perfluorinated carboxylates exposure in Arctic breeding black-legged kittiwakes[END_REF][START_REF] Tartu | The stress of being contaminated? Adrenocortical function and reproduction in relation to persistent organic pollutants in female black legged kittiwakes[END_REF]38 and significant effects on organism's oxidative status 40 , which could affect the relationships between PFAS and oxidative status markers. We focused on males because this investigation on oxidative stress is part of a larger project aiming at assessing the overall consequences (ornament coloration, fecundity, oxidative stress, sexual hormones) of PFASs exposure in males during the pre-laying stage (nest site defence, pair-bonding, copulation, nest building) a period during which males appear to be sensitive to pollutants. 38 We have also examined whether the effect size of the association between each oxidative status marker and each PFAS compound varies according to their carbon-chainlength (C8-14) because the toxicity of PFASs may increase with carbon-chain-length.

MATERIALS AND METHODS

Sampling

Fieldwork was conducted in 2016 on a colony of Arctic kittiwakes at Kongsfjord (78° 54' N; 12° 13' E), Svalbard. Blood samples were collected from 50 adult males during the pre-laying period (courtship and mating period), from 25 th May to 6 th June. Birds were caught on their nest with a loop at the end of a long pole. Within 3 minutes since capture 0.5 ml of blood were taken from the brachial vein using a heparinized syringe and a 25gauge needle. This blood sample was used to measure oxidative status markers and hormones. Straightaway, a second sample of venous blood (ca. 2 ml) was collected using another syringe and this blood sample was used to assess the PFASs burden. Then, tarsus, skull (head + bill) and wing length were measured using a calliper (nearest 0.1 mm) and body mass was taken using a Pesola spring balance (nearest 5 grams). Blood samples were stored on ice in the field. On average, blood samples were stored on ice for 3h40min 38s (min: 1h30; max 9h55) before being centrifuged and stored at -80°C. Plasma and red blood cells, obtained after centrifugation, were kept frozen separately, either at -80°C for subsequent oxidative status markers or at -20°C for PFAS analyses. All samples were analysed within 4 months since collection.

Hormone assays

Plasma levels of testosterone, baseline corticosterone and luteinizing hormone (LH) were measured by radioimmunoassay at the Centre d'Etudes Biologiques de Chizé following protocols previously validated. Briefly, testosterone and corticosterone were extracted using diethyl ether and ethyl ether, respectively. Plasma concentrations of all three hormones were measured by radioimmunoassay. As with corticosterone, a commercial antiserum against corticosterone-3-(O-carboxy-methyl) oxime bovine serum albumin conjugate (Biogenesis, UK) was used. The lowest detectable quantities significantly different from zero at a 90% confidence level were 0.05 ng/ml for testosterone, 0.4 ng/ml for corticosterone and 1.7 ng/ml for LH. All samples were analysed in duplicate.

Corticosterone and LH were analysed in a single run and the mean coefficient of variation was 4.9 and 12.0%, respectively. Testosterone was analysed in two runs and the coefficients of variation were 11.2 and 11.5%.

Oxidative status markers

One marker of plasma oxidative damage, one marker of plasma non-enzymatic antioxidant capacity and one red blood cell antioxidant enzyme were measured at the Centre d'Etudes Biologiques de Chizé using standard methods. Protein carbonyls (marker of oxidative protein damage) were measured using the Protein Carbonyl Colorimetric assay (Cayman Chemical Company, Ann Arbor, USA). This assay is based on the colorimetric method proposed by [START_REF] Levine | Determination of carbonyl content in oxidatively modified proteins[END_REF] . A same volume of plasma was used for all samples and the amount of carbonyls was standardised by the plasma protein concentration according to manufacturer's instructions. Protein carbonyls were derivatized to 2,4dinitrophenylhydrazone by reaction with 2,4-dinitrophenylhydrazine (DNPH). The absorbance was read at 370 nm. The extinction coefficient for DNPH (0.022/µM/cm) was used to calculate the concentration of protein carbonyls, which was expressed as nmol/mg protein (amount of carbonyls generated per unit of protein) or as total nmol obtained by multiplying the concentration of carbonyls by the concentration of plasma proteins (i.e., total amount of carbonyls in the sample, which is also dependent on the amount of substrates available, i.e., proteins). The mean coefficient of variation of duplicates was 11.5%. The metric expressed as nmol/mg indicates the amount of carbonyls that occurs in a same amount of protein, thus this is standardised by the amount of substrates (i.e., proteins) that can be carbonylated. The second metric expressed as total amount of carbonyls indicates the total amount of carbonyls that occurs in the tissue, which is influenced by the amount of proteins available. This second metric is also important because accumulation of carbonyls is detrimental for the cells 50 . The OXY-Adsorbent test (Diacron International, Italy) was used to quantify the non-enzymatic antioxidant capacity of plasma against HOCl. Values were expressed as either mM of HOCl neutralised or as mM of HOCl neutralised/mg protein to estimate the antioxidant potential of micromolecular antioxidants (e.g., vitamins, carotenoids, glutathione) without the contribution of proteins (i.e., non-enzymatic micro-molecular antioxidant capacity). The correlation between OXY values and protein concentration was actually high and significant (r = 0.76, p < 0.001), which is to be expected because plasma proteins, such as albumin, are prone to react with HOCl. Although free-radical trapping properties vary among proteins, standardising OXY values by the concentration of total plasma proteins gave us some kind of control about the contribution of proteins to OXY. The mean coefficient of variation of duplicates was 9.4%. The Ransel assay (RANDOX Laboratories, UK) was used to measure the activity of the antioxidant enzyme glutathione peroxidase (GPX) in haemolysates (red blood cells diluted with distilled water). Values were expressed as Units of GPX/mg of protein of haemolysate. The mean coefficient of variation of duplicates was 8.1%. The Bradford protein assay (Bio-Rad Laboratories, Hercules, USA) with bovine albumin as a reference standard was used to measure the concentration of proteins in both plasma samples and haemolysates.

PFAS analyses

Perfluoroalkyl carboxylic acids (i.e., PFASs with 8 or more carbons) were analysed in plasma at the Norwegian Institute for Air Research (NILU) in Tromsø, Norway. The following compounds were analysed in each plasma sample: perfluorooctanesulfonamide (PFOSA), perfluorobutanesulfonic acid (PFBS), perfluoropropanesulfonic acid (PFPS), perfluorohexanesulfonic acid (PFHxS), perfluoroheptanesulfonic acid (PFHpS), perfluorooctane sulfonic acid (PFOSlin), branched perfluorooctane sulfonic acid (PFOSbr), perfluorononane sulfonic acid (PFNS), perfluorodecane sulfonic acid (PFDcS), perfluorohexanoic acid (PFHxA), perfluoroheptanoic acid (PFHpA), perfluorooctanoic acid (PFOA), perfluorononanoic acid (PFNA), perfluorodecanoate (PFDcA), perfluoroundecanoic acid (PFUnA), perfluorododecanoic acid (PFDoA), perfluorotridecanoic acid (PFTriA), perfluorotetradecanoic acid (PFTeA) and two precursor compounds, the fluorotelomer sulfonates (6:2 FTS and 8:2 FTS). PFASs with concentrations below the limit of quantification (LOQ) were replaced with a value equal to (LOQ × detection frequency) when the detection frequency (percentage of detection) was > 50%. A 0.2 ml aliquot of plasma spiked with internal standards (carbon labeled PFAS) [START_REF] Hanssen | Partition of perfluoroalkyl substances (PFASs) in whole blood and plasma, assessed in maternal and umbilical cord samples from inhabitants of arctic Russia and Uzbekistan[END_REF] was extracted in methanol (1 ml) by repeated sonication and vortexing. The supernatant was cleaned-up using ENVICarb graphitized carbon absorbent and glacial acetic acid. Extracts were analysed by UPLC/MS/MS. Recovery of the internal standards ranged between 86.3% and 120%. Results were validated with blanks (clean and empty glass tubes treated like a sample) and standard reference material (1957 human serum from NIST) run with every 10 samples. The deviation of the target concentrations in the SRMs were within the laboratory's accepted range (69 -119%). Blanks varied between concentrations below the instrument detection limits and 30 pg/g and were applied as the LOD in the form of 3 times the average concentration.

Body size and body condition

A body size index was estimated using the PC1 from a PCA of tarsus, skull and wing length (loadings, 0.73, 0.83 and 0.45, respectively). As far as the body condition, we did not use the ratio of body mass onto body size nor the residuals of a linear regression of body mass onto body size because both indices have been criticized. [START_REF] García-Berthou | On the misuse of residuals in ecology: testing regression residuals vs. the analysis of covariance[END_REF] Rather we included both body mass and body size index as factors in the models. In doing so, the coefficient estimate of body mass is calculated considering the effect of body size, thus the outcome reflects the effect of body condition on the given marker of oxidative status. [START_REF] García-Berthou | On the misuse of residuals in ecology: testing regression residuals vs. the analysis of covariance[END_REF]

Statistical analyses

Generalized linear models were performed using the software STATISTICA 10 (StatSoft. Inc., Tulsa, OK, USA) to assess relationships between each oxidative status marker (protein carbonylation, non-enzymatic antioxidant capacity, non-enzymatic micro-molecular antioxidant capacity and glutathione peroxidase) and the following predictor variables: PFASs congener (PFOSlin, PFNA, PFDcA, PFUnA, PFDoA, PFTriA or PFTeA), body size index, body mass, hormonal status, day of blood sampling and time of blood sampling. All predictor variables were scaled to mean of 0 and standard deviation of 1. These predictor variables were included in all models because prior work showed that each of them can be significantly associated with oxidative status markers (reviewed in 40 ). The two metrics of protein oxidative damage (protein carbonyls per mg of protein and total protein carbonyls) were combined using the PC1 from a Principal Components Analysis (PCA) because they were highly correlated (r = 0.79, p < 0.001). The hormonal status was estimated using the PC1 from a PCA of corticosterone, testosterone and luteinizing hormone (loadings, 0.39, 0.61 and 0.84, respectively). A normal error function and an identity-link function were applied to models of non-enzymatic antioxidant capacity and GPX. A gamma error function and an identity-link function were applied to models of protein carbonyls and non-enzymatic micro-molecular antioxidant capacity. These functions were selected because the model had the best fitting to the dataset according to the Akaike Information Criterion. Preliminary analyses showed that the time elapsed from the collection of blood to its storage was not significantly correlated (pvalue > 0.45) with any marker of oxidative status nor hormones, thus it was not further considered in the analyses. Visual inspection of residuals, Q-Q plots and Cook's distance did not highlight any violation of normality and homogeneity of variance nor the presence of outliers (all samples were below a 0.5 Cook's distance). The variance inflation factor was always below 2, indicating that multicollinearity was low. The multicollinearity is thought to be high and problematic when the variance inflation factor is higher than 5.

The "compute.es package" [START_REF] Del Re | Compute.es: compute effect sizes. R package version 0[END_REF] in R [START_REF] Core Team | R: A language and environment for statistical computing[END_REF] was used to calculate the standardized effect size Hedges' g from test statistics of oxidative status markers that had significant associations with PFASs (i.e., protein oxidative damage and non-enzymatic micromolecular antioxidant capacity). The "forestplots function" of the "metafor package" in R was used to visualise boxplots of effect size and 95% confidence interval. Effect sizes were considered to be small (Hedges' g = 0.2, explaining 1% of the variance), intermediate (g = 0.5, explaining 9% of the variance) or large (g = 0.8, explaining 25% of the variance) according to 46 . PFDcS, PFHxA, PFHpA, PFOA, and the two precursor fluorotelomer sulfonates (6:2 FTS and 8:2 FTS) were below the detection limit in all the investigated samples. Protein oxidative damage was significantly higher in those birds having higher plasma concentration of PFDoA, PFTriA or PFTeA (Table 2). Effect size estimates increased with chain length of PFASs (indicating an increase of protein damage with chain length) and were significantly different from zero ) for PFDoA (95% confidence interval: 0.06 to 1.25), PFTriA (95% confidence interval: 0.01 to 1.20) and PFTeA (95% confidence interval: 0.29 to 1.54;Fig. 1). The non-enzymatic micro-molecular antioxidant capacity was significantly lower in those birds having higher plasma concentration of PFUnA, PFDoA or PFTeA (Table 2). There was also a near-significance tendency of the non-enzymatic micro-molecular antioxidant capacity to be lower in birds with higher plasma PFTriA (Table 2). Effect size estimates were larger for longer PFASs (indicating a decrease of micro-molecular antioxidants with chain length) and were significantly different from zero for PFUnA (95% confidence interval: -1.20 to -0.02), PFDoA (95% confidence interval: -1.35 to -0.14) and PFTeA (95% confidence interval: -1.23 to -0.04; Fig. 2). The non-enzymatic antioxidant capacity including the contribution of antioxidant of protein origin was significantly lower in those birds having higher plasma concentration of PFOSlin, but it was not associated with any other PFAS congener (Table 2). The activity of GPX was not associated with any PFAS compounds (Table 2). Finally, our models showed that (i) kittiwakes in poorer body condition had more plasma protein carbonyls, (ii) the non-enzymatic antioxidant capacity was higher in kittiwakes sampled later in the day, and (iii) the non-enzymatic micro-molecular antioxidant capacity was higher in kittiwakes having lower concentrations of hormones.

RESULTS

Concentrations of detectable PFASs are reported in

DISCUSSION

Our results provide the first evidence in wild vertebrates that the correlation between oxidative status markers and PFASs is stronger for long-chain congeners. We found that male kittiwakes having higher plasma concentrations of long-chain PFASs had more protein oxidative damage and less plasma antioxidants after controlling statistically for potentially confounding variables. The non-enzymatic antioxidant capacity (including antioxidants of protein origin) was significantly lower in kittiwakes having higher plasma concentration of PFOSlin. On the other hand, the activity of glutathione peroxidase in erythrocytes was not related to any PFAS congener. Effect size estimates were intermediate to large, indicating that PFASs explained from 9 to more than 25% of the variance in protein oxidative damage and non-enzymatic micro-molecular antioxidant capacity of plasma. 46 Intermediate effect sizes are suggested to be biologically meaningful because average proportions of variance explained in ecological, evolutionary and physiological studies is usually below 7%. [START_REF] Møller | How much variance can be explained by ecologists and evolutionary biologists[END_REF] Our effect size estimates were also larger than those found in the comparison of oxidative status markers between animals living in polluted (e.g., air pollution, heavy metals) and unpolluted sites. [START_REF] Isaksson | Pollution and its impact on wild animals: a meta-analysis on oxidative stress[END_REF] PFOSlin, PFUnA and PFTrA concentrations were the highest of all PFASs measured in the investigated samples, with a percentage contribution for each kittiwake ranging from 11.8 to 54.5%. The concentration of PFOSlin was higher than that previously recorded in males from the same kittiwake population in 2012 (13.4 vs. 10.6 pg/g ww in [START_REF] Blévin | Perfluorinated substances and telomeres in an Arctic seabird: Cross-sectional and longitudinal approaches[END_REF] ). In contrast, the average concentrations of PFUnA (10.3 vs. 12.1 pg/g ww in [START_REF] Blévin | Perfluorinated substances and telomeres in an Arctic seabird: Cross-sectional and longitudinal approaches[END_REF] ) and of PFTrA (8.6 vs. 11.6 pg/g ww in [START_REF] Blévin | Perfluorinated substances and telomeres in an Arctic seabird: Cross-sectional and longitudinal approaches[END_REF] ) were both lower in our study than in prior work. [START_REF] Blévin | Perfluorinated substances and telomeres in an Arctic seabird: Cross-sectional and longitudinal approaches[END_REF] One reason for such differences might be because [START_REF] Blévin | Perfluorinated substances and telomeres in an Arctic seabird: Cross-sectional and longitudinal approaches[END_REF] measured PFASs of male kittiwakes caught during the chick rearing phase. Work on male glaucous gulls during the incubation period in Svalbard found levels of PFOSlin similar to ours, while those of PFUnA (4.4 pg/g ww) and of PFTrA (3.9 pg/g ww) were much lower than those we recorded. [START_REF] Melnes | Dissimilar effects of organohalogenated compounds on thyroid hormones in glaucous gulls[END_REF] The strength of the correlation between oxidative status markers and PFASs increased with the chain length of the congener. While persistent in the environment, PFASs with fewer than eight carbons, such as PFHxA, and PFSAs with fewer than six carbons, such as PFBS, are generally less toxic and bioaccumulative in wildlife and humans, while exposure of laboratory animals to long-chain congeners produces detrimental reproductive, developmental, and systemic effects. [START_REF] Jensen | Emerging endocrine disrupters: perfluoroalkylated substances[END_REF][START_REF] Lau | Perfluoroalkyl acids: a review of monitoring and toxicological findings[END_REF][START_REF] Conder | Are PFCAs bioaccumulative? A critical review and comparison with regulatory criteria and persistent lipophilic compounds[END_REF][START_REF] Concawe | Environmental fate and effects of poly-and perfluoroalkyl substances (PFAS)[END_REF] For example, the toxic effects of PFASs on rat brain cells decreased with increasing carbon chain length. [START_REF] Berntsen | Time-dependent effects of perfluorinated compounds on viability in cerebellar granule neurons: dependence on carbon chain length and functional group attached[END_REF] Interestingly, the toxic effects of PFASs were attenuated by the antioxidant vitamin E, indicating a possible involvement of oxidative stress in the reduction of cell viability. [START_REF] Concawe | Environmental fate and effects of poly-and perfluoroalkyl substances (PFAS)[END_REF] Further work showed that, compared to short-chain PFASs, long-chain PFASs are stronger inducers of the response of genes regulating the cell oxidative status. 35 Effects of PFAS exposure on oxidative status have been moderately investigated in laboratory models and almost unexplored in wild animals. Thus, limited information is available for a comparison with our results and interpretation. Protein carbonylation arises from overproduction of ROS by metabolic reactions that use oxygen and shift the balance between oxidant/antioxidant statuses in favour of the oxidants. 50 Protein carbonylation also occurs when carbonyls are introduced into proteins through the reactions with lipid oxidative damage products (malondialdehyde and hydroxynonenal). 50 Carbonylation is mostly irreversible and results in alteration of protein structure and function. Only a small fraction of carbonylated proteins can be removed through proteasome-dependent proteolysis. Work carried out on laboratory models found evidence that exposure to PFASs may change expression of genes regulating proteasome activation and proteolysis. [START_REF] Lau | Perfluoroalkyl acids: a review of monitoring and toxicological findings[END_REF][START_REF] Nilsen | Comparative hepatic gene expression profiling of rats treated with 1H,1H,2H,2Hheptadecafluorodecan-1-ol or with pentadecafluorooctanoic acid[END_REF][START_REF] Zhang | Proteomic analysis of male zebrafish livers chronically exposed to perfluorononanoic acid[END_REF] It is, however, unclear, whether such changes in gene expression make proteins one important target of the pro-oxidant effects of PFASs. This is important because when protein carbonyls accumulate, they tend to aggregate leading to cell death, tissue injury and development of disorders. Several studies found higher amounts of plasma protein carbonyls in individuals affected by a given disease, [START_REF] Winterbourn | Elevated protein carbonyls as plasma markers of oxidative stress in acute pancreatitis[END_REF][START_REF] Hlaváčková | Enhanced plasma protein carbonylation in patients with myelodysplastic syndromes[END_REF] suggesting a potential role of protein carbonylation in disease progression.

Depletion of circulating non-enzymatic antioxidants might reflect increased oxidation due to reaction with ROS, reduced intake from diet or mobilisation of antioxidants from blood to other target tissues. Irrespective of the reason, prior work on other bird species found evidence that circulating antioxidants may be linked to important individual or population fitness-related traits. For example, [START_REF] Saino | Antioxidant defenses predict long-term survival in a passerine bird[END_REF] found that barn swallows (Hirundo rustica) with lower plasma non-enzymatic antioxidants had reduced probability of survival. [START_REF] Beaulieu | Integrating oxidative ecology into conservation physiology[END_REF] found that Gentoo (Pygoscelis papua) and Adélie (Pygoscelis adeliae) penguins from increasing populations had higher plasma non-enzymatic antioxidant capacity than those from decreasing populations. Metabolic activity is one important source of ROS production. Thus the association we found between PFASs and oxidative status markers might mirror an effect of PFASs on metabolism. Prior work on the same kittiwake population found a positive association between the long-chain PFTriA and resting metabolic rate in females but not in males. [START_REF] Blévin | Contaminants and energy expenditure in an Arctic seabird: Organochlorine pesticides and perfluoroalkyl substances are associated with metabolic rate in a contrasted manner[END_REF] Thus, the association between PFASs and oxidative status markers does not appear, at least for males, to be due to a dysregulation of general body metabolism. The lack of an effect on metabolic rate does not reject the hypothesis that ROS production might have been higher in the more contaminated birds. For example, PFASs might have localised effects on important ROS generators (e.g., mitochondria of red blood cells or of other target tissues) without compromising the metabolism of the whole organism. In vitro studies found that PFAS exposure may impair mitochondrial activity and lead to increased rates of ROS production. [START_REF] O'brien | Mitochondrial permeability transition as the critical target of N-acetyl perfluorooctane sulfonamide toxicity in vitro[END_REF] It is unclear why the activity of glutathione peroxidase was not associated with any PFAS compound. It might be that up-regulation of this enzyme might have been too costly for the birds given the imminent start of breeding activity or that any effects of PFASs on oxidative status did not come through the pathways involving glutathione peroxidase. The biochemical function of glutathione peroxidase is to reduce hydrogen peroxide to water and organic hydroperoxides to their corresponding alcohols. 50 Thus, we cannot exclude that results would have been different if another antioxidant enzyme with a different biochemical function would have been measured. There are, however, many discrepancies in the literature about the response of antioxidant enzymes to PFAS exposure. For example, prior work did not find any association between PFASs exposure and whole-body catalase activity in the planktonic crustacean Daphnia magna [START_REF] Li | Chronic effects of perfluorooctane sulfonate and ammonium perfluorooctanoate on biochemical parameters, survival and reproduction of Daphnia magna[END_REF] or plasma superoxide dismutase activity in white-tailed eagle (Haliaeetus albicilla) nestlings. 36 In contrast, exposure to PFASs affected catalase activity in hepatocytes of freshwater tilapia Oreochromis niloticus [START_REF] Liu | The thyroid-disrupting effects of long-term perfluorononanoate exposure on zebrafish (Danio rerio)[END_REF] and expression of antioxidant genes Sod1, Sod2, Gpx2 and Nqo1 in mouse pancreas. [START_REF] Kamendulis | Perfluorooactnoic acid exposure triggers oxidative stress in the mouse pancreas[END_REF] Irrespective of the mechanisms involved, the activity of GPX in erythrocytes does not appear to be an informative marker about the impact on oxidative status of the PFASs we have measured in this work.

In conclusion, our work shows that higher protein oxidative damage was found in those birds having higher concentrations of PFDoA, PFTriA and PFTeA. Lower plasmatic non-enzymatic micro-molecular antioxidants was found in those birds having higher concentrations of PFUnA, PFDoA and PFTeA. Experimental work will be needed to ascertain whether the correlation between individual PFAS burden and oxidative status markers reflects a direct toxic effect of PFASs on oxidative homeostasis. It will also be important to determine whether increased oxidative damage or decreased antioxidant defences turn into a reduction in survival probability or lifetime reproductive success.

Resistance against oxidative stress may also decrease with chronological age. [START_REF] Marasco | Environmental conditions can modulate the links among oxidative stress, age, and longevity[END_REF] Thus, we highlight the importance of assessing in future studies whether the effects of PFASs on oxidative status markers are stronger in older individuals.

  (i.e., oxidative status marker) is calculated considering the effect of body size, thus it actually indicates the covariation between a given marker and the individual body condition.[START_REF] García-Berthou | On the misuse of residuals in ecology: testing regression residuals vs. the analysis of covariance[END_REF] GPX = glutathione peroxidase; N-E Antioxs = non-enzymatic antioxidant capacity of plasma; N-E Micromol Antioxs = non-enzymatic micro-molecular antioxidant capacity of plasma; ce ± se = coefficient estimate ± standard error.

Figure captions Figure 1 .

 captions1 Figure captions

Figure 2 .

 2 Figure 2. Mean estimates of effect size and 95% confidence interval were calculated from

Figure 2

 2 Figure 2

  Table 1 together with other variables measured in kittiwakes. Six out of 20 PFASs (i.e. PFOSlin, PFNA, PFDcA, PFUnA, PFDoA, PFTriA) were detectable in all individuals, while one PFAS (i.e. PFTeA) was detectable in 33 out of 50 individuals. PFOSlin, PFUnA and PFTriA concentrations were the highest of all PFASs measured in the investigated samples, with a percentage contribution for each kittiwake ranging from 23.4 to 54.5%, from 20.3 to 34.8% and from 11.8 to 35.2%, respectively. The percentage contribution of all other detected PFASs ranged between 0.1 and 10.6%. PFOSA, PFBS, PFPS, PFHxS, PFHpS, PFOSbr, PFNS,

Table 1 .

 1 Descriptive statistics of all poly-and perfluorinated alkyl substances congeners, oxidative status markers (i.e., responde variables) and several individual metrics (potential confounding factors) measured in 50 male black-legged kittiwakes from Svalbard.

	Variable

Table 2 .

 2 The table shows the outcomes of generalized linear models performed to test the effect of each PFASs congener on oxidative status markers. Each model also included a number of potential confounding factors that may affect oxidative status markers independently from PFASs. Significant effects are shown in bold type. Note that the coefficient estimate for the relationship between body mass and the dependent variable

ACKNOWLEDGEMENTS

This project was financially and logistically supported by Institut Polaire Français (IPEV project 330 to O. Chastel) and by Research Council of Norway (RCN, Arctic Field Grant n° 256934 from the Research Council of Norway provided to P. Blévin). P. Blévin was funded by a PhD grant from University of La Rochelle. Additional funding was provided by the Hazardous Substances Flagship (the Multistress project) at the Fram center in Tromsø and by the Contrat de Plan État-Région (CPER) ECONAT, at the CEBC. We thank the Polar Norwegian institute for their logistic help in the field. We also thank the Littoral Environnement et Sociétés (Université La Rochelle) for allowing us to use the plate reader to measure one of the markers. This study was approved by the Norwegian Food Safety Authority (FOTS ID 8679) and the Governor of Svalbard. We thank Ségolène Humann-Guilleminot, Bruno Michaud, Solveig Nilsen and Ingrid Gabrielsen for their great help in the field. We thank all the staff of the NILU for their assistance during PFASs analysis; C. Trouvé and C. Parenteau from CEBC for hormone assay.

Finally, we thank four anonymous reviewers for providing their comments on our work.

COMPETING INTERESTS

The authors declare no competing financial interests.

PFNA and PFHxA in human HepG2 cells. Mutation Res./Genet. Toxicol. Environ.

Mutag. 2010, 700, 39-43.

(35). Nobels, I. ; Dardenne, F. ; De Coen, W.; Blust, R. Application of a multiple endpoint bacterial reporter assay to evaluate toxicological relevant endpoints of perfluorinated compounds with different functional groups and varying chain length.

Toxicol. In Vitro 2010,


Dj Boussaa

K Dang Van

P Labbe

H T Tang�

FINITE PURE BENDING OF CURVED PIPES

Abs�act-We present � n � riginal treatment for the fi nite bending of curved pipes with arbitrary cross sectiOns. The curved pipe IS successively regarded as a three-dimensional continuum and a shell and a formulati � m is proposed for eac? model. We show that, from a numerical point of view, the finite bending problem IS reductble to an axisymmetric analysis augmented with I d.f. We also show how to take advantage of this analogy to solve the bending problem using standard axisymmetric FE routines.

l. INTRODUCTION

Curved pipes are among the most vulnerable com ponents of piping systems. When compared to straight pipes of the same cross section and length, real-life curved pipes prove much more flexible and suffer much more severe stresses under the same bending moment. This contrast is due to the ovaling of the curved pipe cross section during bending, a phenomenon that thus cannot be accounted for by the usual beam theory_ Even though the experimental discovery of the high flexibility of curved pipes is generally credited to Bantlin (1910), the above basic facts were actually known by the end of the 19th century. Piping systems of steamers were designed using beam theory, the extra fl exibility of the curved parts being acounted for by multiplying their beam-stiffness by an empirical coefficient of 6/ 10, well-suited to the curved pipe geometries used at that time in naval engineering [I].

As regards modeling, the earliest conclusive contribution, in the small displacement setting, is due to von Karman (191 1 ). Many attempts however preceded this work. In 190 I, as is reported in its Comptes Rendus de l'Academie des Sciences Always in the small displacement setting, Clark et al. [START_REF] Clark | Bending of curved tubes[END_REF][START_REF] Clark | Stress and deformations of toroidal shell problems[END_REF] proposed general formulae which are commonly used in the design of piping systems. An illustration is provided by the rigidity factor l'f. Let 1/ EI denote the beam flexibility of the curved pipe and lll'fEI its effective flexibility, respectively. Clark t To whom correspondence should be addressed.

and Reissner [START_REF] Clark | Bending of curved tubes[END_REF] showed that IJ, for a wide range of geometries, is given by

(I)
where fJt is the radius of curvature of the curved pipe, b the radius of its cross section assumed to be circular h its thickness assumed to be constant and v it� Poisson's ratio.

In the large displacement setting, many researchers have paid attention to the problem ((4-7] and refer ences therein). Except for the last reference, the treatment is purely analytical, involves many assump tions, and is based on Fourier series expansions which impose serious limitations (e.g. perfect ge ometries required) and suggests that more flexible solution techniques, such as the FE method, could prove better suited_ Guidelines are given in Ref. [START_REF] Greenbaum | Pure moment loading of axisymmetric fi nite element models[END_REF] to treat the curva ture controlled case using a standard axisymmetric FE analysis. A full formulation restricted to the small displacements case is developed in Ref. [START_REF] Cook | Axisymmetric fi nite element analysis for pure moment loading of curved beams and pipe bends[END_REF]. Moreover, it is shown in this reference that the FE dicretized pure bending problem may be regarded as an axisym metric analysis augmented with I d.f. In some re spects, our work may be inscribed as an extension of these two references, although we followed a different approach from those developed therein.

The paper is organized as follows. The second section is devoted to the statment of the problem in the full three-dimensional framework. The simi larities between this problem and an axisymmetric analysis are emphasized by a systematic decompo sition of the kinematics of the bending problem into an axisymmetric part and additional terms. In the third section we reduce this three-dimensional formu lation to a shell formulation. In the fourth section, by resorting to a FE discretization, we derive a matrix statement of the problem. The similarities between the present problem and an axisymmetric analysis are accounted for and the discretized formulation is shown to be reducible to an axisymmetric problem augmented with I d.f. Finally, we present some numerical comparisons with existing results, followed by a brief conclusion.

THE CURVED PIPE AS A THREE-DIMENSIONAL CONTINUUM

Notations

Throughout the paper, the symbol ® will denote the standard tensor product of two vectors ((x, y)�--+ x®y) or two second order tensors ((X, Y)�--+ X®Y), the dot the Euclidean inner product of two vectors ((x, y)�--+ x, • y) or two second order tensors ((X, Y)t--+ X • Y = trace(XYT)).

Kinematics

In its initial configuration assumed stress free, the curved pipe under study, shown in Fig. I with the main notations, is supposed to be a three-dimensional body occupying a sector no of an axisymmetric body.

Let 1:0 denote a typical cross section, 1:0°1 and I:i;" the parts of 1:0 occupied by the matter and the hole, respectively, 9'�"' and .'l'g" the inner and outer skins,

m = 0 + re,(8) + zez. ( 4
)
The pure in-plane bending of the curved pipe can be described conveniently with the following relations:

{r= r(R, Z, t),

8 = (1 + a(t))e, z = z(R, Z, t),
where a(t) is a scalar function of time.

Remarks

(5)

(I)

No assumption was made on the cross section: the wall may be thin or thick, with constant or varying thickness.

(2) The kinematics, eqn (5), includes the usual assumptions for the in-plane bending problem. As an illustration, the assumption that cross sections remain plane is accounted for by the dependence of () on e solely, the assumption that all the cross sections deform likewise in their plane by the inde pendent of rand z of e, the assumption of uniformity of the stretching of the parallels by the linearity between () and e' etc.

(3) Fixing a(t) to zero in eqn ( 5) yields an axisym metric evolution. Accordingly, releasing a(t) adds only I d.f. to those of this companion axisymmetric evolution.

(4) In the small deformation setting, the above assumptions will make it possible to exhibit one solution to the problem; the existence and uniqueness of the solution will ensure that the exhibited solution is the solution. In the large displacement setting, imposing eqn (5) beforehand means that only rotationally symmetric instability modes can be accounted for. These modes are closer to those observed experimentally. 4) with respect to its arguments and insertion of the results into eqn ( 6) yields

The expression of F-1 can be obtained from that of F by substituting in eqn [START_REF] Boyle | The finite bending of curved pipes[END_REF] upper case letters to lower case letters and vice versa.

From now on, all the kinematical quantities will be decomposed into axisymmetric and additional terms.

Let Ft be the deformation gradient associated with the companion axisymmetric evolution, that is

The deformation gradient F may be decomposed as follows:

A similar multiplicative decomposition can also be applied to J, the determinant of F:

with bmt the VVF associated with the axisymmetic companion transformation.

If the initial configuration is considered as the reference configuration, the gradient of a VVF may be written as or + bO-eO ®Ez. az [START_REF] Destuynder | Modelisation des Coques Minces Elas tiques[END_REF] If, on the contrary, it is the current configuration that is taken as the reference confi guration, eqn ( 14) is simplified to

To simplify the notations, we introduce where lt is the determinant of Ft.

The virtual velocity fi elds (VVF) compatible with the kinematics [eqn [START_REF] Crandall | The infl uence of pressure on the bending of curved tubes[END_REF]] are of the form bm=c>re,+rb!1e0+bze.,

(II)

with the rotational symmetry conditions:

lc> r = c>r(R, Z}, c>O=c>ae, c>z = bz(R, Z), ( 1 2)
where c>r and c>z are arbitrary scalar functions of R and Z, and c>a an arbitrary constant. Note that the right-hand side of eqn (l l) can be written as The applied loads are of two types: those (typical in three-dimensional continuum descriptions) given pointwise, and those (less common) prescribed on average [START_REF] Ericksen | Special topics in elastostatics[END_REF], which, under the present context, are the bending moment .A and the normal force .AI, defi ned on the current confi guration by

(22) (2 3)
The fi rst type of boundary conditions in our problem is the vanishing of the stress vector on the outer and inner skins of the curved pipe and the shear force at the end cross sections:

Tn, = 0 on .9'int and .9'�xt,

(2 4)
The boundary condition given on average corre sponding to a pure bending is the vanishing normal force, and a prescribed moment in the case of moment controlled loading. In the case of curvature controlled loading, the moment is an unknown variable of the problem which needs to be calculated.

The forms ofT and S are needed in the sequel. The rotational symmetry and the vanishing shear at both ends of the pipe imply that at each point in the body, Equations ( 26), ( 19), [START_REF] Boyle | The finite bending of curved pipes[END_REF] and the remark on F-1 that follows eqn [START_REF] Boyle | The finite bending of curved pipes[END_REF] yield that at each point in the body:

2.4.

Constitutive equations

The form of the constitutive equations is crucial for the quality of the predictions of the model as well as for the numerical strategy to be implemented. We deliberately choose a simple form which has the advantage of involving only the standard axisymmet ric small displacement stiffness matrix and the initial stress matrix, which are easily computable using standard FE routines. Discussion of other types of constitutive equations and the corresponding implementations can be found in Ref. [11]. We assumed the following constitutive equation with the current configuration as the reference confi gur ation [START_REF] Bathe | Static and dynamic-geometric and material nonlinear analysis[END_REF]:

P =CD, ( 28
)
where C is the standard constant elastic modulus tensor of the material which is assumed to be homo geneous and isotropic. (31)

Jno

Differentiating this equation with respect to time and then choosing the current configuration as the reference configuration yields the following rate equation:

-a I 8DI:. (D o ®S)Do dO, Jn,

-ba l DI: • (D0 ®S)D0 dO, Jn,

In obtaining eqn (32), we made use of the following relation:

which can be proved by a direct application of the principle of virtual work with the (noncompatible) virtual velocity fi eld r8 2e,.

Inserting eqns (2 1) and (2 8) into eqn (32) and applying the decomposition, eqn [START_REF] Dhatt | Une Presentation de Ia Methode des Elements Finis[END_REF], to D and taking into account the equality between P and S Fig. 2 . Portion of the cross section. Some notations.

since the current configuration is chosen as a refer ence, yields the expression of the principle of virtual work given in Box 1. We will now develop a shell model for our prob lem. This can be done in several ways. One approach consists of developing a formulation ab initio stated in a two-dimensional framework. An alternative approach, which we have adopted, consists in starting from the three-dimensional formulation in Box I and reducing it to a shell formulation. Note that, by doing so, and owing to eqn [START_REF] Axelrad | Elastic tubes[END_REF] and its consequences, this amounts to assuming that bm� and not 8m taken as a whole is a shell velocity field (say of Kirchhoff-Love type). This contrasts with the usual formulations in which bm is entirely taken as a shell velocity field. This latter assumption gives rise to coupling terms between the longitudinal and hoop directions in the strain rate tensor which are usually neglected without justification as discussed in Ref. [START_REF] Axelrad | Elastic tubes[END_REF].

0= -L. bD� . CD� dO, -i L�P•bL�dO, a, -ba i D� • CD0d0, -it i bD� • CD0dn, n, n, -2ba i D� • (D0®P)D0d!l,
The notations to be used are defined in Fig. 2. We shall denote the reference surface by !/', the outward normal to !/' by n, the spatial derivative defined over !/' by V y and � as the third coordinate measured from !/. Moreover, we define the tensor p as:

(33)

The derivation of a two-dimensional model from a three-dimensional one is achieved by reducing the three-dimensional fields and integrals defined over n to surface fi elds and integrals. Kinematical fields Dt, Lt and their virtual counterparts <5 Dt and b Lt are treated fi rst. In the shell framework, these fields are usually reduced to affine fu nctions of �.

(

) 34
where y and 'f.. , respectively the membrane strain rate and the curvature rate tensors, are fi elds defi ned over the reference surface !/'. We have adopted the Kirchhoff-Love approximation. However, the form of eqn ( 34) is common to many shell models starting from a three-dimensional description.

The integrals defined over n are reduced to surface integrals by the following approximation on p. The equality L ( . ) d !l, = LJ�1 :1 2 ( . ) det(p) d� dY'. (35) where ( • ) is any suitable argument, is approximated by

r <.) dn, � f f h / 2 <.) d� dY. (36) Jnt �,

� h t 2

i.e. det(p) = I. This approximation is compatible with many models (Koiter, Donne!, etc.) [START_REF] Destuynder | Modelisation des Coques Minces Elas tiques[END_REF].

A last approximation, classical in shell theory, consists in approximating C by the plane-stress elastic modulus tensor which is denoted hereafter by the same symbol. On the other hand, it is convenient to introduce

A shell version of the three-dimensional formu lation

We once again assume that we can solve the companion axisymmetric problem and will focus our attention on the additional terms 3-7 in Box I. To make them shell terms, one has to approximate volume integrals in Terms {3, 4 and 6} by surface integrals. Equation (34) will make it possible to write .:5DI and DI in terms of surface fields.

If we designate by y0 the restriction of D0 to the reference surface !/, it is easy to prove that This reduces Term {3} to The second term of {4} is likewise computable. So we have

{3} = -ba f Y I • CmYo dY ,
{4} = -2.:5a( L , Y I • (Yo®N)yodY, + L . li • (Yo®M)yodY) -u(L. by); • CYo ®N)yo dY, + L • .:5zi • (Yo ®M)Yo d!/ ) (42)
Term { 5} is to be kept in its original form in Box l.

Replacing C by its value in the term { 6} yields {5} = -baa(� f h dY, ) .

FINITE ELEMENT SOLUTION PROCEDURE

We give in the sequel the discretized version of the full three-dimensional formulation in Box I, and develop a numerical solution procedure based on FE techniques for spatial discretization. A discretized version of the shell formulation in Box 2 can be obtained in the same manner.

Spatial discretization

We have mentioned that, if a and ba are assumed to be zero, the problem is reduced to an axisymetrical analysis, and then the kinematical unknowns are the velocities of the particles at any cross section. If moreover, the cross section is discretized into finite elements, the set of unknowns reduce to the set of mesh nodes velocities, represented as usual by a vector v� = { . . . , r, z, ... }, (44) where f; and i, are the components of the velocity vector of the ith node.

Releasing a adds I d. f. Accordingly, the set of unknowns of the problem is VI augmented by a; that is,

(45)
The following notations are needed. Let .:5 VI be the virtual counterpart of VI and F0 be the force vector

About the numerical implementation

In this section, we restrict ourselves to the discus sion of two points related to the grafting of the numerical procedure described above onto standard FE codes. The first is concerned with the "push forward" of the second Piola-Kirchhoff stress tensor between two consecutive configurations. This oper ation is to be achieved through (68) Using eqns [START_REF] Cook | Axisymmetric fi nite element analysis for pure moment loading of curved beams and pipe bends[END_REF] Using a thin shell theory developed by Reissner, Boyle [START_REF] Boyle | The finite bending of curved pipes[END_REF] proposed a combined analytical-numerical formulation. In the latter, equilibrium was written in a strong form: the system of partial differential equations which express this equilibrium was reduced to a system of first order nonlinear differential equations which is solved by means of the multi-point shooting method.

In this respect, our approach is notably different. Among the differences, in addition to the strong dissimilarity between the two approaches to establish the governing equations, one can note weak formu lation against strong formulation, and FE against multi-point shooting method.

Comparison with existing results

Figures 3456789show the relations between m and a we obtained by the shell version of our formulation. In these figures are also reported the results obtained by Boyle, and those corresponding to eqn (76). On these figures, we have also added the results that were obtained by skipping the push-forward procedure of the second Piola-Kirchhoff stress tensor. The motivation for the introduciton of this approximation is the way the behavior is formulated in [START_REF] Reissner | On fi nite bending of pressurised tubes[END_REF]. It seems to us that in this reference, the constitutive equation amounts to relating directly the Cauchy stress tensor T to the Green-Lagrange strain tensor (E = l/2(FTF-1)) through T = CE. Accordingly, our formulation could be better compared to the existing ones by adopting the same constitutive equations. When the push-forward procedure is skipped, in which case the second Piola-Kirchhoff and the Cauchy stress tensors are identical, the various approaches give very similar results. However, with the complete version of our algorithm, the limit points tend to occur for lower values of the parameter a. Note that the coordinates of the calculated limit points are outside the range of elasticity of real life curved pipes.

Accordingly, a more realistic modeling of the bending problem should include material nonlineari ties, and, if need be, nonlinear effects due to internal pressure (follower forces). In this paper, an original formulation for the in-plane bending of curved tubes including geometri cal nonlinearities has been proposed. It has been shown that, from a numerical point of view, the bending problem could be reduced to an axisym metric problem augmented with I d.f. This similarity has been turned into account for the numerical treatment of the problem of bending using standard axisymmetric FE routines.

(

  pp. l057-1058), the French Academy of Sciences acknowledged the contribution of naval engineer Marbec to justify the flexibility coefficient of 6/ 10.

1/ 10

 10 Fig. I. General view and typical cross section of the curved pipe.

  of curvature defined by and M the position of a typical particle with cylindri cal coordinates (R, e, Z), and a normalized natural basis (ER, E8, E2):Under external loads, the body deforms and a par ticle with initial position M occupies the new position m(r, 8, z) with a normalized natural basis (e, e0, ez):

( 5 )

 5 Rigid body displacements compatibile with eqn (5) are translations along the Z-axis. A simple means to get rid of them is to fix the displacements along the Z -direction of one parallel. Let V no denote the spatial derivative referred to the initial configuration. The deformation gradient, denoted by F, is by definition: Direct differentiation of eqn (

( 16 )

 16 Let b D and 8 Dt denote the symmetric part of b L and bLt, respectively. Then the following relation holds: (17) where (18) (IO) By substituting the real velocity field riJ for bm, we define in the same manner L, D, Lt and Dt.

2. 3 .

 3 External loadingThe Cauchy, first and second Piola-Kirchhoff stress tensors will be denoted by T, Sand P, respect ively. They are related by s = JTF-IT = FP.

  the current confi guration is considered as the reference confi guration, the relations, eqn (19), reduce to T=S = P.

  (20) Differentiating eqn (19) with respect to time and (1 3) choosing the current confi guration as the reference configuration leads to (21)

2. 5 .

 5 Weak form of equilibrium The local equilibrium equation written in the current configuration with zero body forces reads div0 , T=0. (29) Multiplying this equation by an arbitrary virtual velocity field and using the usual arguments leads to Inserting the boundary conditions, eqns (24) and (2 5), into eqn (30), the vanishing of the normal force and the value of the moment [eqn (22)], and writing the virtual work of the internal forces on the initial confi guration, gives l S•bFdO+ba.AI/1,=0.

-

  2a i bD� • (D0®P)D0 dO,a, +baa .Itt/!, -baa l D0 • CD0 do, a, +ba.itt/1 , Box I. Pnnciple of vtrtual work.3. THE CURVED PIPE AS A SHELL

h3

  Cm = h C and Cr = -C 12 '(37) the membrane and fl exural stiffnesses, respectively.

• 6 aFig. 3 .

 63 Fig. 3. A.* = 0.2.

2 +

 2 fl', -a L .:5y i . CmyodY,. (39) Resorting to eqn (34) once again yields where f f XI• (Yo ®M)y0 d Y, (40)

{6}Box 2 .

 2 (43) 1v .'1', With the hypothesis det(p) = l, the integral of the right-hand side of eqn (43) is the volume '1"501• Finally, Term {7} is to be kept unchanged. Putting Terms {3}-{6} together yields the formu lation in Box 2. "axisymmetrical" terms -2t5 a (L, Yr . (Yo ®N)yo dSI', + L . Xr • (Yo®M)yodSI') { 2} -2a(L , t5yr . (Yo ®N)yo dSI' + L . t5xr • (Yo®M)yo dSI') {3} {4} -t5 aa ( � f h dSI',) 1-v y , {5} Shell version of fo rmulation m Box I.

Fig. 4 . 2 •

 42 Fig. 4. 2 • = 0.5.

6 8 thenFig. 5 .

 685 Fig. 5. 2 * = !.

Fig. 6 .

 6 Fig. 6. A* = 2.

  provided b «fll and A.*= fll hjb2 � )12(1 -v2)j2.

Fig. 7 .

 7 Fig. 7. A*= 5.

  Fig. 8

8

 8 Fig. 8

.

�Fig. 9 .

 9 Fig. 9. A*= 20.

associated with the "initial stress" CD 0 in a purely axisymmetric evolution: the vector F 0 is such that for every velocity fi eld bVr, [ <}Dr• CD 0 dil. = FJbVr.

Jo, (46)

The expression for F0 is well known and is given by (47) where Br is the strain-displacement matrix for a purely axisymmetric transformation Jo, • Let U0 be a fi ctitious displacement associated with Fo:

(48) These vectors will be used to restate Terms { 1 }-{7} in Box 1. Term { l} now reads (49) where K� is the standard axisymmetric small displace ment stiffness matrix. By analogy with this equation, Term {2} can be rewritten as (50

where K� is the standard initial stress matrix [15,[START_REF] Bathe | Finite Element Procedures in Engineering Analysis[END_REF].

Term {3} reads {3} = -FJ(baVr +a<5Vr).

(51)

Term {4} can be evaluated in the same way as F0• Let P0 be defined as follows: Physically, J appears to be the instantaneous stiff ness to bending of the curved pipe.

Time discretization

A one-iteration incremental scheme with very small increments is implemented for time-discretization. This scheme yields the displacement increment L\U resulting from a force increment L\Q, as follows [START_REF] Dhatt | Une Presentation de Ia Methode des Elements Finis[END_REF]: (67)