Simon Pouil
email: simon.pouil1@gmail.com

Paco Bustamante

Michel Warnau

Marc Metian

Overview of trace elements trophic transfer in fish through the concept of assimilation efficiency

Keywords: Assimilation efficiency, Fish, Food, Metals, Experimental studies

Among the different accumulation pathways of trace elements, water has initially retained the attention of the scientific studies on fish but the trophic transfer gradually gained considerations for now being generally identified as the major contribution pathway for trace element intake.

The experimental approach is currently the most appropriate way to precisely quantify the trophic transfer of trace elements in fish. Thus, the assimilation efficiency (AE) of trace elements from ingested food is a commonly-determined parameter. However, there are still some discrepancies in the literature regarding the definition and the determination of AE in aquatic organisms and especially in fish. In this context, this review gathers the information about this concept as well as a description of the methods and protocols used to quantify the AE of trace elements thanks to experimental studies. It also looks over the main results concerning trace element AE in fish from the available literature. Most studies reporting AE considered the effects of biotic factors, especially the influence of the quality of the food.

Abiotic factors have received less attention although they affect fish physiology and by extension potentially affect AE of trace elements. The need for further investigations is thus rising from the review, especially looking at the influence of abiotic factors such as temperature, salinity or pH on trace element AE or in the context of multiple stressors co-occurrence; this will help the better understanding trophic transfer of trace elements in fish and thus the overall their bioaccumulation in fish.

INTRODUCTION

In the field of ecotoxicology, the first use of fish in scientific studies originated in the 1930s with the purpose of testing the effect of various chemicals on them, including toxic trace elements usually released in aquatic environments by anthropogenic activities [START_REF] Valavanidis | Metal pollution in ecosystems, ecotoxicology studies and risk assessment in the marine environment[END_REF]. Since then, fish have proved their suitability for ecotoxicological studies [START_REF] Braunbeck | Fish Ecotoxicology[END_REF] given their broad species diversity, the wide range of diets (from algae to other fish) and their broad geographical distribution in various environments. Furthermore, the relevance of fish in ecotoxicology is also connected to their ecological and economic importance [START_REF] Holmlund | Ecosystem services generated by fish populations[END_REF][START_REF] Tidwell | Fish as food: aquaculture's contribution[END_REF].

Fish accumulate trace elements through both the dissolved and particulate pathways but the diet appears to be the predominant source for a series of elements [START_REF] Xu | Exposure and potential food chain transfer factor of Cd, Se and Zn in marine fish Lutjanus argentimaculatus[END_REF][START_REF] Mathews | Dominance of dietary intake of metals in marine elasmobranch and teleost fish[END_REF]. Therefore, understanding the trophic transfer of trace elements is a key aspect to assess the accumulation capacities in fish and their exposure to contaminants. Since distinction of the food contribution to the global bioaccumulation is complex to perform on individuals collected in the field, experimental approach appears to be the best option to assess unambiguously the trophic transfer of trace elements in fish [START_REF] Wang | Assimilation efficiencies of chemical contaminants in aquatic invertebrates: a synthesis[END_REF].

One of the most relevant parameters to quantify trophic transfer of a contaminant its the assimilation efficiency (AE) from ingested food. AE is a first-order physiological parameter that can be quantitatively compared among trace elements, fish species, diets and environmental conditions [START_REF] Wang | Assimilation of trace elements and carbon by the mussel Mytilus edulis: effects of food composition[END_REF][START_REF] Croteau | Determining metal assimilation efficiency in aquatic invertebrates using enriched stable metal isotope tracers[END_REF]. Because dietary trace elements bioaccumulation is directly related to AE, this parameter is important to understand and predict global trace elements uptake [START_REF] Wang | Assimilation of trace elements and carbon by the mussel Mytilus edulis: effects of food composition[END_REF][START_REF] Luoma | Why is metal bioaccumulation so variable? Biodynamics as a unifying concept[END_REF][START_REF] Croteau | Determining metal assimilation efficiency in aquatic invertebrates using enriched stable metal isotope tracers[END_REF]). This parameter is thus widely used in modern ecotoxicological studies. However, the concept of AE appears sometimes unclear in the literature due to some discordances in the way it is defined.

This review provides a general definition of the concept of AE, critically examines the methodologies used for AE measurements in fish to date and discusses the recent improvements made on the different methods. It also extensively analyzes results of trace elements' AE in fish reported in the literature. This review finally presents a summary of perspectives for guiding future studies on the subjecting and in complements the review made 18 years ago on AE in invertebrates [START_REF] Wang | Assimilation efficiencies of chemical contaminants in aquatic invertebrates: a synthesis[END_REF].

The need to clearly define the concept of assimilation efficiency

Assimilation efficiency (AE) is a physiological parameter determined to understand the trophic transfer of chemicals in organisms. However, as [START_REF] Wang | Assimilation efficiencies of chemical contaminants in aquatic invertebrates: a synthesis[END_REF] have already pointed out in their review, there are still discrepancies in experimental studies regarding the definition of the AE. According to these authors, "In bioenergetic studies, absorption of an element or compound equals total ingestion of the substance minus its quantity in faecal matter and is the sum of assimilation and post-digestive soluble excretion (i.e., loss of material into the dissolved phase after post-ingestive metabolism)". According to this definition, AE is the fraction of ingested element or compound that is incorporated into biological tissue, whereas absorption efficiency is the fraction of the ingested element or compound that passes through the gut epithelium by passive and active transports [START_REF] Brett | Physiological Energetics[END_REF][START_REF] Penry | Applications of efficiency measurements in bioaccumulation studies: Definitions, clarifications, and a critique of methods[END_REF]. Assimilation thus equals absorption minus defecation and excretion. This AE definition is in line with [START_REF] Warnau | Biokinetics of selected heavy metals and radionuclides in the common Mediterranean echinoid Paracentrotus lividus: sea water and food exposures[END_REF], which, in essence, indicates that the AE could be defined as the fraction of the ingested material that is tightly bound (i.e. incorporated) in the organs and tissues of a given organism. From a theoretical point of view, the difference between absorption and assimilation is obvious, but in practice, it is difficult to delineate quantitatively these two mechanisms at the whole-body level because during the gut transit, these physiological processes can occur at the same time. Thus, another physiological parameter is used to determine the required time to assess AE (e.g. [START_REF] Ni | Transfer of Cd, Cr and Zn from zooplankton prey to mudskipper Periophthalmus cantonensis and glassy Ambassis urotaenia fishes[END_REF][START_REF] Xu | Exposure and potential food chain transfer factor of Cd, Se and Zn in marine fish Lutjanus argentimaculatus[END_REF]: the gut transit time (GTT; i.e. duration that a food ration spends in the digestive tract between its ingestion and its defecation). Indeed, it is during that phase that the absorption of chemicals takes place. This method has some limitations which must be taken into account. Indeed, during the GTT, it is difficult to guaranty that only absorption of the ingested compounds takes place since excretion can also already intervene; hence a part of the absorbed fraction can be already excreted. Indeed, after intestinal absorption compounds such as trace elements are conveyed through the bloodstream to the liver and can be directly excreted via the biliary secretions discharged in the intestine or, latter through the gills and the urine [START_REF] Wood | An introduction to metals in fish physiology and toxicology[END_REF]. Furthermore, there are some assumptions that egestion directly from the gut can occur through compounds secreted with digestive juices or sloughed inside detached enterocytes and then evacuated via the faeces or rectal fluid [START_REF] Wood | An introduction to metals in fish physiology and toxicology[END_REF]. In addition, we assume that part of the non-assimilated fraction might remain a bit longer in the digestive tract, associated to the intestinal mucus that can play a regulatory role in the absorption of ingested elements such as trace elements [START_REF] Warnau | Biokinetics of selected heavy metals and radionuclides in the common Mediterranean echinoid Paracentrotus lividus: sea water and food exposures[END_REF][START_REF] Bury | Nutritive metal uptake in teleost fish[END_REF]. This situation might thus impact the accuracy for determining AE. This fact raises the crucial importance of the design and duration of experiments (i.e. the duration of the feeding period and the time during which depuration will be followed after ingestion of food) in order to accurately determine AE (see also section 3.2.3).

Determination of the assimilation efficiency in fish

Assimilation efficiency of macromolecules by fish

Since AE of a given element or compound is defined as its absorption minus its excretion, it could be calculated as the difference between its quantity ingested (quantity presents in the food) minus quantity egested (quantity in the faeces). This method, the so-called mass-balance, has been used to study the AE of nutrients such as proteins and lipids in farmed fish. Using this method, AE can be calculated as follow:

AE (%) = ( ingested -fecal ingested ) x 100
However, urinary and branchial excretions are not taken into account in this calculation, which limits its accuracy. Furthermore, to be efficient, the mass-balance approach requires to be able to get an accurate quantification of the studied compound in the food and faeces. Some challenges may appear at this stage such as the ability to collect the faeces before their complete or partial dissolution in the water, that could lead to the loss or partial loss of the studied element [START_REF] Choubert | La digestibilité des nutriments chez les poissons : Aspects de méthodologie (in french)[END_REF].

Another method, based on the same mass-balance principle, uses an inert tracer, such as chrome oxide Cr2O3 [START_REF] Austreng | Digestibility determination in fish using chromic oxide marking and analysis of contents from different segments of the gastrointestinal tract[END_REF][START_REF] Austreng | Evaluation of selected trivalent metal oxides as inert markers used to estimate apparent digestibility in salmonids[END_REF], titanium oxide TiO2 [START_REF] Weatherup | Comparison of estimates of digestibility of two diets for rainbow trout, Oncorhynchus mykiss (Walbiaum), using two markers and two methods of faeces collection[END_REF][START_REF] Vandenberg | Apparent digestibility comparison in rainbow trout (Oncorhynchus mykiss) assessed using three methods of faeces collection and three digestibility markers[END_REF][START_REF] Richter | Evacuation of pelleted feed and the suitability of titanium (IV) oxide as a feed marker for gut kinetics in Nile tilapia[END_REF] or acid-insoluble ash [START_REF] Sarker | Nile tilapia (Oreochromis niloticus) show high digestibility of lipid and fatty acids from marine Schizochytrium sp. and of protein and essential amino acids from freshwater Spirulina sp. feed ingredients[END_REF]. Incorporated in the compounded feed or ingredients/constituents of the food matrix [START_REF] Tacon | Comparison of chromic oxide, crude fibre, polyethylene and acid-insoluble ash as dietary markers for the estimation of apparent digestibility coefficients in rainbow trout[END_REF][START_REF] Morales | Re-evaluation of crude fibre and acid-insoluble ash as inert markers, alternative to chromic oxide, in digestibility studies with rainbow trout (Oncorhynchus mykiss)[END_REF], the inert tracer allows correcting AE measurement for possible post egestion loss. In this case, AE can be calculated using the following equation [START_REF] Maynard | Animal nutrition, 6st Edition[END_REF]:

AE (%) = ( % inert marker in the food % inert marker in the feces x % element in the feces % element in the food ) x 100

This ratio is widely used in aquaculture nutrition since it does not require a complete recovery of faeces, as it is the case for the original approach. Its use is nevertheless limited nowadays given the fact that the selected inert tracer/marker must fulfil several characteristics, which are not easily met. Indeed, the inert marker, in principle, should: (1) be absolutely inert, without physiological effect on fish; (2) not be absorbed or metabolized; (3) not influence absorption and/or digestion; (4) be easily and quickly measurable [START_REF] Choubert | La digestibilité des nutriments chez les poissons : Aspects de méthodologie (in french)[END_REF]. To the best of our knowledge, no marker perfectly fits all these conditions at once. Furthermore, this method does not take into account urinary and branchial excretion. Despite some disadvantages, this method is however still used in aquaculture studies to determine the assimilation efficiency of macromolecules such as proteins and lipids in fish (e.g. Zhang et al. 2015;[START_REF] Sarker | Nile tilapia (Oreochromis niloticus) show high digestibility of lipid and fatty acids from marine Schizochytrium sp. and of protein and essential amino acids from freshwater Spirulina sp. feed ingredients[END_REF].

With an increasing research interest in the trophic transfer of trace elements in fish, other methods for AE determination, developed specifically for these elements, have emerged.

3.2. Assimilation efficiency of trace elements in fish

Use of radiotracers

One of the most efficient method to determine AE of trace elements in fish is the use of radiotracers. As isotopes of a given share the same properties among each other, radioactive isotope of a trace element can be used as tracer of that element. Thus, the two approaches described in the previous section (mass-balance and ratio) can be applied in the determination of AE for trace elements, using radiotracers in aquatic organisms such as fish. For example, [START_REF] Ni | Transfer of Cd, Cr and Zn from zooplankton prey to mudskipper Periophthalmus cantonensis and glassy Ambassis urotaenia fishes[END_REF] have already compared AE of Cd, Cr and Zn in the mudskipper Periophthalmus modestus and the glassy Ambassis urotaenia obtained, using radiotracers with mass-balance and ratio approaches. In that study, the authors concluded that to two approaches give similar results. Since then, the ratio approach has not been tested again in the determination of trace element AE in fish.

In addition to the two previous methods, the use of radiotracers and particularly gammaemitting radiotracers allowed developing a new approach in the determination of the AE of trace elements: the pulse-chase feeding mathod. This technique has many advantages that explain its widespread use in the literature (e.g. [START_REF] Xu | Exposure and potential food chain transfer factor of Cd, Se and Zn in marine fish Lutjanus argentimaculatus[END_REF][START_REF] Wang | Dietary bioavailability of cadmium, inorganic mercury, and zinc to a marine fish: effects of food composition and type[END_REF][START_REF] Pouil | Influence of food on the assimilation of essential elements (Co, Mn, and Zn) by turbot Scophthalmus maximus[END_REF]. The use of gamma-emitting radioisotopes allows radiocounting fish alive, thus limiting the number of individuals to sacrifice and generating data with reduced biological variability [START_REF] Warnau | Radiotracer techniques: a unique tool in marine ecotoxicological studies[END_REF]. In the pulse-chase feeding method, fish are fed with radiolabelled food (natural prey or compounded feeds) are radiocounted just after the radiolabelled feeding. Then, fish are regularly counted alive in order to describe the depuration kinetic of the radiotracers and, thereby, to determine the AE (see details in section 3.2.3). The determination of AE based on a kinetic approach is done from a unique feeding with a radiolabelled food item. The fish are allowed to feed on radiolabelled food for a short period of time (shorter than their GTT; usually from 5 min to 2 h) to ensure that the radioactivity ingested can be accurately quantified without any possible radiotracer recycling from seawater due to leaching from the radiolabelled food, leading to an overestimation of AE. Recently, Pouil et al.

(2017) provided an experimental validation of the single-feeding approach for the determination of Co, Cd, Mn and Zn AEs in the turbot Scophthalmus maximus fed with radiolabeled compounded food.

Improvements in the AE calculation

Two methods are commonly used to calculate trace element AE using gamma-emitters. For both methods, the proportion of trace elements retained in the fish during the depuration period is followed using regular gamma-countings of live organisms. In the first method, AE is determined at a given time and expressed as a percentage of trace element retained after the GTT from the total ingested fraction (e.g. [START_REF] Xu | Exposure and potential food chain transfer factor of Cd, Se and Zn in marine fish Lutjanus argentimaculatus[END_REF][START_REF] Van Campenhout | Assimilation efficiencies of Cd and Zn in the common carp (Cyprinus carpio): effects of metal concentration, temperature and prey type[END_REF][START_REF] Goto | Influences of prey-and predator-dependent processes on cadmium and methylmercury trophic transfer to mummichogs (Fundulus heteroclitus)[END_REF]. Usually, in this method, the depuration is followed over a short time (i.e. few hours or few days; Table 1); it provides therefore a rapid insight on the transfer of trace elements in fish from their food. The second method is based on the actual determination of the trace element depuration kinetics. This technique has been extensively used in radioecological studies and improved by the use of multi exponential models, which parameters are solved by iterative adjustment. Depuration of trace elements are typically expressed as the percentage of remaining radioactivity (radioactivity at time t divided by the initial radioactivity measured in the organism at the beginning of the depuration period*100). Depuration kinetics are genrally best fitted by a two-component exponential model:

A t = A 0s x e -k es t + A 0l x e -k el t
where At and A0 are the remaining activities (%) at time t (d) and 0, respectively; ke is the depuration rate constant (d -1 ). "s" and "l" subscripts are related to the shortand long-lived component, respectively. The "s" component represents the depuration of the radiotracer fraction that is weakly associated with the organisms and rapidly eliminated (i.e. proportion associated with the faeces). The "l" component describes the depuration of the radiotracer fraction that is actually absorbed by the organism and eliminated slowly [START_REF] Hubbell | Radiotracer, gravimetric and calorimetric studies of ingestion and assimilation rates of an isopod[END_REF][START_REF] Reichle | Radioisotope turnover and energy flow in terrestrial isopod populations[END_REF][START_REF] Reichle | Turnover and concentration of radionuclides in food chains[END_REF][START_REF] Whicker | Radioecology: nuclear energy and the environment[END_REF][START_REF] Warnau | Biokinetics of selected heavy metals and radionuclides in the common Mediterranean echinoid Paracentrotus lividus: sea water and food exposures[END_REF]. The longlived component allows estimating the assimilation efficiency (AE), by calculating the y-axis intercept of the "l" component, of the radiotracer ingested with food (AE = A0l; Reichle 1967; [START_REF] Fowler | High absorption efficiency for ingested plutonium in crabs[END_REF][START_REF] Miramand | Comparative study of vanadium biokinetics in three species of echinoderms[END_REF]. In some studies, the depuration of the assimilated fraction of trace elements was shown to be very slow (e.g. [START_REF] Pouil | Trophic transfer of 110m Ag in the turbot Scophthalmus maximus through natural prey and compounded feed[END_REF][START_REF] Pouil | Influence of food on the assimilation of essential elements (Co, Mn, and Zn) by turbot Scophthalmus maximus[END_REF]. When the long-term depuration rate constant (kel) is not significantly different from 0, the "l" component of the exponential model can therefore be simplified and replaced by a constant (e.g. [START_REF] Pouil | Trophic transfer of 110m Ag in the turbot Scophthalmus maximus through natural prey and compounded feed[END_REF]2016) and the equation becomes:

A t = A 0s x e -k es t + A 0l with A0l = AE
This method requires that the fish be depurated for a sufficiently long period of time to get an accurate determination of the slope of the slowest depurating compartment. Usually, the depuration of the fish is followed for several weeks (Table 1). Because all the excretion processes (urinary, branchial and biliary) are taken into account, it is the most robust method to accurately determine AE (see section 3.2.3).

How the duration of depuration influences AE determination

In fish, the depuration can be usually described in three different phases. The first phase, usually few hours after the feeding, very rapid, corresponds to the passage of the ingested food from the stomach to the intestine where absorption process occurred [START_REF] Baines | Assimilation and retention of selenium and other trace elements from crustacean food by juvenile striped bass (Morone saxatilis)[END_REF][START_REF] Dutton | Bioaccumulation of As, Cd, Cr, Hg (II), and MeHg in killifish (Fundulus heteroclitus) from amphipod and worm prey[END_REF]. The second phase, usually in the first week of depuration, is dominated by the occurrence of the absorption and rapid excretion processes [START_REF] Baines | Assimilation and retention of selenium and other trace elements from crustacean food by juvenile striped bass (Morone saxatilis)[END_REF][START_REF] Dutton | Bioaccumulation of As, Cd, Cr, Hg (II), and MeHg in killifish (Fundulus heteroclitus) from amphipod and worm prey[END_REF][START_REF] Pouil | Influence of food on the assimilation of essential elements (Co, Mn, and Zn) by turbot Scophthalmus maximus[END_REF]. During phases 1 and 2, as shown by Pouil et al. (2017), almost all the trace elements ingested were distributed in the stomach and the intestine. Then, the third phase reflects the physiological turn-over from the slowest depurating compartment after absorption and excretion [START_REF] Wang | Assimilation efficiencies of chemical contaminants in aquatic invertebrates: a synthesis[END_REF]. The loss of trace elements during this phase is reduced and the body burden of trace elements is stabilizing. From a practical point of view, the duration of follow-up period of the depuration, defined by the experimenters, is therefore decisive to catch these biological processes. As explained in Section 2, GTT can be used to estimate the duration of the experiments in order to determine AE accurately. Some authors estimate the GTT by the frequent collection and the radiocounting of faeces subsequently to single-feeding and thus GTT ends at the moment when the last radioactive faeces have been collected (e.g. [START_REF] Ni | Transfer of Cd, Cr and Zn from zooplankton prey to mudskipper Periophthalmus cantonensis and glassy Ambassis urotaenia fishes[END_REF]. Thus, the duration of depuration is chosen to cover the GTT (i.e.

in general from 24 to 72 hours in fish; e.g. [START_REF] Xu | Exposure and potential food chain transfer factor of Cd, Se and Zn in marine fish Lutjanus argentimaculatus[END_REF][START_REF] Van Campenhout | Assimilation efficiencies of Cd and Zn in the common carp (Cyprinus carpio): effects of metal concentration, temperature and prey type[END_REF][START_REF] Goto | Influences of prey-and predator-dependent processes on cadmium and methylmercury trophic transfer to mummichogs (Fundulus heteroclitus)[END_REF]. When depuration follow-up is made over a period close to GTT ("short-term" approach), AE is determined at a given time and expressed as a percentage of trace element retained. This approach does not allow taking into consideration the third phase depuration kinetics (i.e. when physiological turn-over occurs after absorption and excretion) as it is usually achievable when the duration of depuration extends over several weeks (i.e. "longterm" approach).

In order to compare AEs obtained using both "short-term" and "long-term" approaches, from data provided by Pouil et al. (2017b, supplementary material), statistical comparison (i.e.

Wilcoxon-Mann-Whitney non-parametric test) was done on 54 Mn remaining activities at different times throughout a 21-d depuration in turbot Scophthalmus maximus fed with compounded pellets (Fig. 1). Remaining activities were stable from day 2 (i.e. less than 24h after the GTT) up to day 21 after the beginning of the depuration (p>0.05). Nevertheless, statistical comparison between individual AE estimated as the percentage of remaining activity after 2 days ("short-term" approach) and individual AE obtained by fitting a model (i.e. "longterm" approach) indicated a significant overestimation of AE by "short-term" approach (p=0.04). This example shows that, on a given dataset, "short-term" and "long-term" approaches may lead to different AE estimations. Such bias can be avoided using a sufficiently long period of depuration that encompasses both the absorption and the excretion processes and allows an accurate delineation of the AE. In the "short-term depuration" approach, a part of the excretion processes occurring during the last phase of the depuration are assumed negligible, which is obviously not correct. Therefore, this approach can only be considered after a careful investigation of the depuration processes in given experimental conditions.

Review of trace element assimilation efficiencies in fish

Results of AE related to trace elements and depuration duration

Figure 2 shows reported range of AEs of different essential (i.e. metabolically required) and non-essential (no biological role) elements in fish. This overview of results from 35 experimental studies reveals that the findings regarding trace element AE are overall similar regardless of the method of determination (i.e. "short-term" and "long-term" approaches, Fig 2A and2B). However, using Zn, one of the most studied elements, an analysis of the coefficients of variation (i.e. allowing to estimate the dispersion of the values from the average) for AE values reveals that the "short-term" approach leads to a higher AE variability than the "long-term" approach. This analysis provides an overall picture of AE variability according to the approach adopted for its determination. These findings, however, must be nuanced by the fact that other experimental factors that can also affect the AE variability (e.g. objectives of the study, number of organisms, etc.) are not taken account.

Non-essential elements, such as Ag, Am, Cd, Cs, Hg(II), MeHg and Po, are the most studied trace elements with, in particular, Cd AE values available for 15 species of fish (Fig. 2, Table 1). Among the 7 studied essential elements: As, Co, Cu, Cr, Mn, Se, and Zn, the latter element is the one with the most AEs values available (more than 180 data expressed as Means  SD).

The analysis of the AEs for the different trace elements shows that there is no obvious relation between the essential character of a trace element and its assimilation by the fish, in contrast to what has been observed in invertebrates [START_REF] Wang | Assimilation efficiencies of chemical contaminants in aquatic invertebrates: a synthesis[END_REF]. Interestingly, the trace elements with the highest AEs are MeHg and Cs, which are both non-essential elements very efficiently assimilated by fish. These high AEs values explain for a large part why Hg and Cs biomagnify in aquatic food webs in both freshwater and marine ecosystems (e.g., [START_REF] Garnier-Laplace | Experimental kinetic rates of food-chain and waterborne radionuclide transfer to freshwater fish: A basis for the construction of fish contamination charts[END_REF][START_REF] Zhao | Biomagnification of radiocesium in a marine piscivorous fish[END_REF][START_REF] Harmelin-Vivien | Differential biomagnification of PCB, PBDE, Hg and radiocesium in the food web of the European hake from the NW Mediterranean[END_REF][START_REF] Lavoie | Biomagnification of mercury in aquatic food webs: a worldwide meta-analysis[END_REF][START_REF] Pan | Radiocesium uptake, trophic transfer, and exposure in three estuarine fish with contrasting feeding habits[END_REF]). Among the most efficiently assimilated elements, Se is an essential trace element known to have an antagonistic action with Hg in aquatic organisms [START_REF] Belzile | The effect of selenium on mercury assimilation by freshwater organisms[END_REF]). Thus, field investigations have shown that high Se concentrations may force a preferential assimilation of this element over Hg through a competitive adsorption on binding sites. The occurrence of Se at high concentrations may also restricts the solubility and bioavailability of Hg to aquatic organisms or reduce it methylation in freshwater ecosystems (Cuvin-Aralar and Furness 1991; [START_REF] Belzile | The effect of selenium on mercury assimilation by freshwater organisms[END_REF][START_REF] Yang | Selenium and mercury in organisms: interactions and mechanisms[END_REF]. To the best of our knowledge, no experimental study has investigated such effect in fish.

Factors influencing trace element AEs in fish

In theory, AE can be influenced by both abiotic and biotic factors because the latter factors potentially affect fish physiology and bioavailability of, or bioaccessibility to trace elements.

The biotic factors have been the most studied in the literature (Fig. 3). The AE of trace elements in fish depends on the relation between the prey and their predators (Fig . 3). Thus, it is possible to distinguish two types of biotic factors: those related to prey and those related to predators.

Numerous studies have investigated the influence of food quality (type of natural prey and compounded food) on AE in fish. Several studies have shown that, in the same predator species, AEs can be very different depending on the type of food ingested (e.g. [START_REF] Dutton | Bioaccumulation of As, Cd, Cr, Hg (II), and MeHg in killifish (Fundulus heteroclitus) from amphipod and worm prey[END_REF][START_REF] Wang | Dietary bioavailability of cadmium, inorganic mercury, and zinc to a marine fish: effects of food composition and type[END_REF][START_REF] Pouil | Influence of food on the assimilation of essential elements (Co, Mn, and Zn) by turbot Scophthalmus maximus[END_REF]. By a mechanistic approach, some authors have studied the factors related to the prey (i.e. bivalves and oligochaetes) that could explain these differences.

In particular, based on studies initiated with invertebrates (i.e. crustaceans, [START_REF] Wallace | Relationship between subcellular cadmium distribution in prey and cadmium trophic transfer to a predator[END_REF][START_REF] Wallace | Subcellular compartmentalization of Cd and Zn in two bivalves. II. Significance of trophically available metal (TAM)[END_REF], the relationship between subcellular fraction of trace elements in food and AE observed in predators has been investigated in several species (Dang andWang 2010, Zhang and[START_REF] Zhang | Significance of subcellular metal distribution in prey in influencing the trophic transfer of metals in a marine fish[END_REF]. However, the results of these studies are contrasted. Some studies highlighted a positive relation between the cytosolic fractionation of Cd, MeHg, Se, and Zn in the prey and the AEs of these elements in different species of fish [START_REF] Seebaugh | Bioenhancement of cadmium transfer along a multi-level food chain[END_REF][START_REF] Zhang | Significance of subcellular metal distribution in prey in influencing the trophic transfer of metals in a marine fish[END_REF][START_REF] Dang | Subcellular controls of mercury trophic transfer to a marine fish[END_REF]. However, more recently, [START_REF] Pouil | Influence of food on the assimilation of essential elements (Co, Mn, and Zn) by turbot Scophthalmus maximus[END_REF] have shown that no obvious relationship was observed for essential elements (Co, Mn, and Zn) in juvenile Scophthalmus maximus fed more complex food matrices (complex pluricellular natural prey).

Interspecific comparisons of trace element AEs have also been made (e.g. [START_REF] Ni | Transfer of Cd, Cr and Zn from zooplankton prey to mudskipper Periophthalmus cantonensis and glassy Ambassis urotaenia fishes[END_REF], Pouil et al. 2017a). The differences observed were often related to the trophic ecology of the organisms or their phylogeny. The influence of the predator size (i.e. allometry) on their AE was also investigated in black seabream Acanthopagrus schlegeli [START_REF] Zhang | Size-dependence of the potential for metal biomagnification in early life stages of marine fish[END_REF]. In this study, Cd AE was independent of body size, whereas Se and Zn AE increased with the predator size. Regarding the feeding behavior of predators, although this parameter appears to be important in the understanding of trace element assimilation, there are still only few studies that have tackled this aspect. Among these, [START_REF] Van Campenhout | Assimilation efficiencies of Cd and Zn in the common carp (Cyprinus carpio): effects of metal concentration, temperature and prey type[END_REF] demonstrated in common carp Cyprinus carpio that frequency and rate of ingestion have a significant impact on the AE for Cd and Zn. In the same study, the influence of water temperature on AE has also been investigated. Authors observed that decreasing the temperature from 25°C to 15°C did not influence Cd AE, while a significant decrease of Zn AE was measured. The influence of trace element concentrations in the environment on AE in fish has been considered in few studies. It was for instance shown that Ag AEs were higher in waters highly contaminated by this element (Long and Wang 2005b;[START_REF] Boyle | Physiological response to a metal-contaminated invertebrate diet in zebrafish: Importance of metal speciation and regulation of metal transport pathways[END_REF]. However, no effect was observed for Cd or Zn [START_REF] Zhang | Effects of Zn pre-exposure on Cd and Zn bioaccumulation and metallothionein levels in two species of marine fish[END_REF][START_REF] Boyle | Physiological response to a metal-contaminated invertebrate diet in zebrafish: Importance of metal speciation and regulation of metal transport pathways[END_REF]. Besides temperature or element concentrations, there is still a lack of knowledge regarding the possible effects of other abiotic factors on AE in fish.

Salinity, which is a key parameter in brackish and marine environments that influences both bioavailability of trace elements and fish physiology, has been investigated only in fish by Ni et al. (2005). These authors found no significant differences in Cd, Se, and Zn AEs in Periophthalmus modestus acclimated from 10 to 30 psu. Recently, environmental pH, known to influence the digestive physiology of fish [START_REF] Zhang | Significance of subcellular metal distribution in prey in influencing the trophic transfer of metals in a marine fish[END_REF][START_REF] Dang | Subcellular controls of mercury trophic transfer to a marine fish[END_REF], was considered to explore the possible effects of ocean acidification on stomach pH and the assimilation of essential elements (Co, Mn, and Zn) in the clownfish Amphiprion ocellaris (Jacob et al. 2017). Another study investigated the influence of pH and temperature on the AE of Ag, Co, and Zn in turbot Scophthalmus maximus (Pouil et al. 2017b). These studies showed no significant effect of environmental pH.

Conclusion

AE is a key parameter in the trophic transfer of trace elements in fish and is therefore widely investigated in ecotoxicology and aquaculture research. Despite it is intensively used, there are still divergences in the definition of the AE concept, as highlighted in this review, which may affect its experimental determination. Thus, we provided a critical analysis of the methods used to determine AE in fish in order to provide guidance for further studies. In complement, the emphasis on trace element AE in fish reveals that among the 35 experimental studies identified

from the available open literature, the influence of environmental variables in the trophic transfer of these elements has received little attention. This research topic continues to offer exciting and challenging scientific questions for ecotoxicology and fish nutrition research. For comparison, AE observed after 2 days of depuration ("short-term" approach) and AEs estimated using kinetic modeling ("long-term" approach) are indicated as bullet-points. Letters indicated significant differences (p<0.05)

La partie de l'image avec l'ID de relation rId2 n'a pas été trouvée dans le fichier.

• AE observed after 2 days : 30±6% a

• AE modeling : 23±3% b (Mathews et al. 2008)

Figure 1 .

 1 Figure 1. Remaining activities of 54 Mn during a 21-d depuration in turbot (n=12) fed radiolabelled with pellets. Data from Pouil et al. (2017b, supplementary material).For comparison, AE observed after 2 days of depuration ("short-term" approach) and AEs estimated using kinetic modeling ("long-term" approach) are indicated as bullet-points. Letters indicated significant differences (p<0.05)

Figure 2 .Figure 3 .

 23 Figure 2. Comparison of AEs values of essential (white bars) and non-essential (grey bars) elements assessed in fish. Duration of experiments are either :(A) "short-term depuration" and (B) "long-term depuration" approaches. The width of the boxes is proportional to the number of observations. Extreme values are not represented. Data extracted from the literature are detailed in Tables1 and 2

Table 1 .

 1 Assimilation efficiencies of trace elements in freshwater and marine fish reported in experimental studies 1

	Species	Objectives	Metal	Food	Depuration (d)	AE (%)	References
	Acanthopagrus schlegeli	Allometry	Ag	Crustaceans	2	10-41	(Long and Wang 2005b)
		Food composition	Cd	Crustaceans -Fish -Molluscs -Pellets	1.5-2	2-38	(Wang et al. 2012)
		Trace element pre-exposure	Cu	Crustaceans -Molluscs	2	2-11	(Zhang and Wang 2005)
			Hg(II)	Pellet	2	3-55	(Zhang and Wang 2007)
			Z n	Crustaceans -Pellets	1.5-2	2-50
	Ambassis urotaenia	Interspecific comparison	Cd	Crustaceans	1	10-43	(Ni et al. 2000)
		Food composition	Cr	Crustaceans	1	1-10
			Zn	Crustaceans	1-2.1	2-32
	Amphiprion ocellaris	Water pH	Mn	Pellets	20	1-10	(Jacob et al. in press)
			Zn	Pellets	20	24-35
	Cyprinodon variegatus variegatus	Pharmacokinetic model	MeHg	Phytoplankton -Pellets	0.1-35	38-100	(Leaner and Mason 2002)
	Cyprinus carpio	Food composition and quantity	Cd	Insects -Oligochaetes -Molluscs	2	9-80	(Van Campenhout et al. 2007)
		Water temperature	Zn	Insects -Oligochaetes -Molluscs	2	20-97
	Danio rerio	Food composition	Ag	Polychaetes	3	1-7	(Boyle et al. 2011)
		Trace element pre-exposure	Cd	Crustaceans -Polychaetes	2.5-3	3-18	(Liu et al. 2002)
			Cr	Crustaceans	2.5	2-47
			Zn	Crustaceans	2.5	12-54
	Dicentrarchus labrax	Food chain	Am	Fish	24	4-8	(Mathews and Fisher 2008)
			Cd	Fish	24	14-31
			Co	Fish	24	13-28
			Cs	Fish	24	76-82
			Mn	Fish	24	24-42
			Se	Fish	24	52-76
			Zn	Fish	24	28-48
	Fundulus heteroclitus heteroclitus	Food chain	As	Crustaceans	9	9-10	(Dutton and Fisher 2011)
		Food composition	Cd	Crustaceans -Fish -Insects -Polychaetes	1-13	3-70	(Seebaugh et al. 2005)
		Subcellular control	Cr	Crustaceans -Polychaetes	9	0-4	(Goto and Wallace 2009)
			Hg(II)	Crustaceans -Polychaetes	9	10-26	(Mathews and Fisher 2008)
			MeHg	Crustaceans -Fish -Insects -Polychaetes	1-13	47-96
			Po	Crustaceans	13	25-37
	Gambusia affinis	Food chain	Hg(II)	Crustaceans	6	25-78	(Pickhardt et al. 2006)6)
		Trace element pre-exposure	MeHg		6	81-98
	Ictalurus punctatus	In-vitro digestion	MeHg	Polychaetes	1.5	52-65	(Leaner and Mason 2002)
	Lepomis microlophus	Food chain	Hg(II)	Crustaceans	6	0-18	(Pickhardt et al. 2006)
		Trace element pre-exposure	MeHg		6	84-94
	Lutjanus argentimaculatus	Food composition	Cd	Crustaceans -Molluscs	3	4-33	(Xu and Wang 2002)
		Ingestion rate	Cs	Crustaceans -Fish -Molluscs	3	82-99	(Zhao et al. 2001)
			Se	Crustaceans -Molluscs	3	27-60
			Zn	Crustaceans -Molluscs	3	13-53
	Menidia sp	Food composition	Cd	Crustaceans	0.8	2-4	(Reinfelder and Fisher 1994)
			Co	Crustaceans	0.8	1-3
			Se	Crustaceans	0.8	25-33
			Zn	Crustaceans	0.8	4-8

Acknowledgments

The International Atomic Energy Agency (IAEA) is grateful to the Government of the Principality of Monaco for the support provided to its Environment Laboratories. M.W. is an Honorary Senior Research Associate of the National Fund for Scientific Research (NFSR, Belgium). The Institut Universitaire de France (IUF) is acknowledged for its support to P.B.

Thanks also to Peter Swarzenski for helpful comments which improved the presentation of this work.


N Tardieu

A Constantinescu

On the determination of elastic coefficients from indentation experiments

The main result of this paper is the extension of the adjoint state method to variational inequalities. This is done for the Signorini contact problem (Kikuchi N and Oden J T 1988 Contact Problems in Elasticity: a Study of Variational Inequalities and Finite Element Methods (Philadelphia: SIAM)) and used for the identification of elastic coefficients from an indentation test. The result is obtained by two independent approaches based on the penalized and respectively, mixed formulations of the direct problem, a Signorini contact problem. An important and astonishing result is that the obtained adjoint problem is a linear problem with Dirichlet boundary conditions. This is expected for problems described with variational equalities (Bui H D 1993 Introduction Aux Problèmes Inverses en Mécanique des Matériaux (Paris: Eyrolles) (Engl. Transl. (Boca Raton, FL: CRC Press)), Lions J L 1968 Contrôle Optimal des Systèmes Gouvernés par des Équations aux Dérivées Partielles (Dunod)), but is a new result for problems described with variational inequalities. As an application, the elastic coefficients of an isotropic body have been identified from the knowledge of a displacement-force curve measured during an indentation test. The efficiency of the method is illustrated on numerical examples for the identification of a bimaterial and a functional gradient material.

Introduction

The indentation test consists in pressing a punch of arbitrary shape (conical, spherical, pyramidal, etc) into a material sample (figure 2) and recording the penetration depth versus the reaction force (figure 1). The underlying identification problem seeks to recover the material coefficients of the constitutive law from the knowledge of an indentation test result. The importance of this testing method stems from its experimental simplicity. However, the interpretation of the test results is not complete in spite of the large amount of interest shown by the engineering community over the last decades.

From a mathematical point of view, the direct indentation problem of a linear elastic material corresponds to an elliptic partial differential equation with unilateral constraints. The variational formulation of this problem therefore conducts to the use of variational inequalities with variables in closed convex sets [START_REF] Kikuchi | Contact Problems in Elasticity: a Study of Variational Inequalities and Finite Element Methods[END_REF]. The associated inverse problems inherit this mathematical complexity.

The existing solutions to the identification problem of the material parameters (i.e. coefficients of the associated partial differential equation) from indentation tests are on the one hand semi-empirical formulae in some particular cases of material behaviour [START_REF] Doener | A method of interpreting the data from depth-sensing indentation measurements[END_REF][START_REF] Hainsworth | Analysis of nanoindentation load-displacement loading curves[END_REF][START_REF] Hill | The theory of wedge indentation of ductile materials[END_REF][START_REF] Johnson | Contact Mechanics[END_REF][START_REF] Tabor | Hardness of Metals[END_REF]. On the other hand, one finds attempts to solve a related minimization problem for a given cost functional without any reference to a gradient computation [START_REF] Hasanov | Inverse coefficient problems for monotone potentials operators[END_REF][START_REF] Hasanov | Inverse coefficient problems for elliptic variational inequalities with a nonlinear monotone operator[END_REF][START_REF] Koguchi | Determination of mechanical properties of thin films and functional gradient materials using an inverse technique[END_REF] approach is obviously of primal interest and would be enhanced by the possibility to compute the gradient of the cost functional with respect to the material parameters.

The purpose of this paper is to show that the gradient of the cost functional can be determined for an inverse elastic indentation problem. With this aim, the adjoint state method has been modified to apply to variational inequalities. As a consequence, an exact gradient is obtained from the knowledge of the solutions to the direct and the adjoint problems. The adjoint problem has been deduced using two different approaches based on the penalized and respectively, mixed variational formulations of the Signorini contact problem. An important and astonishing result is that the obtained adjoint problem is a linear problem with Dirichlet boundary conditions. This is expected for problems described with variational equalities [START_REF] Bui | Introduction aux Problèmes Inverses en Mécanique des Matériaux[END_REF][START_REF] Lions | Contrôle Optimal des Systèmes Gouvernés par des Équations aux Dérivées Partielles[END_REF], but is a new result for problems described with variational inequalities.

In section 2, the direct contact problem is presented using three different weak formulations [START_REF] Kikuchi | Contact Problems in Elasticity: a Study of Variational Inequalities and Finite Element Methods[END_REF]: primal, penalized and mixed. In section 3, the inverse problem is expressed as a constrained minimization problem.

In section 4, the application of the adjoint state method for variational inequalities is presented. The direct and adjoint problems are obtained as sufficient conditions for the stationarity of a Lagrangian functional.

Finally, a series of numerical examples illustrates the efficiency of the method by identifying coated and functional gradient materials from an indentation test with a conical punch.

The direct contact problem

Let us consider an axisymmetric body, with its section occupying in its reference configuration an open subset ⊂ R 2 with smooth boundary (figure 2). The boundary is partitioned in three parts = D ∪ F ∪ C : the part D where displacements are imposed, the free surface F and the surface C where contact might occur. We shall denote by eff C ⊂ C the part where contact is effective.

The problem considered in what follows is the indentation of the body by a rigid punch, whose profile is characterized by the gap g (the distance between the punch and the surface C ). The contact is considered without friction. One can note that the effective contact region eff C is not known in advance, causing the nonlinearity of the direct problem.

In order to simplify the presentation without restricting the generality of the method, the configuration is supposed axisymmetric and no body forces or imposed surface tractions are considered.

The real and the virtual displacement fields u, v belong to the following functional space:

V = {v ∈ (H 1 ( )) 2 | u = 0 on D }. (1)
Under the assumptions of small strains and an elastic constitutive law, the displacement vector, the second-order strain and stress tensors are subject to the following set of equations:

ε(u) = 1 2 (∇u + ∇ T u) (2) σ(u) = A(c) : ε(u) (3)
where A(c) is the fourth-order tensor of elastic moduli depending of a vector of material parameters c ∈ L, a closed subset of R n (n 1). The stress tensor σ also satisfies the equilibrium equations:

divσ(u) = 0. (4)
Using the previous notations, we will recall three classical variational formulations of the frictionless contact problem between a rigid punch of arbitrary shape and an elastic foundation, denoted as the Signorini problem (for a detailed presentation see Kikuchi and Oden [15]).

Primal formulation (P)

This problem is driven by the indentation depth U exp and is written as follows:

find u ∈ K such that σ(u) : ε(v -u) d 0 ∀v ∈ K = {v ∈ V | v 2 g + U exp on C }. (5)
The resultant force F on the indenter is computed from the solution u of the contact problem by

F = C σ 22 (u) d . ( 6
)
Within this formulation, the contact conditions are directly imposed to the solution u by searching it in the closed convex set K. This causes a certain number of difficulties from a numerical point of view and therefore equivalent formulations on vector spaces are to be preferred.

Penalized formulation (P )

For an arbitrary small > 0, we consider the following problem: [START_REF] Gladwell | Contact Problems in the Classical Theory of Elasticity[END_REF] where • + denotes the positive part.

find u ∈ V such that σ(u ) : ε(v) d + 1 2 C u 2 -g -U exp 2 + v 2 d = 0 ∀v ∈ V
The reaction force F on the punch is defined as

F = - 1 2 C u 2 -U exp -g 2 + d . ( 8
)
In this case, one can consider in a rough interpretation that the contact conditions are replaced by a nonlinear contact force proportional to the allowed interpenetration and inverse proportional to . Some convergence properties are proven: u converges strongly to u in V and F converges to F in R. Formally, this means that, as → 0, the solution to the penalized problem (P ) converges to the solution to the primal problem (P).

Mixed formulation (P m )

This formulation has the following expression:

find

(u, p) ∈ V × N such that σ(u) : (v) d - C p • v 2 d = 0 ∀v ∈ V C (q -p) • (u 2 -U -g) d 0 ∀q ∈ N . (9)
The contact condition have been relaxed in this case by the introduction of the Lagrange multiplier p. The displacement field u is now supposed to belong to the vector space V and the Lagrange multipliers p to the convex cone N = {q ∈ (H 1/2 ( C )) |q 0}, where (.) denotes the dual of (.).

The Lagrange multiplier p ∈ N shows up to be the pressure distribution under the punch. Therefore, the resultant force F is expressed as

F = C p d . ( 10
)

The inverse problem

The inverse problem addressed here is the identification of the material properties from a displacement-force measurement curve on the indenter. From a practical point of view, it is interesting to consider the inverse problem as a minimization problem of the following form [START_REF] Hasanov | The solution of an axisymmetric inverse elasto-plastic problem using penetration diagrams[END_REF][START_REF] Koguchi | Determination of mechanical properties of thin films and functional gradient materials using an inverse technique[END_REF]:

find c * ∈ L such that J (c * ) = min c∈L J (c) ( 11
)
where J represents a cost functional involving measured and computed quantities. Different cost functionals can be imagined:

• the difference between measured and computed forces F exp , F for given material parameter c and a prescribed indentation depth U exp :

J (c) = 1 2 (F -F exp ) 2 (12)
• the difference between measured and computed displacements U exp , U for imposed indentation force F exp :

J (c) = 1 2 (U -U exp ) 2 (13)
• the reciprocity gap, i.e., the crossed difference between computed and measured forces and displacements:

J (c) = (U exp -U )(F -F exp ). ( 14
)
If the preceding cost functionals have been defined for a single definite indentation depth, one can also imagine cost functionals as sums or time integrals of these quantities. In this paper, we shall consider simple cost functionals based on the squared difference between computed and experimental forces [START_REF] Hilding | Optimization of structures in unilateral contact[END_REF].

The adjoint state method for variational inequalities

In the previous section, the inverse problem has been expressed as a minimization problem. In order to solve it using a descent algorithm, the gradient of the cost functional should be computed. This is not a straightforward operation: on the one hand due to the implicit dependence of the cost functional with regard to the minimization parameters and on the other hand due to the direct contact problem involving non-differential relations between different fields.

One can ask if the non-differentiability relation due to contact implies a nondifferentiability of the cost functional with respect to the material parameters. Nondifferentiability would certainly occur, if the cost functional is minimized with regards to the parameters of the contact itself as shown in a review paper by Hilding et al [START_REF] Hilding | Optimization of structures in unilateral contact[END_REF].

An inspection of some analytical solutions in the classical elastic theory [START_REF] Gladwell | Contact Problems in the Classical Theory of Elasticity[END_REF][START_REF] Johnson | Contact Mechanics[END_REF] shows that the cost functional is differentiable with respect to elastic moduli, the minimization parameters in our case. Numerical computations in a series of configurations involving bimaterial or functional gradient materials exhibit the same conclusion (figure 7).

In the following, we shall propose a gradient computation using the adjoint state method. The adjoint state method permits one to compute the gradient of a constrained cost functional using the solution to an auxiliary problem, called the adjoint problem [START_REF] Bourgeois | Contrôle optimal et problèmes inverses en plasticité[END_REF][START_REF] Bui | Introduction aux Problèmes Inverses en Mécanique des Matériaux[END_REF][START_REF] Lions | Contrôle Optimal des Systèmes Gouvernés par des Équations aux Dérivées Partielles[END_REF]. This technique will be adapted to the contact problem using the penalized and mixed formulations of the direct contact problem.

The adjoint state method based on the penalized formulation (P -1 )

For a small > 0, the inverse problem is expressed as follows:

minimize J = 1 2 (F -F exp ) 2 with respect to c ∈ L,
where F is computed from the solution of (P ).

As we handle a constrained minimization problem, it is natural to introduce a Lagrangian L , by adjoining to the penalized cost functional J the variational equality of the penalized direct problem (P ):

L (u , v , c) = 1 2 (F -F exp ) 2 + σ(u ) : ε(v ) d + 1 2 C u 2 -g -U exp 2 + v 2 d ( 15
)
where F is given by ( 8) and

(u , v , c) ∈ V 2 × L.
All variables are supposed mutually independent, the virtual displacement field v plays the role of a Lagrange multiplier. The construction of the Lagrangian ensures that the saddle point of the L gives the minimum of J and that L ≡ J if u is a solution to the direct problem (P ).

The necessary conditions of stationarity of L can be formally written as

∂L ∂v , w = σ(u ) : ε(w) d + 1 2 C u 2 -g -U exp 2 + w 2 d = 0 ∀w ∈ V ( 16
)
∂L ∂u , w = σ(v ) : ε(w) d + 1 C u 2 -g -U exp + w 2 (v 2 -(F -F exp )) d = 0 ∀w ∈ V (17) ∂L ∂c , d -c = ε(u ) : ∂A ∂c : ε(v ) • (d -c) d 0 ∀d ∈ L ( 18
)
where (. , .) and [. , .] are respectively the duality pairing on V × V and L × L.

As expected, the derivation of the Lagrangian with respect to the adjoint variable v (16) returns the variational formulation of the penalized direct problem. The derivation of the Lagrangian with respect to the direct variable u [START_REF] Lions | Contrôle Optimal des Systèmes Gouvernés par des Équations aux Dérivées Partielles[END_REF] returns the weak formulation of a elastic problem called the penalized adjoint problem (P adj ).

Let us choose a point (u , v , c). If u is solution of (P ), by definition of L , we have L ≡ J . Moreover if v is solution of (P adj ) it follows that

∇ c J = ∂L ∂c = ε(u ) : ∂A ∂c : ε(v ) d . ( 19
)
Thus the explicit expression of the gradient of J has been obtained using the solution to the direct and the adjoint penalized problems.

The previous result has been established within the penalized formulation. It is interesting to know how it evolves as → 0. Without giving a precise mathematical proof, the following results can be conjectured.

• The solution v to the penalized adjoint problem (P adj ) converges to the solution v to the adjoint problem (P adj ) driven by the imposed displacements (F -F exp ) on the effective contact surface eff C :

find v ∈ V adj such that σ(v) : ε(w) d = 0 ∀w ∈ V adj = {w ∈ V | w 2 = (F -F exp )on eff C } (20)
where F , C eff are respectively the force on the indenter and the effective contact surface, computed in the direct contact problem (P).

• The limits and the gradient operations are commutative, i.e.

lim →0 ∇ c J = ∇ c (lim →0 J ) = ∇ c J . ( 21
)
As a consequence the gradient of the cost functional can be computed using the following formula:

∇ c J = ε(u) : ∂A ∂c : ε(v) d (22)
where u and v are the solutions to the direct problem (P) and respectively the adjoint problem (P adj ).

4.2.

The adjoint state method based on the mixed formulation (P -1 m ) This problem is expressed as follows: 2 with respect to c ∈ L, where F is computed from the solution to (P m ).

minimize J = 1 2 (F -F exp )
The corresponding Lagrangian is expressed as

L(u, v, p, q, c) = 1 2 (F -F exp ) 2 - σ(u) : ε(v) d + C p • v 2 d + C q • (u 2 -U -g) d ( 23
)
where F is given by ( 10), (u, v, p, q, c)

∈ V 2 × Q 2 × L and Q = {q ∈ (H 1/2 ( C )) | q = 0 on C / eff C }.
Let us remark that, opposite to the classical application of the adjoint state method, q has not been sought in the closed convex N . If q was in N , the stationarity conditions would turn out to be a set of coupled variational inequalities, which are ineffective for practical search of the adjoint state.

With q ∈ Q, the necessary conditions of stationarity of L can be formally written as follows:

∂L ∂v , w = σ(u) : ε(w) d - C p • w 2 d = 0 ∀w ∈ V (24) ∂L ∂u , w = σ(v) : ε(w) d - C q • w 2 d = 0 ∀w ∈ V (25) ∂L ∂q , s = C s • (u 2 -U -g) d = 0 ∀s ∈ Q (26) ∂L ∂p , s = C s • (v 2 -F exp + F ) d = 0 ∀s ∈ Q (27) ∂L ∂c , d -c = ε(u) : ∂A ∂c : ε(v) • (d -c) d 0 ∀d ∈ L ( 28
)
where {. , .} denotes the duality pairing on Q × Q.

In this case, the derivation of the Lagrangian with respect to the adjoint variables v (24) and q (26) does not return the mixed formulation (P m ) of the direct problem. Nevertheless, the solutions to (P m ) are also solutions to (24) and (26).

The derivation of the Lagrangian with respect to the direct variables u (25) and p (27) returns an equivalent formulation of the adjoint problem (P adj ) (20).

Therefore if (u, p) is solution to the direct problem (P) and (v, q) is solution to the adjoint problem (P adj ), then (24)-( 27) are verified. As a consequence, in the penalized case, the gradient can be expressed as

∇ c J = ∂L ∂c = ε(u) : ∂A ∂c : ε(v) d . ( 29
)

Remarks on the adjoint problem (P adj )

For problems with Dirichlet or Neumann boundary conditions described by variational equalities, the application of the adjoint state method leads to a linear adjoint problem described by variational equalities [START_REF] Bourgeois | Contrôle optimal et problèmes inverses en plasticité[END_REF][START_REF] Bui | Introduction aux Problèmes Inverses en Mécanique des Matériaux[END_REF][START_REF] Lions | Contrôle Optimal des Systèmes Gouvernés par des Équations aux Dérivées Partielles[END_REF].

eff : u = F -F 2 exp Γ F = 0 n j : σ i j Γ F = 0 n j : σ i j Γ D : u = 0 Γ D : u = 0 Γ D : u = 0 Γ C Figure 3. Adjoint problem (P adj ).
Table 1. Gradient comparisons between finite difference method (FDM) and adjoint state method (ASM). It is important to point out that, for problems described by variational inequalities, the obtained adjoint problem (P adj ) is linear with Dirichlet boundary conditions (see figure 3), and therefore described by a variational equality (20).

E exp E ∇J (E) ∇J (E) GPa GPa (ASM) (
From a practical point of view, this implies that the overburden of computing the adjoint problem, and implicitly the gradient of the cost functional, is very small.

Numerical examples

The gradient expressions obtained previously have been applied in a series of numerical examples:

• comparison of gradient evaluations of the cost functional J using the adjoint state method (ASM) and the finite difference method (FDM) (see table 1). • identification of the elastic moduli of a bimaterial • identification of the elastic moduli of a functional gradient material with Young's moduli depending linearly on radius or depth.

For the identification problems, all indentation experiments have been simulated by direct computations.

The direct and the adjoint problems have been programmed in the object oriented language gibiane of the finite element code CASTEM 2000 [START_REF]Castem2000 Finite Element Code webpage[END_REF]. From a numerical point of view the contact conditions have been treated in an exact manner, i.e. without use of penalization. Fournoded linear quadrangular elements have been employed and a typical mesh is presented in figure 4.

As an indenter, spheres of different radiuses and cones with different semi-angles have been used without changing the quality of the final results. Computations have been performed on HP 9000 workstation, on which a contact computation for the mesh shown took 30 s CPU time.

In the next examples, two parameters (c 1 , c 2 ) characterizing the material were identified from two pairs of displacement-force measurements from the identation curve (see figure 1). Therefore two cost functionals were defined as follows:

J 1 (c 1 , c 2 ) = 1 2 (F -F exp 1 ) 2 J 2 (c 1 , c 2 ) = 1 2 (F -F exp 2 ) 2 .
The relaxation algorithm used can be described as follows:

(i) Choose a initial pair (c 1 , c 2 ) and two identification precisions

ε 1 > 0, ε 2 > 0 (ii) While J 1 ε 1 or J 2 ε 1 :
(a) minimize J 1 with respect to c 1 , until ∂J 1 ∂c 1 < ε 2 (b) minimize J 2 with respect to c 2 , until ∂J 2 ∂c 2 < ε 2 where the minimization is performed using a Newton method.

This method proved to be more efficient than minimizing directly J 1 , J 2 , or J 1 + J 2 with respect to both E 1 and E 2 using a BFGS method [START_REF] Gill | Practical Optimization[END_REF].

The bimaterial

Identifications have been performed in the case of a cylinder composed of two perfectly bonded elastic coatings (figure 5). The thickness of the coating has been considered as a priori known and only the values of the two Young's moduli, i.e. c = (E 1 , E 2 ), have been identified from simulated measurements.

The two values choosen from the complete indentation curve (U exp (t), F exp (t)), correspond to indentation depths smaller and respectively larger than the coating (U exp 1 = 0.2 mm and U exp 2 = 2.0 mm). From a mechanical point of view, this means that the load will be carried in different proportions by the coating and the substrate. This can be observed from the shape of the cost functionals J 1 (E 1 , E 2 ) and J 2 (E 1 , E 2 ): in figure 7, we observe that both cost functionals present a narrow valley and that J 1 is essentially sensitive to variations of E 1 (substrate) and that J 2 is essentially sensitive to variations of E 2 (coating).

In all tested cases, the identification converges after some 30 iterations (figure 6) with identified moduli within 0.001% of the experimental values. A series of results corresponding to different starting points are presented in table 2.

The functional gradient materials

Identifications have been performed for a cylinder with Young's modulus depending on depth (E(r, z)

= E 1 z + E 2 ) or on radius (E(r, z) = E 1 r + E 2 ).
The previous identification strategy has been used for these problems. In all cases, the identification converges after some 30 iterations with identified moduli within 0.001% of the experimental values. Identification results corresponding to different starting points are show in tables 3 and 4.

In the case of a Young's modulus depending on depth z, convergence required more iterations than in the radius dependency case (50 iterations) and results were slightly less accurate. This can be explained by the important radial strain under a blunt conical indenter [START_REF] Gladwell | Contact Problems in the Classical Theory of Elasticity[END_REF][START_REF] Johnson | Contact Mechanics[END_REF].

Conclusion

This paper presented the application of the adjoint state method to variational inequalities, through the identification of elastic coefficients from an indentation test. Using two different reasonements, based on the mixed and penalized formulations of the contact Signorini problem, it has been shown that the adjoint problem is completely linear with Dirichlet boundary conditions on the effective contact area. The effective contact area has been previously detected in the direct problem. The effectiveness of the method is illustrated on numerical examples using the finite element method. An extension of the method to evolution equations, corresponding to nonlinear constitutive laws and large strains is also possible. An application for viscoplastic materials is currently under development [START_REF] Constantinescu | On the identification of elastoviscoplastic constitutive laws from indentation tests Inverse Problems in Engineering at press[END_REF][START_REF] Tardieu | On the identification of nonlinear constitutive laws from indentation tests[END_REF].

In spite of promising numerical results, the presented method needs a precise mathematical proof. Other fundamental questions related to the indentation identification problem, such as uniqueness, stability, choice of a convex cost functional etc, are also open problems.

FFigure 1 .Figure 2 .

 12 Figure 1. Indentation curve.

Figure 4 .

 4 Figure 4. An axisymmetric mesh used for identification simulation.

Figure 5 .

 5 Figure 5. Bimaterial used for the identification.

Figure 6 .

 6 Figure 6. Convergence of the normalized elastic coefficients for the bimaterial.

Figure 7 .

 7 Figure 7. Contourplots of J 1 for a deep (U exp = 0.2 mm) and J 2 for a small indentation (U exp = 2.0 mm).

Table 2 .

 2 Identification results for the bimaterial.

	Starting	Final	Reference
	(E 1 , E 2 )	(E 1 , E 2 )	( E 1 , E 2 )
	(10 5 MPa) (10 5 MPa)	(10 5 MPa)
	(0.4, 0.4)	(1.200, 2.801) (1.200, 2.800)
	(0.4, 4.0)	(1.200, 2.801) (1.200, 2.800)
	(4.0, 0.4)	(1.200, 2.799) (1.200, 2.800)
	(4.0, 4.0)	(1.200, 2.799) (1.200, 2.800)

Table 3 .

 3 Identification results for the radius functional gradient material. MPa m -1 , MPa) 10 5 (MPa m -1 , MPa) 10 5 (MPa m -1 , MPa)

	Starting	Final	Reference
	(E 1 , E 2 )	( E 1 , E 2 )	( E 1 , E 2 )
	10 5 ((0.4, 0.4)	(1.200, 2.801)	(1.200, 2.800)
	(0.4, 4.0)	(1.200, 2.801)	(1.200, 2.800)
	(4.0, 0.4)	(1.201, 2.798)	(1.200, 2.800)
	(4.0, 4.0)	(1.200, 2.799)	(1.200, 2.800)

Table 4 .

 4 Identification results for the depth functional gradient material. MPa m -1 , MPa) 10 5 (MPa m -1 , MPa) 10 5 (MPa m -1 , MPa)

	Starting	Final	Reference
	(E 1 , E 2 )	( E 1 , E 2 )	( E 1 , E 2 )
	10 5 ((0.4, 0.4)	(1.199, 2.805)	(1.200, 2.800)
	(0.4, 4.0)	(1.199, 2.806)	(1.200, 2.800)
	(4.0, 0.4)	(1.201, 2.791)	(1.200, 2.800)
	(4.0, 4.0)	(1.200, 2.797)	(1.200, 2.800)

Acknowledgments

The authors would like to thank Huy Duong Bui, Stephane Andrieux, Laurent Bourgeois and Said Taheri for helpful discussions during the fullfilment of this work.