Adeline Bourgeault
email: adeline.bourgeault@cea.fr

Philippe Ciffroy

Cédric Garnier

Carole Cossu-Leguille

Jean-François Masfaraud

Rayna Charlatchka

Jean-Marie Garnier

Speciation and bioavailability of dissolved copper in different freshwaters: Comparison of modelling, biological and chemical responses in aquatic mosses and gammarids

Keywords:

Introduction

The Environmental Quality Standards (EQS) for metals established by the Water Framework Directive (WFD) for ensuring a good water quality status of freshwater systems are related to the total dissolved concentrations of the metals (directive 2008/105/CE of the European Parliament and Council). However, the bioavailability and toxicity of dissolved metals are not only related to their concentrations but also, and primarily, to their speciation, which depends on the water composition [START_REF] Campbell | Metal speciation and bioavailability in aquatic systems[END_REF]Meylan et al., 2004a). For instance, the concentration and type of dissolved organic carbon (DOC) have been shown to modify the bioaccumulation of Cu via chemical complexation [START_REF] Lores | The effect of humic acid on uptake/adsorption of copper by a marine bacterium and two marine ciliates[END_REF][START_REF] Luider | Influence of natural organic matter source on copper speciation as demonstrated by Cu binding to fish gills, by ion selective electrode, and by DGT gel sampler[END_REF][START_REF] Ferreira | More than inorganic copper is bioavailable to aquatic mosses at environmentally relevant concentrations[END_REF][START_REF] Zhong | Uptake of dissolved organic carbon-complexed Cu-65 by the green mussel Perna viridis[END_REF]. Furthermore, some studies have shown that speciation alone is insufficient for predicting accumulation and toxicity because competition with cations can affect the processes. For instance, the presence of competing cations such as Ca 2+ , Mg 2+ , H + , Na + or K + reduces the bioavailability of Cu [START_REF] De Schamphelaere | A biotic ligand model predicting acute copper toxicity for Daphnia magna: the effects of calcium, magnesium, sodium, potassium, and pH[END_REF][START_REF] Ferreira | Modelling exchange kinetics of copper at the water-aquatic moss (Fontinalis antipyretica) interface: influence of water cationic composition (Ca, Mg, Na and pH)[END_REF]. Considering this background information, three categories of approaches or techniques are candidates for improving the risk assessment of metals in water bodies.

First, analytical and sampling methods have been developed to approach metals in freshwater systems. The free metal ion activity can be measured using an ion selective electrode (ISE) (e.g., [START_REF] Rachou | Use of an ion-selective electrode for free copper measurements in low salinity and low ionic strength matrices[END_REF]. Differential pulse anodic stripping voltammetry (DPASV) and diffusive gradient in thin film (DGT) techniques can provide operational concentrations of labile metals, which are defined as the addition of free metal ions and easily dissociating complexes (e.g., Meylan et al., 2004a;[START_REF] Rachou | Use of an ion-selective electrode for free copper measurements in low salinity and low ionic strength matrices[END_REF][START_REF] Priadi | Spatio-temporal variability of solid, total dissolved and labile metal: passive vs discrete sampling evaluation in river metal monitoring[END_REF][START_REF] Feldmann | Critical review perspective: elemental speciation analysis methods in environmental chemistrymoving towards methodological integration[END_REF].

Second, integrative biological indicators (or biomonitors) have been used because they can accumulate metals to high concentrations, thereby facilitating the measurement of these contaminants. Furthermore, potential biological effects can be assessed in these biomonitors through the measurement of various biomarkers as an early response to chemical contamination [START_REF] Vasseur | Biomarkers and community indices as complementary tools for environmental safety[END_REF]. With respect to freshwater, the use of aquatic bryophytes and gammarids has been proposed for water monitoring [START_REF] Say | Accumulation of heavy-metals by aquatic mosses. 1. Fontinalis antipyretica Hedw[END_REF][START_REF] Siebert | The use of the aquatic moss Fontinalis antipyretica L ex Hedw as a bioindicator for heavy metals[END_REF][START_REF] Geffard | Influence of biotic and abiotic factors on metallothionein level in Gammarus pulex[END_REF]. Indeed, these two species are widespread in freshwater systems throughout France and Europe and are net accumulators of trace metals which make them reliable biomonitors of metal bioavailability.

Finally, models have been developed to predict metal speciation, bioaccumulation and/or toxicity. One of the most commonly used models to estimate metal speciation is the WHAM (Windermere Humic Aqueous Model) [START_REF] Tipping | Modelling the chemical speciation of trace metals in the surface waters of the Humber system[END_REF]. This model allows the prediction of different metal species as long as the input data (pH, dissolved metal, DOC and ion concentrations) are available. However, the variable nature of organic matter adds uncertainty to the aquatic speciation model. To consider competition with cations that affects toxicity processes, speciation models such as the WHAM have also been coupled to models that simulate the accumulation of metals onto the biotic membrane, and the result of this coupling is the so-called Biotic Ligand Models (BLM) [START_REF] Paquin | The biotic ligand model: a historical overview[END_REF]. In the complete version of the BLM, the metal toxicity is predicted according to 13 water chemistry parameters. More recently, the freely downloadable Bio-Met Bioavailability Tool (www.bio-met.net) was designed to use a limited number of variables (dissolved Cu concentration, DOC, Ca and pH). However, some studies have reported some limits of the BLM. For instance, the Cu-BLM do not adequately predict the toxicity of Cu to rainbow trout in soft water [START_REF] Ng | Can the biotic ligand model predict cu toxicity across a range of pHs in softwater-acclimated rainbow trout?[END_REF].

In the present study, tools belonging to each of these three categories (i.e., monitoring speciation, biological tools and modelling tools) were used to investigate how the speciation of copper affects its accumulation and toxicity in freshwater organisms. The following techniques were compared on samples obtained from common ex-situ field experiments: to fully characterise the copper speciation, ISE, DPASV and DGT techniques were used; as biomonitors able to provide information about the bioaccumulation potential, two species were selected (Gammarus pulex and Fontinalis antipyretica), and oxidative stress measurements were performed to assess the toxicity of copper in these two species; finally, the WHAM and BLM-BioMet models were applied to the investigated waters. Furthermore, the physicochemical parameters of the waters (especially concentrations of ions and dissolved organic matter) were measured to evaluate their impact on the accumulation and toxicity of Cu.

Exposures with ex-situ flow-through field mesocosms were used to compare these techniques on a common basis and to provide better insight into the complexation and competing processes involved in natural freshwaters. Indeed, unlike in-situ experiments, the mesocosms used here, which were continuously supplied with natural water spiked with various environmental Cu concentrations, allowed the comparison of processes along a Cu gradient in waters that had the same physicochemistry. To better understand the factors that affect the bioavailability of Cu and to improve the risk assessment of this metal, three different freshwaters were studied (with low, medium and high ionic concentration). A lower Cu bioavailability is expected with a higher concentration of competing cations and organic matter in the exposure water. Accordingly, the goal of the present study was to investigate the predictive powers of modelling and speciation analysis and to consequently determine the Cu fraction (e.g., free Cu, weakly complexed Cu, and total dissolved Cu) that best correlated with bioaccumulation and toxicity.

Materials and methods

Collection and handling of organisms

Aquatic mosses (F. antipyretica) and gammarids (G. pulex) were collected from July to August 2011 from two low-contaminated sites: the Sauldre River (Ménétréol-sur-Sauldre, France) for the collection of aquatic mosses and the rill Ru de L'Etang (Doue, France) for the gammarids. New green tips (2-3 cm) of aquatic mosses and male gammarids (12-15 mm) were selected and returned to the laboratory in the field water. Precopulas of the gammarids were collected in the field to easily identify males. Alder leaves were also collected near the Sauldre River and dried for 3 days at 60 °C. The organisms were acclimated in the laboratory in aerated water at 18 °C with a 14 h:8 h light:dark photoperiod for 3 days before the experiments. During this acclimation, the field water in the containers was progressively replaced with exposure water and the gammarids were fed ad libitum with alder leaves.

Fig. 1. Exposure system composed of a pump set at 0.5 L min -1 (1), a lamellar decanter (2), a purge valve (3), a 6 L Cu stock solution (10 or 30 mg L -1 ) (4), a peristaltic pump set at 0.25 mL min -1 (5), a 20 L reservoir for Cu-contaminated water (6), a 20 L tank (7), a multi-parameter probe (8), an aeration system (9) and a system overflow (10). Adapted from (Ferreira et al., in press).

Experimental design

Exposures were conducted in 3 natural waters with different chemical compositions: a mineral water (MontCalm) (low ionic concentration, referred to as Site 1), the Loire River water from Belleville/France (moderate ionic concentration, referred to as Site 2) and the Moselle River water from Cattenom/France (high ionic concentration, referred to as Site 3). For each site, three exposure systems were deployed: the first one for natural concentration and two others spiked with Cu to reach a concentration of 5 or 15 μg L -1 . These concentrations are environmentally realistic. For instance, the mean concentration of dissolved copper in the water of Loire River is 3.3 μg L -1 (min: 0.6 μg L -1 , max: 45 μg L -1 ) (data obtained from the network monitoring of the French Water Agency Loire-Bretagne available at http://www.eauloire-bretagne.fr/). The exposure device used in the field at Sites 2 and 3 is described in Fig. 1.

This device is composed of a lamellar decanter that is continuously fed with field water. The decanted water is then continuously spiked using a peristaltic pump in a 20 L reservoir with a Cu stock solution (CuNO 3 , laboratory reagent grade, Fisher Scientific) to reach a final spiked dissolved Cu concentration of 0, 5 or 15 μg L -1 . The Cucontaminated water is then directed to a 20 L tank where the gammarids and aquatic mosses are exposed to the water. The acid-washed exposure device was conditioned for 24 h before the start of the experiment, and the tank was continuously aerated. In the case of the laboratory experiment with the MontCalm mineral water (Site 1), the exposure system was used in a semi-static way. The Cu-contaminated water was supplied from a 100 L reservoir and continuously circulated in the tank using peristaltic pumps. The concentration of dissolved Cu was measured daily in each of the exposure systems and adjusted, if necessary, with a Cu stock solution (CuNO 3 , purity> 95%, Fisher chemicals, 120 mg L -1 ) to the nominal Cu concentration (i.e., 5 or 15 μg L -1 ).

Each experiment was conducted outdoors to provide similar exposure conditions (natural light/dark cycle, daily variation of temperature). For the Site 1 exposure, the 100 L reservoir of Cu contaminated water was stored in a thermostated room at 18 °C to avoid a significant warming of the exposure medium in the tanks that were exposed outdoors. In addition, the three exposure experiments were performed on a short time scale (from July 5th to August 12th, 2011) with calibrated organisms (i.e., male gammarids and new green tips of aquatic mosses) to limit biological variability. Male gammarids were selected since [START_REF] Correia | Age-related changes in antioxidant enzyme activities, fatty acid composition and lipid peroxicdation in whole body Gammarus locusta (Crustacea: Amphipoda)[END_REF] show that level of lipid peroxide in Gammarus locusta is 40% higher in adults males compared to adults females or juveniles.

Adult male gammarids (75 organisms) were divided into 3 pools and each pool was placed into polypropylene cylinder (diameter ×length: 3×10 cm) whose ends were capped with pieces of net (mesh size: 1 mm) and positioned parallel to the water flow. Dried alder leaves (0.16 g) were supplied to each cage to feed the gammarids and to provide a resting surface to minimise cannibalism. Aquatic mosses (300 green tips) were directly exposed in the tank.

Water samples were automatically collected from the tank every 30 min by autosampling devices (ISCO 6700) to obtain 4 composite samples of 500 mL per day for each exposure system. To prevent contamination, the bottles of the autosampling devices were cleaned with 10% nitric acid and then rinsed with Milli-Q water (Millipore). Before autosampling, 10 mL of NaN 3 (0.05 M) was added to each bottle to prevent any biological activity in the collected samples.

Water chemical analyses

Physico-chemical parameters

The temperature, conductivity, pH, chlorophyll and oxygen concentrations were measured in the tanks every 10 min using a multiparameter probe (YSI 6600 V2). Composite water samples collected by the autosampling devices were used to monitor the physicochemical parameters (anions, cations, and dissolved organic and inorganic carbon concentrations) and metal contamination (total and dissolved Cu concentrations). The major cation and anion concentrations were determined using inductively coupled plasma atomic emission spectroscopy (ICP-AES Ultima C, Jobin-Yvon Horiba) and capillary electrophoresis (CIA type, Waters), respectively, after filtration through a 0.45 μm cellulose nitrate filter. The dissolved organic and inorganic carbon (DOC/DIC) concentrations were measured using a TOC-analyser (Shimadzu TOC-V CSH analyser) after filtration through a pre-combusted GFF filter (Whatman). Raw water samples for the analysis of total copper (45 mL) were acidified with 1 mL of HNO 3 (65% suprapur Merck) and 4 mL of HCl (30% suprapur, Merck). These samples were then microwave digested in a Teflon vessel (Mars Xpress, CEM) during a 15 min ramp to 180 °C at 1600 W and a 15 min stable period at 180 °C at 1600 W. The samples for the analysis of dissolved Cu were filtered through a 0.45 μm cellulose nitrate filter (Whatman) and acidified with 1% (v/v) HNO 3 before the metal analysis with a graphite furnace atomic absorption spectrometer (GF-AAS, iCE 3000 series, ThermoFisher).

DGT-labile copper concentration

The 5-day integrative concentration of labile Cu was monitored using DGT technique [START_REF] Davison | In situ speciation measurements of trace components in natural waters using thin-film gels[END_REF]. Three preassembled DGT open-pore units (DGT research, Landcaster, UK) were used for each water sample. The DGT devices are composed of Chelex resins and open-pore diffusive gels that are covered with a 0.45 μm polyether sulfone (PES) filter. After retrieval of the DGTs, the Chelex resins were eluted in 1 mL of 1 M HNO 3 (65% suprapur Merck) in which the Cu analysis was performed by GF-AAS after sample dilution. The integrated labile concentrations and the associated standard deviations were calculated using the method described by [START_REF] Tusseau-Vuillemin | Dissolved and bioavailable contaminants in the Seine River basin[END_REF].

DPASV-labile copper concentration

Water samples were analysed by pseudopolarography to assess the concentration of the most available forms of Cu, following an analytical protocol adapted from previous studies [START_REF] Louis | Speciation of trace metals in natural waters: the influence of an adsorbed layer of natural organic matter (NOM) on voltammetric behaviour of copper[END_REF][START_REF] Nicolau | Study of interactions of concentrated marine dissolved organic matter with copper and zinc by pseudopolarography[END_REF][START_REF] Bravin | Root-induced changes in pH and dissolved organic matter binding capacity affect copper dynamic speciation in the rhizosphere[END_REF]. Voltammetric measurements were performed using a μAutolabIII potentiostat (EcoChemie) coupled to a Metrohm 663 VA stand (Metrohm). Polynomial baseline subtraction was applied for each obtained voltammogram to determine the Cu peak area [START_REF] Omanović | Significance of data treatment and experimental setup on the determination of copper complexing parameters by anodic stripping voltammetry[END_REF]. Details of the experimental procedure are provided in the Supporting information. Briefly, the following procedure was applied:

(1) Measurements of pseudopolarograms (varying the deposition potential, E dep , from -1.5 to +0.1 V with a step of 0.1 V) at natural pH. (2) Cu standard additions with DPASV measurements (E dep -1.5 V)

to determine the sensitivity of the technique at natural pH.

(3) Measurements of pseudopolarograms (using minimal E dep of -1.1 V) after acidification to fully dissociate the organic Cu complexes (4) Cu standard additions with DPASV measurements (E dep -1.1 V)

to determine the sensitivity of the technique under acidic pH conditions and to calculate the concentration of total dissolved Cu. (5) In addition to the "inorganic" Cu fraction, 2 other fractions, which were called "weak" and "strong" Cu complexes fractions, were operationally defined in reference to the measurements performed at natural pH with an E dep -0.9 V and -1.5 V. Finally, from the difference between the concentration of the total dissolved Cu (pHb 2) and the sum of the 3 Cu fractions ("inorganic", "weak" and "strong" Cu complexes), a 4th fraction was defined as "inert" Cu complexes, which correspond to organic Cu complexes that are not electrochemically detectable in the defined conditions (e.g., [START_REF] Nicolau | Study of interactions of concentrated marine dissolved organic matter with copper and zinc by pseudopolarography[END_REF][START_REF] Feldmann | Critical review perspective: elemental speciation analysis methods in environmental chemistrymoving towards methodological integration[END_REF][START_REF] Bravin | Root-induced changes in pH and dissolved organic matter binding capacity affect copper dynamic speciation in the rhizosphere[END_REF].

Free copper concentration

The free Cu concentration was measured in the 4 water samples obtained daily using an ion selective electrode (ISE, Orion Cupric electrode 96-29, Thermo Electron Corporation) following the methodology described by [START_REF] Rachou | Use of an ion-selective electrode for free copper measurements in low salinity and low ionic strength matrices[END_REF]. Briefly, 9 standard solutions were used for the daily calibration of the electrode. These solutions were made to 500 mL using 1 mM IDA (iminodiacetic acid), 0.1 mM Cu(NO 3 ) 2 , 6 mM NaOH, 2.5 mM potassium acid phthalate and 0.01 M KNO 3 . The ionic strengths of the standard solutions were adjusted to the ionic strength of the water samples using varying amounts of KNO 3 , and the pH was adjusted from 3 to 10 using HNO 3 10% v/v. The measured value in mV (acceptation of the value at b 0.1 mV/min) was converted to free Cu 2+ activity after calculating the speciation of Cu in each standard solution using the chemical equilibrium software PHREEQC [START_REF] Parkhurst | User guide to PHREEQC (Version 2): a computer program for speciation, batch-reaction, one-dimensional transport, and inverse geochemical calculations[END_REF].

Biological responses

Bioaccumulation measurements

The metal content in the organisms was determined at 0 and 5 days of exposure from three pools of 5 gammarids and five pools of 15 new green tips of aquatic mosses. This time of exposure was chosen based on a previous study (Ferreira et al., in press) in which Cu bioaccumulation in mosses reached equilibrium after 5 days. Each pool of the 15 green tips of aquatic mosses was washed for 1 h in 50 mL of 1 mM EDTA (normapur grade, Acros Organics) to evaluate the adsorbed metal [START_REF] Meylan | Accumulation of copper and zinc in periphyton in response to dynamic variations of metal speciation in freshwater[END_REF]. The Cu concentration in the EDTA solution was analysed to measure the adsorbed metal on the biological membrane. Then, the EDTA-washed mosses and gammarids were dried for 4 days at 60 °C and microwave digested (Mars Xpress, CEM) with 5 ml of HNO 3 (65% suprapur Merck) and 1 ml of H 2 O 2 (30% suprapur Merck). The Cu concentrations were determined using GF-AAS. Reference materials (BCR-60 aquatic mosses and ERM-CE278 mussel tissue) were used to validate the microwave digestion. The deviation for Cu from the certified values was b 7% (n =10) and b10% (n = 6) for BRC-60 and ERM-CE278, respectively.

Oxidative stress

Oxidative stress parameters were measured on three pools of 5 gammarids and five pools of 10 tips of aquatic mosses collected at 0, 2 and 5 days of exposure and immediately frozen in liquid nitrogen. In the mosses, the lipid peroxidation level was assessed through the measurement of thiobarbituric acid reactive substances (TBARS), as described by [START_REF] Aravind | Zinc alleviates cadmium-induced oxidative stress in Ceratophyllum demersum L.: a free floating freshwater macrophyte[END_REF]. Moss strands were weighed before being homogenised in a 0.5% TBA-20% TCA solution. After centrifugation (10,000 ×g, 10 min), the samples were incubated at 95 °C for 30 min and finally transferred onto ice to stop the reaction. The non-specific absorbance of the supernatant was measured at 600 nm and subtracted from the absorbance at 532 nm. The TBARS concentration was determined using an extinction coefficient of 155 mM -1 cm -1 .

To determine the guaiacol peroxidase (GPX) activity level, the moss samples were homogenised in a 50 mM phosphate buffer with a pH of 6.0. The supernatant from centrifuging at 15,000 ×g (10 min) was used as the enzyme extract. The GPX level of the mosses was spectrophotometrically assayed according to [START_REF] Roy | Monitoring polycyclic aromatic hydrocarbons using 'moss bags': bioaccumulation and responses of antioxidant enzymes in Fontinalis antipyretica Hedw[END_REF] with no modifications. For calculating the GPX activity level, a molar extinction coefficient of 26.6 mM -1 cm -1 at 470 nm for tetraguaiacol was used, and the protein concentration in the enzyme extracts was determined using Bradford's method (1976).

The lipid peroxidation level was measured in the gammarids from the malondialdehyde (MDA) levels determined using an HPLC method adapted from [START_REF] Behrens | Malonaldehyde determination in tissues and biological fluids by ion-pairing high performance liquid chromatography[END_REF] with UV detection at 267 nm. Each pool of gammarids was homogenised with a manual Potter Elvejehm tissue grinder in a 50 mM phosphate buffer of KH 2 PO 4 /K 2 HPO 4 (pH 7.6) supplemented with 1 mM phenylmethylsulphonylfluoride (PMSF) and a 1 mM serine-borate mixture as protease inhibitors. The homogenisation buffer was adjusted to a volume two-times greater than the wet weight of the sample pool. Seventy microliters of the total homogenate were diluted four-fold in 95% ethanol (HPLC grade) and cooled on ice for 1.5 h to deproteinise them. The mixture was then centrifuged at 18,000 ×g for 30 min at 4 °C. The resulting supernatant (100 μL) was injected into the HPLC separation system (Merck Itachi, Lachrom).

BLM and WHAM simulations

For each experiment, a user-friendly Cu-BLM (www.bio-met.net) was used to predict a bioavailability factor (BioF) and a bioavailable Cu concentration, which is the total dissolved Cu concentration multiplied by the BioF from a limited number of input parameters (dissolved Cu concentration, DOC, Ca and pH).

To predict the dissolved copper speciation and particularly its complexation with organic ligands, the WHAM VI Model was applied [START_REF] Tipping | WHAMa chemical equilibrium model and computer code for waters, sediments, and soils incorporating a discrete site/electrostatic model of ion-binding by humic substances[END_REF].

Statistical analysis

The results are expressed as the mean± standard deviation. The normal distribution of the data was examined using the Shapiro-Wilk test. Differences in responses were evaluated by ANOVA followed by a post hoc HSD Tukey test of the log of the transformed data. Statistical significance was set at p b 0.05, and analyses were performed using the SigmaStat 3.5 software package.

Principal component analysis (PCA) was applied to evaluate the relationship between the water chemical parameters, copper speciation, Cu-BLM predictions and copper bioaccumulation in the mosses exposed to the 3 types of water. Computations were performed on standardised variables using XLStat 2010 4.03.

Results

Water chemistry and copper speciation

The average conductivity measured over the 5 days of exposure (i.e., 40 μS cm -1 , 280 μS cm -1 and 1870 μS cm -1 for Sites 1, 2 and 3, respectively) covered a wide range of ionic strengths, which is explained by the different contents of Cl -, SO 4 2-, Ca 2+ , Mg 2+ , Na + and K + in each water sample (Table 1). Furthermore, for each experiment, a daily evolution exhibited a decrease in the temperature at night combined with a decrease in the oxygen and pH (Fig. S1).

The dissolved Cu concentrations were very similar to the total ones that included the particulate phase, which indicates that the Physico-chemical parameters measured during the 5-day exposures at Sites 1, 2 and 3. Mean ± SD (n = 20).

Site 1

Site 2 Site 3

Cl -(mg L -1 ) 0.7 ± 0.0 17.2 ± 0.1 377.7 ± 2.2 SO 4

2-(mg L -1 ) 10.3 ± 0.1 13.7 ± 0.1 132.6 ± 1.6 Ca 2+ (mg L -1 ) 2.8 ± 0.1 28.7 ± 0.7 177.6 ± 2.7 Mg 2+ (mg L -1 ) 0.7 ± 0.0 4.9 ± 0.1 21.8 ± 0.4 K + (mg L -1 ) 0.9 ± 0.1 4.6 ± 0.3 17.0 ± 1.1 DOC (mg L -1 ) 2.6 ± 0.9 5.0 ± 1.1 5.3 ± 1.5 DIC (mg L -1 ) 0.7 ± 0.6 15.8 ± 1.4 17.3 ± 2.37 pH 6.9 ± 0.0 8.6 ± 0.2 7.6 ± 0.1 Temperature (°C)

18 ± 1 18 ± 1 23 ± 1 Chlorophyll (μg L -1 ) 0.9 ± 0.7 5 ± 1 -Conductivity (μS cm -1 ) 40 ± 1 280 ± 3 1870 ± 32 Oxygen (mg L -1 ) 9.5 ± 0.2 9.6 ± 0.3 6.3 ± 0.6 DOC: dissolved organic carbon. DIC: dissolved inorganic carbon. Na is not reported here since NaN 3 spiking induced a bias in the measured concentration.

experimental design and especially the efficiency of the lamellar decanter allowed only dissolved exposures. Furthermore, rather stable dissolved copper concentrations were obtained over time (see Fig. S2 for Site 3); the temporal variations of the dissolved Cu concentration were 10% on average in the tank, with a maximum of 15% for the exposures at Site 2. Similarly, the semi-static exposure in the laboratory, which involved a large reservoir (100 L) of MontCalm water (Site 1) and a daily copper spiking, maintained a constant concentration of dissolved copper throughout the exposure (time variation of 10% on average). The measured dissolved copper concentrations were closer to the nominal concentrations for Sites 1 and 3 (4.6± 0.5 μg L -1 and 6.5± 0.6 μg L -1 , respectively, for the 5 μg L -1 exposure; 17.2±1.5 μg L -1 and 15.1±0.6 μg L -1 for the 15 μg L -1 exposure).

For Site 2, the measured total dissolved concentrations were less than the nominal ones (i.e., 1.8± 0.3 μg L -1 and 11.8±1.8 μg L -1 for 5 and 15 μg L -1 , respectively). The concentrations of total dissolved Cu measured by DPASV were very similar to the ones determined by GFAAS (r 2 = 0.98). Additionally, due to its greater sensitivity, DPASV allowed the determination of the total Cu concentrations in the unspiked samples from Site 1 (Table 2). In the unspiked samples, the Cu speciation appeared to be strongly controlled by organic ligands (dominated by inert complexes) as the inorganic fraction represents less than 10% (Fig. 2), which confirms the high affinity of this metal for dissolved organic matter, as observed in various environments [START_REF] Donat | Trace elements in the oceans. Trace elements in natural waters[END_REF]Meylan et al., 2004a;[START_REF] Louis | Speciation of trace metals in natural waters: the influence of an adsorbed layer of natural organic matter (NOM) on voltammetric behaviour of copper[END_REF][START_REF] Bravin | Root-induced changes in pH and dissolved organic matter binding capacity affect copper dynamic speciation in the rhizosphere[END_REF]. When increasing the Cu concentration, the proportion of inorganic Cu also increases, which signifies the progressive saturation of the organic ligands. In the spiked samples from Site 1, Cu appears to be quasi-fully inorganic; therefore, it is likely considerably more bioavailable for the organisms. As an example, Fig. S3 shows the results obtained from Site 3 for individual samples and composite ones (obtained by a mixing of the 5 sub samples). The Cu speciation appeared to be stable between the different samples collected during the timescale of the organism exposition at a defined Cu concentration, which is in contrast to the observed variation when increasing the total Cu concentration. This result is of particular importance when compared to the DGT measurements, which correspond to an integrative value. The DGT-labile Cu accounted for 12% to 82% of the total dissolved Cu (Table 2). As also observed by the DPASV measurements, the fraction of DGT-labile Cu increased when Cu was added to the exposure water.

The free Cu concentration was only measured using the ISE in the exposure water spiked at 5 and 15 μg L -1 because the potential measured in the non-contaminated water was less than the quantification limit (i.e., b 0.01 ng L -1 ) (Table 2). The free Cu concentration ranged between 0.06 ng L -1 in the water spiked at 5 μg L -1 and 4.8 ng L -1 in the water spiked at 15 μg L -1 depending on the site considered. The free Cu fractions were between 0.002% and 0.033% of the total dissolved Cu, and the Cu 2+ concentration fluctuated over time by a factor of 3-6 depending on the exposure.

Bioaccumulation measurement

The Cu concentration in the gammarids slightly increased with the dissolved exposure concentration, although it was not statistically significant. The copper content in the gammarids exposed to 17.2 μg L -1 (Site 1) reached a maximum of 99.6 μg g dw -1

, whereas the initial copper content in the gammarids was 69.3 μg g dw -1 (Table 2).

Table 2 Total, DGT-labile, DPASV-labile, BLM-predicted (μg L -1 ) and free (ng L -1 ) copper concentrations in exposure waters and copper content in mosses and gammarids (μg g dw -1 ). Fig. 2. Evaluation of the different Cu fractions ("inorganic", "weak", "strong" and "inert") measured using DPASV (%) for Sites 1, 2 and 3 as a function of the nominal copper exposure concentrations (0, 5 and 15 μg L -1 ).

A significant Cu accumulation was observed in the aquatic mosses (Table 2). For instance, the total Cu concentration (=C int +C ext ) in the mosses exposed to 1.9, 6.5 and 15.0 μg L -1 of Cu in Site 3 is 17 ± 1, 141 ±19 and 466 ±24 μg g dw -1

, respectively, with a mean initial copper content of 11 μg g dw -1

. The proportion of extracellular Cu in the mosses is a function of the Cu exposure concentration (Table 2). Furthermore, significant correlations were observed between the Cu content of the mosses and the dissolved Cu (r 2 =0.95, p b 0.01), DGT-labile Cu (r 2 = 0.94, p b 0.01) or BLM-predicted concentrations (r 2 =0.73, pb 0.01) but not with the free Cu concentrations (r 2 b 0.24, p >0.29) (Fig. 3).

Oxidative stress

No effect of Cu exposure on lipid peroxidation was observed in the mosses and gammarids (p b 0.05), regardless of the site or the concentration considered (data not shown). The induction of GPX enzymes in the mosses was observed after 5 days of exposure in Site 1 water spiked at 5 and 15 μg L, although the result was only significant (p b 0.05) for the exposure at 15 μg L -1 (Fig. 4).

BLM

The simulations conducted using the BioMet-BLM (Table 2) were determined from the average measured pH, Cu, Ca and DOC concentrations (Table 1). The average pH at Site 2 (i.e., 8.6) and the average Ca concentration at Site 3 (i.e., 178 mg L -1 ) exceeded the range for the use of the BioMet-BLM (i.e., 5.5-8.5 for pH and 3.1-93 mg L -1 for Ca). Therefore, the maximal pH and Ca concentration for the range of applications were used for the simulations.

Discussion

Copper speciation

The characterisation of Cu speciation is an important step to understand bioaccumulation. This characterisation can be performed due to the different tools, such as ISE, DGT, DPASV and WHAM, which were used here to measure the different Cu fractions.

The DGT-labile Cu concentration and the sum of the inorganic Cu and weak organic Cu complexes obtained by DPASV exhibited a similar response pattern; the slope was 1.002 ± 0.047 (r 2 = 0.76), which is in agreement with a previous study [START_REF] Bravin | Root-induced changes in pH and dissolved organic matter binding capacity affect copper dynamic speciation in the rhizosphere[END_REF].

The WHAM model was applied to predict the copper speciation. Typically, 50% of dissolved organic matter (DOM) is considered to be composed of fulvic acid (i.e., active DOM). Under this consideration, the agreement between the WHAM predictions and the experimental inorganic and free Cu concentrations was poor (data not shown). [START_REF] Craven | Copper(II) binding by dissolved organic matter: importance of the copper-to-dissolved organic matter ratio and implications for the biotic ligand model[END_REF] also observed a poor correlation when comparing Cu 2+ predicted by WHAM to the measured concentration and suggested that the WHAM predictions could be improved by refining the Cu-DOM binding constant. With respect to the inorganic fraction, a mean difference of 2 orders of magnitude was observed for the 3 exposure waters. A primary input parameter in the WHAM is the concentration of active DOM because it determines the number of binding sites available to copper. Because the composition of DOM ) as a function of the copper concentration in water (free Cu 2+ , DGT-labile, and total dissolved concentrations are represented as temporal mean±SD) or bioavailable copper concentration predicted by BioMet-BLM (μg L -1 ). was not known, the WHAM was conducted to adjust the percentage of active DOM (i.e., fulvic acids) that best fitted the inorganic copper concentration measured by DPASV. In this manner, the binding capacities of DOM were characterised. The best fit was obtained when the percentage of active DOM was set to 2%, 13% and 16% for Site 1 (r 2 = 0.988), Site 2 (r 2 = 0.988) and Site 3 (r 2 = 0.940), respectively. These values are low when compared to other studies. For instance, [START_REF] Dwane | Testing a humic speciation model by titration of copper-amended natural waters[END_REF] observed that the best match between the WHAM-calculated and ISE-measured Cu 2+ was obtained when 40% to 80% of the DOM was considered to be active fulvic acid. Despite the fact that Cu speciation is primarily controlled by organic ligands (Fig. 2), especially at ambient total Cu concentration, the WHAM modelling suggested that this distribution of the Cu form is assessed by a few organic ligands, which can be oversaturated when increasing the Cu concentration (e.g., during a pollution event). Furthermore, these Cu organic complexes appear to be weak (in terms of binding strength) because they are dissociated by DGT and electrochemically reducible by DPASV at -0.9 V; therefore, they are potentially bioavailable for organisms.

With the optimised percentages of active DOM (i.e., 2%, 13% and 16% for Sites 1, 2 and 3, respectively), the Cu 2+ concentrations predicted by WHAM are 1-3 orders of magnitude less than the Cu 2+ concentrations measured using the ISE. Several other studies that have compared the WHAM-predicted and measured Cu 2+ concentrations have also observed a poor agreement and a predicted stronger binding of DOC to Cu [START_REF] Nolan | Chemical speciation of Zn, Cd, Cu, and Pb in pore waters of agricultural and contaminated soils using Donnan dialysis[END_REF]. Despite this poor agreement, the WHAM successfully predicts the relative daily variation of Cu 2+ measured using the ISE. Indeed, the WHAM-predicted values vary by 0.4-1.7 log units on a daily basis, and the measured ones vary by 0.4-1.6 log units (data not shown). This daily variation could be explained as a decrease in photosynthetic activity at night that leads to an acidification of the water and consequently to an increase of the Cu 2+ concentration [START_REF] Superville | An adaptable automatic trace metal monitoring system for on line measuring in natural waters[END_REF].

Comparison of biological responses, chemical measurements and modelling

A principal component analysis (PCA) was performed to determine the variables that explain the majority of the data variability and to evaluate the contribution of each variable (Fig. 5). Axis 1 reflects the ionic and DOC gradient, whereas axis 2 reflects the copper concentration. Axes 1 and 2 account for 49% and 34%, respectively, of the total variance. The PCA revealed that the bioaccumulation of copper in mosses is highly correlated with the copper concentration (either total dissolved, DGT-labile, inorganic or predicted with BioMet-BLM) and less so with the chemical parameters of water.

With respect to the bioaccumulation, the PCA indicated that bryophytes display a high sensitivity to the dissolved concentration of Cu and its speciation and, to a lesser extent, to physico-chemical characteristics (Fig. 5). In contrast, the oxidative stress measured in the mosses via their GPX activity did not reveal a strong influence of the Cu concentration, but it is inversely correlated with the gradient of ions and DOC. In comparison to the waters from Sites 2 and 3, the water from Site 1 has very low ionic concentrations. Nutrient deficiencies may lead to oxidative stress in terrestrial plants [START_REF] Tewari | Macronutrient deficiencies and differential antioxidant responses-influence on the activity and expression of superoxide dismutase in maize[END_REF], notably in the case of Mg [START_REF] Anza | Time course of antioxidant responses of Capsicum annuum subjected to progressive magnesium deficiency[END_REF][START_REF] Tewari | Magnesium deficiency induced oxidative stress and antioxidant responses in mulberry plants[END_REF], but this has not yet been described for aquatic plants. To investigate whether any nutrient deficiency resulted from exposure to the Site 1 water, a growth test was conducted. The size of the green tips was followed during a 40 day exposure in unspiked waters from Sites 1, 2 and 3. After 40 days, the growth rate was 30% on average, but no significant size difference was observed between the three exposure waters. This result suggests that no deficiency occurred, but the hypothesis of an oxidative stress resulting from the low concentration of Mg in the water from Site 1 remains to be investigated in the moss because [START_REF] Chou | Effect of magnesium deficiency on antioxidant status and cadmium toxicity in rice seedlings[END_REF] observed an oxidative stress in rice submitted to a Mg deprivation for 12 days without altering the plant growth. However, in the present study, we observed that the GPX activity only increased in the water from Site 1 spiked with 15 μg L -1 and not the other two groups. In this condition, the internalised Cu reached maximal values (113.4 μg g dw -1

), which suggests a threshold effect of Cu int on the GPX activity.

Effect of the ionic gradient and DOC on copper uptake

Because the DOC and ions displayed similar gradients among the exposure waters, the PCA did not allow the effect of the DOC to be distinguished from the effect of the ions. However, a primary objective of this study was to investigate how the speciation of Cu (driven by DOC) and competing processes (depending on the concentrations of ions) affect bioaccumulation. To accurately distinguish the effect of the ionic gradient from the effect of the DOC gradient, the results were expressed as uptake rate constants (k u ) (Fig. 6), as explained in the details of [START_REF] Bourgeault | Bioaccumulation of waterborne Ni in Dreissena polymorpha: a stable isotope experiment to assess the effect of zinc, calcium and dissolved organic material[END_REF]. Briefly, the uptake rate constant (L g -1 d -1 ) is calculated as the slope of Cu influx (which is the copper content in organisms at the end of the exposure divided by the exposure time) (μg g -1 d -1 ) vs. the exposure concentrations of Cu (μg L -1 ). Expressing the results as k u allows a comparison of the bioaccumulation measurements in the 3 sites while ignoring that the Cu concentrations are not precisely similar, and also allows the k u values to be calculated from various Cu fractions (e.g., total dissolved, inorganic or labile fraction). Therefore, the effect of DOC The uptake rate constant was not calculated for the gammarids because the accumulation was not significant along the Cu gradient. For the mosses, when the k u values were calculated from the total dissolved Cu concentration or the DGT-labile Cu concentration, no significant differences were observed along either the DOC gradient or the ionic gradient (Fig. 6). In contrast, when k u was calculated from the inorganic copper concentration (i.e., the DPASV concentration measured at -0.3 V), the results widely varied. This variation in the results can be attributed to the bioavailability of weak DOC-complexes of copper that are not considered when the k u values are calculated from the inorganic Cu concentrations, which leads to higher k u values. Some studies have also reported that organisms can access weakly complexed Cu (Meylan et al., 2004b;[START_REF] Ferreira | More than inorganic copper is bioavailable to aquatic mosses at environmentally relevant concentrations[END_REF][START_REF] Ytreberg | Influence of salinity and organic matter on the toxicity of Cu to a brackish water and marine clone of the red macroalga Ceramium tenuicorne[END_REF]. Note that the k u values are similar for Site 1 (i.e., the values at low DOC concentrations) regardless of the Cu fraction considered because >80% of Cu is inorganic in this exposure water (Fig. 2). When the k u values were plotted with the ionic concentration on the X-axis, the same pattern was observed as that for the DOC concentration on the X-axis (Fig. 6). No additional decrease of Cu uptake was observed as the ionic concentration increased, which indicates that no competition process occurs. Therefore, thanks to a full characterisation of Cu speciation, it appears that the uptake is controlled more by organic copper complexation than through competition by major ions.

A potential effect of moss physiology on copper accumulation?

A previous study has shown that bioaccumulation in mosses is governed by ionic competition for membrane ligands, and the impact has been modelled using a two-compartment model calibrated under a wide range of water compositions (Ferreira et al., 2009, in press). However, in the present study, the copper bioavailability appears to be controlled more by organic complexation than major competing ions. Similarly, [START_REF] Peters | Using biotic ligand models to help implement environmental quality standards for metals under the Water Framework Directive[END_REF][START_REF] Peters | Regulatory consideration of bioavailability for metals: simplification of input parameters for the chronic copper biotic ligand model[END_REF] mentioned that calcium have less influence on Cu toxicity, and thus on the overall result of the BLM, than either pH or DOC. As a result, the bioaccumulation model described by Ferreira et al. (in press) did not succeed to predict the Cu content measured in the present study in mosses (Fig. S4). The predicted Cu accumulation in mosses is overestimated by a factor 2 to 75. This observation is quite surprisingly since similar exposure design was used. This poor predictive capacity of the model can be attributed to a difference in Cu (i) adsorption/desorption or (ii) internalisation/elimination capacities among seasons (see below). Indeed, the model described by Ferreira et al. (in press) was validated in the field in winter whereas the present study was conducted in summer. As noticed by [START_REF] Vazquez | Uptake of heavy metals to the extracellular and intracellular compartments in three species of aquatic bryophyte[END_REF], the physiological characteristics of mosses (i.e. external binding sites and detoxification potency) are probably subject to seasonal modification.

We further investigated the physiological processes (either adsorption/desorption or internalisation/elimination) at the root of accumulation changes. In Ferreira et al. (in press) model, the internalisation/elimination processes, formalised as the ratio of an internalisation rate constant and an elimination rate constant, is constant and can be assess through the intra/extra-cellular distribution. In the present study, this intra/extra-cellular distribution is actually fairly constant when mosses are exposed to 5-20 μg L -1 of copper and is comparable to the one reported by Ferreira et al. (in press) (Fig. S5). Thus, the overestimation of Cu accumulation by the model for the exposures at 5-20 μg L -1 could not be explained by a seasonal effect on internalisation/elimination processes. Nevertheless, it can be observed that the intra/extra ratio decreased when the exposure concentration was less than 5 μg L -1 (Fig. S5), which suggests a poorer predictive capacity of the model for low Cu concentrations because the model considers that the ratio is constant regardless of the concentration. Nonetheless, as the poor predictive capacity of the model cannot be attributed to internalisation/elimination processes (at least for 5-20 μg L -1 exposure concentrations), the season likely modifies the adsorption/desorption processes by acting on extra cellular binding sites of the biological membrane. We are utilising a relatively small data set. It is likely that the inferred hypothesis could be validated from experiments that cover the entire life cycle of mosses.

The present study emphasises the major role of speciation on the bioavailability of Cu in natural waters and supports the incorporation of BLM-BioMet into regulations. However, it is also of considerable importance to consider the biological behaviour of organisms. Indeed, differences in the metal uptake capacity of aquatic organisms can be observed among different seasons (resulting either from physiological modification and/or from the physico-chemical characteristics of water systems evolution). A modification of the organisms' physiology can also occur [START_REF] Vazquez | Uptake of heavy metals to the extracellular and intracellular compartments in three species of aquatic bryophyte[END_REF][START_REF] Bourgeault | Modeling the effect of water chemistry on the bioaccumulation of waterborne cadmium in zebra mussels[END_REF], which is not explicitly considered in the BioMet-BLM. For instance, an increase in the DOC concentration from 0.2 to 1.1 mgC L -1 increases the filtration rate of freshwater mussels by a factor of 6 and consequently enhances the accumulation of metals [START_REF] Bourgeault | Modeling the effect of water chemistry on the bioaccumulation of waterborne cadmium in zebra mussels[END_REF]. Furthermore, because Cu speciation is related to the strength of organic binding sites, a seasonal variation of Cu bioavailability could also be expected with the seasonal type of DOC. For instance, [START_REF] Luider | Influence of natural organic matter source on copper speciation as demonstrated by Cu binding to fish gills, by ion selective electrode, and by DGT gel sampler[END_REF] have shown that allochthonous natural organic matter (NOM) (terrestrial derived) decreases the Cu bioaccumulation better than autochthonous NOM (algal derived).

Conclusions

Chemical speciation and biological responses were analysed in mesocosms to obtain insight into the site-specific bioavailability of copper. The mesocosms allowed organisms to be exposed to different environmental Cu concentrations (natural level and spiked with 5 and 15 μg L -1 ) in the same natural water, which consequently ) Fig. 6. Uptake rate constant of copper (L g -1 d -1 ) (k u ) calculated in moss from the total dissolved copper concentration ( ), the inorganic copper concentration determined using DPASV (Δ) or the labile copper concentration determiner using DGT (■) as a function of the conductivity (μS cm -1 ) and the dissolve organic carbon (mg L -1 ).

allowed an accurate estimation of the confounding factors on the uptake of Cu. Aquatic mosses exhibited high enrichment rates that were above the background levels compared to gammarids. WHAM calculations using the default parameters failed to simulate the distribution of Cu measured by DPASV. However, a simple adjustment of the percentage of reactive organic matter allows the actual reactivity of dissolved organic ligands present in the studied systems to be obtained. This adaptation appears to be necessary to avoid inaccurate evaluations of metal speciation in novel ecosystems. The DGT and DPASV techniques were demonstrated to be efficient tools for assessing Cu speciation and for providing useful information to understand the uptake processes. Indeed, by determining the uptake rate constants from the DGT or DPASV-Cu concentrations, the results suggest that the major ions exert no significant effect on the bioaccumulation in mosses and that organic complexation accounts for the bioavailability of Cu. Furthermore, weak organic complexes of Cu were identified as being bioavailable for aquatic mosses. To assess the hazard and risk posed by Cu to aquatic organisms, BLM appears to be a promising approach for predicting the site-specific bioavailability of Cu. However, to fully validate the predictive capacity of the user-friendly BLM, a study implying seasonal variations should be conducted to assess an eventual effect of the physiological state of aquatic organisms. Indeed, the adsorption/desorption process of Cu is likely modified in mosses by the season.

Fig. 3 .

 3 Fig. 3. Total copper concentration in moss (μg g dry weight -1

Fig. 4 .

 4 Fig.4. GPX activity (as % of D2 value of unexposed group) measured in Fontinalis antipyretica after 2 and 5 days of exposure at 0, 5 and 15 μg L -1 of copper for the 3 sites. * Significantly different from the values of D5 group exposed at 15 μg L -1 of Cu (pb 0.05).

Fig. 5 .

 5 Fig. 5. PCA plot of chemical and biological responses measured after exposures fromSite 1 (S1), Site 2 (S2) and Site 3 (S3). BB: Bioaccumulation Bryophytes, DOC: dissolved organic carbon, DIC: dissolved inorganic carbon. in: inorganic, w: weak complexes, s: strong complexes, tot: total, BioMet: predicted by BioMet BLM model.

Table 1

 1

A. Bourgeault et al. / Science of the Total Environment 452-453 (2013) 68-77

Acknowledgement

The authors thank S. Maestri, H. Salhi, S. Gaillard and V. Landrault for their support in field work and sample analysis.

Conflict of interest

There is no conflict of interest.

Appendix A. Supplementary data

Supplementary data to this article can be found online at http:// dx.doi.org/10.1016/j.scitotenv.2013.01.097.


Marius Cocou
email: cocou@lma.cnrs-mrs.fr

A class of dynamic contact problems with Coulomb friction in viscoelasticity

Keywords: Dynamic problems, pointwise conditions, Coulomb friction, viscoelasticity, set-valued mapping 35Q74, 49J40, 74A55, 74D05, 74H20

The aim of this work is to study a class of nonsmooth dynamic contact problem which model several surface interactions, including relaxed unilateral contact conditions, adhesion and Coulomb friction laws, between two viscoelastic bodies of Kelvin-Voigt type. An abstract formulation which generalizes these problems is considered and the existence of a solution is proved by using Ky Fan's fixed point theorem, suitable approximation properties, several estimates and compactness arguments.

Introduction

This paper is concerned with the study of a class of dynamic contact problems which describe various surface interactions between two Kelvin-Voigt viscoelastic bodies. These interactions can include some relaxed unilateral contact, pointwise friction or adhesion conditions.

Existence and approximation of solutions to the quasistatic elastic problems have been studied for different contact conditions. The quasistatic unilateral contact problems with local Coulomb friction have been studied in [START_REF] Andersson | Existence results for quasistatic contact problems with Coulomb friction[END_REF][START_REF] Rocca | Existence and approximation of a solution to quasistatic Signorini problem with local friction[END_REF][START_REF] Rocca | Numerical analysis of quasistatic unilateral contact problems with local friction[END_REF] and adhesion laws based on the evolution of intensity of adhesion were analyzed in [START_REF] Raous | A consistent model coupling adhesion, friction, and unilateral contact[END_REF][START_REF] Cocou | Existence results for unilateral quasistatic contact problems with friction and adhesion[END_REF]. The normal compliance models, which are particular regularizations of the Signorini's conditions, have been investigated by several authors, see e.g. [START_REF] Kikuchi | Contact Problems in Elasticity : A Study of Variational Inequalities and Finite Element Methods[END_REF][START_REF] Han | Quasistatic Contact Problems in Viscoelasticity and Viscoplasticity[END_REF][START_REF] Shillor | Models and Analysis of Quasistatic Contact[END_REF] and references therein.

1

A unified approach, which can be applied to various quasistatic problems, including unilateral and bilateral contact with nonlocal friction, or normal compliance conditions, has been considered recently in [START_REF] Badea | Internal and subspace correction approximations of implicit variational inequalities[END_REF].

The corresponding dynamic contact problems are more difficult to solve than the quasistatic ones, even in the viscoelastic case. Dynamic frictional contact problems with normal compliance laws for a viscoelastic body have been studied in [START_REF] Martins | Existence and uniqueness results for dynamic contact problems with nonlinear normal and friction interface laws[END_REF][START_REF] Kikuchi | Contact Problems in Elasticity : A Study of Variational Inequalities and Finite Element Methods[END_REF][START_REF] Kuttler | Dynamic friction contact problems for general normal and friction laws[END_REF][START_REF] Chau | A dynamic frictional contact problem with normal damped response[END_REF][START_REF] Migórski | Nonlinear Inclusions and Hemivariational Inequalities[END_REF]. Nonlocal friction laws, obtained by suitable regularizations of the normal component of the stress vector appearing in the Coulomb friction conditions, were considered for viscoelastic bodies in [START_REF] Jarušek | Dynamic contact problems with given friction for viscoelastic bodies[END_REF][START_REF] Kuttler | Dynamic bilateral contact with discontinuous friction coefficient[END_REF][START_REF] Kuttler | Dynamic contact with Signorini's condition and slip rate depending friction[END_REF][START_REF] Eck | Unilateral Contact Problems -Variational Methods and Existence Theorems[END_REF][START_REF] Cocou | Existence of solutions of a dynamic Signorini's problem with nonlocal friction in viscoelasticity[END_REF][START_REF] Cocou | Analysis of a dynamic unilateral contact problem for a cracked viscoelastic body[END_REF]. Dynamic frictionless problems with adhesion have been studied in [START_REF] Chau | Dynamic frictionless contact with adhesion[END_REF][START_REF] Kuttler | Existence and regularity for dynamic viscoelastic adhesive contact with damage[END_REF][START_REF] Sofonea | Analysis and Approximation of Contact Problems with Adhesion or Damage[END_REF] and dynamic viscoelastic problems coupling unilateral contact, recoverable adhesion and nonlocal friction have been analyzed in [START_REF] Cocou | A dynamic unilateral contact problem with adhesion and friction in viscoelasticity[END_REF][START_REF] Cocou | A class of implicit evolution inequalities and applications to dynamic contact problems[END_REF].

Note that, using the Clarke subdifferential, the variational formulations of various contact problems can be given as hemivariational inequalities, which represent a broad generalization of the variational inequalities to locally Lipschitz functions, see [START_REF] Migórski | Nonlinear Inclusions and Hemivariational Inequalities[END_REF][START_REF] Migorski | A unified approach to dynamic contact problems in viscoelasticity[END_REF][START_REF] Naniewicz | Mathematical Theory of Hemivariational Inequalities and Applications[END_REF][START_REF] Panagiotopoulos | Hemivariational Inequalities: Applications in Mechanics and Engineering[END_REF] and references therein.

A static contact problem with relaxed unilateral conditions and pointwise Coulomb friction was studied in [START_REF] Rabier | Fixed points of multi-valued maps and static Coulomb friction problems[END_REF], based on new abstract formulations and on Ky Fan's fixed point theorem. The extension to an elastic quasistatic contact problem was investigated in [START_REF] Cocou | Sur la modélisation mathématique de conditions unilatèrales en mécanique du contact[END_REF].

This work extends the results in [START_REF] Rabier | Fixed points of multi-valued maps and static Coulomb friction problems[END_REF] to a new class of nonsmooth dynamic contact problems in viscoelasticity, which constitutes a unified approach to study some complex surface interactions.

The paper is organized as follows. In Section 2 the classical formulation of the dynamic contact problem is presented and the variational formulation is given as a two-field problem. Section 3 is devoted to the study of a more general evolution variational inequality, which is written as an equivalent fixed point problem, based on some existence and uniqueness results proved in [START_REF] Cocou | Analysis of a dynamic unilateral contact problem for a cracked viscoelastic body[END_REF]. Using the Ky Fan's theorem, the existence of a fixed point is proved. In Section 4 this abstract result is used to prove the existence of a variational solution of the dynamic contact problem.

Classical and variational formulations

We consider two viscoelastic bodies, characterized by a Kelvin-Voigt constitutive law, which occupy the reference domains

Ω α of R d , d = 2 or 3, with Lipschitz boundaries Γ α = ∂Ω α , α = 1, 2. Let Γ α U , Γ α F and Γ α C be three open disjoint sufficiently smooth parts of Γ α such that Γ α = Γ α U ∪ Γ α F ∪ Γ α C
and, to simplify the estimates, meas(Γ α U ) > 0, α = 1, 2. In this paper we assume the small deformation hypothesis and we use Cartesian coordinate representations. Let y α (x α , t) denote the position at time t ∈ [0, T ], where 0 < T < +∞, of the material point represented by x α in the reference configuration, and u α (x α , t) := y α (x α , t)x α denote the displacement vector of x α at time t, with the Cartesian coordinates u α = (u α 1 , ..., u α d ) = (ū α , u α d ). Let ε α , with the Cartesian coordinates ε α = (ε ij (u α )), and σ α , with the Cartesian coordinates σ α = σ α ij , be the infinitesimal strain tensor and the stress tensor, respectively, corresponding to Ω α , α = 1, 2.

Assume that the displacement U α = 0 on Γ α U × (0, T ), α = 1, 2, and that the densities of both bodies are equal to 1. Let

f 1 = (f 1 1 , f 2 1
) and

f 2 = (f 1 2 , f 2
2 ) denote the given body forces in Ω 1 ∪ Ω 2 and tractions on Γ 1 F ∪ Γ 2 F , respectively. The initial displacements and velocities of the bodies are denoted by

u 0 = (u 1 0 , u 2 0 ), u 1 = (u 1 1 , u 2 1
). The usual summation convention will be used for i, j, k, l = 1, . . . , d.

Suppose that the solids can be in contact between the potential contact surfaces Γ 1 C and Γ 2 C which can be parametrized by two C 1 functions, ϕ 1 , ϕ 2 , defined on an open and bounded subset Ξ of

R d-1 , such that ϕ 1 (ξ)-ϕ 2 (ξ) ≥ 0 ∀ ξ ∈ Ξ and each Γ α C is the graph of ϕ α on Ξ that is Γ α C = { (ξ, ϕ α (ξ)) ∈ R d ; ξ ∈ Ξ}, α = 1, 2.
Define the initial normalized gap between the two contact surfaces by

g 0 (ξ) = ϕ 1 (ξ) -ϕ 2 (ξ) 1 + |∇ϕ 1 (ξ)| 2 ∀ ξ ∈ Ξ.
Let n α denote the unit outward normal vector to Γ α , α = 1, 2. We shall use the following notations for the normal and tangential components of a displacement field v α , α = 1, 2, of the relative displacement corresponding to v := (v 1 , v 2 ) and of the stress vector σ α n α on Γ α C :

v α (ξ, t) := v α (ξ, ϕ α (ξ), t), v α N (ξ, t) := v α (ξ, t) • n α (ξ), v N (ξ, t) := v 1 N (ξ, t) + v 2 N (ξ, t), [v N ](ξ, t) := v N (ξ, t) -g 0 (ξ), v α T (ξ, t) := v α (ξ, t) -v α N (ξ, t)n α (ξ), v T (ξ, t) := v 1 T (ξ, t) -v 2 T (ξ, t), σ α N (ξ, t) := (σ α (ξ, t)n α (ξ)) • n α (ξ), σ α T (ξ, t) = σ α (ξ, t)n α (ξ) -σ α N (ξ, t)n α (ξ),
for all ξ ∈ Ξ and for all t ∈ [0, T ]. Let g := -[u N ] = g 0u 1 Nu 2 N be the gap corresponding to the solution u := (u 1 , u 2 ). Since the displacements, their derivatives and the gap are assumed small, by using a similar method as the one presented in [START_REF] Boieri | Existence, uniqueness, and regularity results for the two-body contact problem[END_REF] (see also [START_REF] Cocou | Analysis of a dynamic unilateral contact problem for a cracked viscoelastic body[END_REF]) we obtain the following unilateral contact condition at time t in the set Ξ: [u N ] (ξ, t) = -g(ξ, t) ≤ 0 ∀ ξ ∈ Ξ.

Classical formulation

Let A α , B α denote two fourth-order tensors, the elasticity tensor and the viscosity tensor corresponding to Ω α , with the components A α = (A α ijkl ) and B α = (B α ijkl ), respectively. We assume that these components satisfy the following classical symmetry and ellipticity conditions:

C α ijkl = C α jikl = C α klij ∈ L ∞ (Ω α ), ∀ i, j, k, l = 1, . . . , d, ∃ α C α > 0 such that C α ijkl τ ij τ kl ≥ α C α τ ij τ ij ∀ τ = (τ ij ) verifying τ ij = τ ji , ∀ i, j = 1, . . . , d, where C α ijkl = A α ijkl , C α = A α or C α ijkl = B α ijkl , C α = B α ∀ i, j, k, l = 1, . . . , d, α = 1, 2.
We choose the following state variables: the infinitesimal strain tensor

(ε 1 , ε 2 ) =(ε(u 1 ), ε(u 2 )) in Ω 1 ∪ Ω 2 , the relative normal displacement [u N ] = u 1 N + u 2
Ng 0 , and the relative tangential displacement u T = u 1 Tu 2 T in Ξ. Let κ, κ : R → R be two mappings with κ lower semicontinuous and κ upper semicontinuous, satisfying the following conditions:

κ(s) ≤ κ(s) and 0 / ∈ (κ(s), κ(s)) ∀ s ∈ R, (1)
∃ r 0 ≥ 0 such that max(|κ(s)|, |κ(s)|) ≤ r 0 ∀ s ∈ R. (2)
Consider the following dynamic viscoelastic contact problem with Coulomb friction.

Problem P c : Find u = (u 1 , u 2 ) such that u(0) = u 0 , u(0) = u 1 and, for all t ∈ (0, T ), üα -div σ α (u α , uα ) = f α 1 in Ω α , (3) σ α (u α , uα ) = A α ε(u α ) + B α ε( uα ) in Ω α , (4)
u α = 0 on Γ α U , σ α n α = f α 2 on Γ α F , α = 1, 2, ( 5
) σ 1 n 1 + σ 2 n 2 = 0 in Ξ, (6) κ([u N ]) ≤ σ N ≤ κ([u N ]) in Ξ, (7) |σ T | ≤ µ |σ N | in Ξ and (8)
|σ T | < µ |σ N | ⇒ uT = 0, |σ T | = µ |σ N | ⇒ ∃ θ ≥ 0, uT = -θσ T ,
where

σ α = σ α (u α , uα ), α = 1, 2, σ N := σ 1 N , σ T := σ 1 T , and µ ∈ L ∞ (Ξ), µ ≥ 0 a.e. in Ξ, is the coefficient of friction.
Different choices for κ, κ will give various contact and friction conditions as can be seen in the following examples.

Example 1. (Adhesion and friction conditions)

Let s 0 ≥ 0, M ≥ 0 be constants, k : R → R be a continuous function such that k ≥ 0 with k(0) = 0 and define

κ(s) =    0 if s ≤ -s 0 , k(s) if -s 0 < s < 0, -M if s ≥ 0, κ(s) =    0 if s < -s 0 , k(s) if -s 0 ≤ s ≤ 0, -M if s > 0.

Example 2. (Friction condition)

In Example 1 we set k = s 0 = 0 and define

κ M (s) = 0 if s < 0, -M if s ≥ 0, κ M (s) = 0 if s ≤ 0, -M if s > 0.
The classical Signorini's conditions correspond formally to M = +∞.

Example 3. (General normal compliance conditions)

Various normal compliance conditions, friction and adhesion laws can be obtained from the previous general formulation if one considers κ = κ = κ, where κ : R → R is some bounded Lipschitz continuous function with κ(0) = 0, so that σ N is given by the relation

σ N = κ([u N ]).

Variational formulations

We adopt the following notations:

H s (Ω α ) := H s (Ω α ; R d ), α = 1, 2, H s := H s (Ω 1 ) × H s (Ω 2 ), v, w -s,s = v 1 , w 1 H -s (Ω 1 )×H s (Ω 1 ) + v 2 , w 2
H -s (Ω 2 )×H s (Ω 2 ) ∀ v = (v 1 , v 2 ) ∈ H -s , ∀ w = (w 1 , w 2 ) ∈ H s , ∀ s ∈ R.
Define the Hilbert spaces (H, |.|) with the associated inner product denoted by (. , .), (V , . ) with the associated inner product (of H 1 ) denoted by . , . , and the closed convex cones L 2 + (Ξ), L 2 + (Ξ × (0, T )) as follows:

H := H 0 = L 2 (Ω 1 ; R d ) × L 2 (Ω 2 ; R d ), V := V 1 × V 2 ,
where

V α = {v α ∈ H 1 (Ω α ); v α = 0 a.e. on Γ α U }, α = 1, 2, L 2 + (Ξ) := {δ ∈ L 2 (Ξ); δ ≥ 0 a.e. in Ξ}, L 2 + (Ξ × (0, T )) := {η ∈ L 2 (0, T ; L 2 (Ξ)); η ≥ 0 a.e. in Ξ × (0, T )}.
Let a, b be two bilinear, continuous and symmetric mappings defined on

H 1 × H 1 → R by a(v, w) = a 1 (v 1 , w 1 ) + a 2 (v 2 , w 2 ), b(v, w) = b 1 (v 1 , w 1 ) + b 2 (v 2 , w 2 ) ∀ v = (v 1 , v 2 ), w = (w 1 , w 2 ) ∈ H 1 , where, for α = 1, 2, a α (v α , w α ) = Ω α A α ε(v α ) • ε(w α ) dx, b α (v α , w α ) = Ω α B α ε(v α ) • ε(w α ) dx. Assume f α 1 ∈ W 1,∞ (0, T ; L 2 (Ω α ; R d )), f α 2 ∈ W 1,∞ (0, T ; L 2 (Γ α F ; R d )), α = 1, 2, u 0 , u 1 ∈ V , g 0 ∈ L 2 + (Ξ)
, and define the following mappings:

J : L 2 (Ξ) × H 1 → R, J(δ, v) = Ξ µ |δ| |v T | dξ ∀ δ ∈ L 2 (Ξ), ∀ v = (v 1 , v 2 ) ∈ H 1 , f ∈ W 1,∞ (0, T ; H 1 ), f , v = α=1,2 Ω α f α 1 • v α dx + α=1,2 Γ α F f α 2 • v α ds ∀ v = (v 1 , v 2 ) ∈ H 1 , ∀ t ∈ [0, T ].
Assume the following compatibility conditions:

[u 0N ] ≤ 0, κ([u 0N ]) = 0 a.e. in Ξ and ∃ p 0 ∈ H such that (p 0 , w) + a(u 0 , w) + b(u 1 , w) = f (0), w ∀ w ∈ V . (9)
The following compactness theorem proved in [START_REF] Simon | Compact sets in the space L p (0, T ; B)[END_REF] will be used several times in this paper.

Theorem 2.1. Let X, Û and Ŷ be three Banach spaces such that X ⊂ Û ⊂ Ŷ with compact embedding from X into Û .

(i) Let F be bounded in L p (0, T ; X), where 1 ≤ p < ∞, and ∂F/∂t := { ḟ ; f ∈ F} be bounded in L 1 (0, T ; Ŷ ). Then F is relatively compact in L p (0, T ; Û ).

(ii) Let F be bounded in L ∞ (0, T ; X) and ∂F/∂t be bounded in L r (0, T ; Ŷ ), where r > 1. Then F is relatively compact in C([0, T ]; Û ).

For every ζ ∈ L 2 (0, T ; L 2 (Ξ)) = L 2 (Ξ × (0, T )), define the following sets:

Λ(ζ) = {η ∈ L 2 (0, T ; L 2 (Ξ)); κ • ζ ≤ η ≤ κ • ζ a.e. in Ξ × (0, T ) }, Λ + (ζ) = {η ∈ L 2 + (Ξ × (0, T )); κ + • ζ ≤ η ≤ κ + • ζ a.e. in Ξ × (0, T ) }, Λ -(ζ) = {η ∈ L 2 + (Ξ × (0, T )); κ -• ζ ≤ η ≤ κ -• ζ a.e. in Ξ × (0, T ) },
where, for each r ∈ R, r + := max(0, r) and r -:= max(0, -r) denote the positive and negative parts, respectively. Since meas(Ξ) < ∞ and κ, κ satisfy (2), it is also readily seen that there exists a constant, denoted by R 0 and depending on meas(Ξ), r 0 and T , such that for all ζ ∈ L 2 (0, T ; L 2 (Ξ)) the sets Λ + (ζ) and Λ -(ζ) are bounded in norm in L 2 (0, T ; L 2 (Ξ)) by R 0 . Moreover, these sets are bounded in L ∞ (0, T ; L ∞ (Ξ)).

A first variational formulation of the problem P c is the following.

Problem P 1 v : Find u ∈ C 1 ([0, T ]; H -ι )∩W 1,2 (0, T ; V ), λ ∈ L 2 (0, T ; L 2 (Ξ)) such that u(0) = u 0 , u(0) = u 1 , λ ∈ Λ([u N ]) and u(T ), v(T ) -u(T ) -ι, ι -(u 1 , v(0) -u 0 ) - T 0 ( u, v -u) dt + T 0 a(u, v -u) + b( u, v -u) -(λ, v N -u N ) L 2 (Ξ) dt (10)
+ T 0 {J(λ, v + k u -u) -J(λ, k u)} dt ≥ T 0 f , v -u dt ∀ v ∈ L ∞ (0, T ; V ) ∩ W 1,2 (0, T ; H), where 1 > ι > 1 2 , k > 0.
The formal equivalence between the variational problem P 1 v and the classical problem (3)-( 8) can be easily proved by using Green's formula and an integration by parts, where the Lagrange multiplier λ satisfies the relation

λ = σ N . Let φ : L 2 + (Ξ) × L 2 + (Ξ) × V → R be defined by φ(δ 1 , δ 2 , v) = -(δ 1 -δ 2 , v N ) L 2 (Ξ) + Ξ µ (δ 1 + δ 2 ) |v T | dξ ∀ (δ 1 , δ 2 ) ∈ (L 2 + (Ξ)) 2 , ∀ v = (v 1 , v 2 ) ∈ V . ( 11
) Since η ∈ Λ(ζ) if and only if (η + , η -) ∈ Λ + (ζ) × Λ -(ζ)
, it follows that the variational problem P 1 v is clearly equivalent with the following problem.

Problem P 2 v : Find u ∈ C 1 ([0, T ]; H -ι )∩W 1,2 (0, T ; V ), λ ∈ L 2 (0, T ; L 2 (Ξ)) such that u(0) = u 0 , u(0) = u 1 , (λ + , λ -) ∈ Λ + ([u N ]) × Λ -([u N ]) and u(T ), v(T ) -u(T ) -ι, ι -(u 1 , v(0) -u 0 ) + T 0 {-( u, v -u) + a(u, v -u) + b( u, v -u)} dt (12)
+ T 0 {φ(λ + , λ -, v + k u -u) -φ(λ + , λ -, k u)} dt ≥ T 0 f , v -u dt ∀ v ∈ L ∞ (0, T ; V ) ∩ W 1,2 (0, T ; H).
The existence of variational solutions to problem P c will be established by using some abstract existence results that will be presented in the following section.

General existence results

Let U 0 , (V 0 , . , . , . ), (U, . U ) and (H 0 , |.|, (. , .)) be four Hilbert spaces such that U 0 is a closed linear subspace of V 0 dense in H 0 , V 0 ⊂ U ⊆ H 0 with continuous embeddings and the embedding from V 0 into U is compact.

To simplify the presentation and in view of the applications to contact problems, L 2 (Ξ) will be preserved in the abstract formulation even if, more generally, the space L 2 ( Ξ) can be considered with ( Ξ, m) a finite and complete measure space, see [START_REF] Rabier | Fixed points of multi-valued maps and static Coulomb friction problems[END_REF] for a time-independent application. Also, we use the notation Ξ T := Ξ × (0, T ).

Let a 0 , b 0 : V 0 × V 0 → R be two bilinear and symmetric forms such that

∃ M a , M b > 0 a 0 (u, v) ≤ M a u v , b 0 (u, v) ≤ M b u v , (13)
∃ m a , m b > 0 a 0 (v, v) ≥ m a v 2 , b 0 (v, v) ≥ m b v 2 ∀ u, v ∈ V 0 . ( 14
)
Let l : V 0 → L 2 (Ξ) and φ 0 :

L 2 + (Ξ) × L 2 + (Ξ) × V 0 →
R be two mappings satisfying the following conditions:

∃ k 1 > 0 such that ∀ v 1 , v 2 ∈ V 0 , l(v 1 ) -l(v 2 ) L 2 (Ξ) ≤ k 1 v 1 -v 2 U , (15)
∀ γ 1 , γ 2 ∈ L 2 + (Ξ), ∀ θ ≥ 0, ∀ v 1 , v 2 , v ∈ V 0 , φ 0 (γ 1 , γ 2 , v 1 + v 2 ) ≤ φ 0 (γ 1 , γ 2 , v 1 ) + φ 0 (γ 1 , γ 2 , v 2 ), (16)
φ 0 (γ 1 , γ 2 , θv) = θ φ 0 (γ 1 , γ 2 , v), (17)
∀ v ∈ V 0 , φ 0 (0, 0, v) = 0, (18)
∀ γ 1 , γ 2 ∈ L 2 + (Ξ), ∀ v ∈ U 0 , φ 0 (γ 1 , γ 2 , v) = 0, (19)
∃ k 2 > 0 such that ∀ γ 1 , γ 2 , δ 1 , δ 2 ∈ L 2 + (Ξ), ∀ v 1 , v 2 ∈ V 0 , |φ 0 (γ 1 , γ 2 , v 1 ) -φ 0 (γ 1 , γ 2 , v 2 ) + φ 0 (δ 1 , δ 2 , v 2 ) -φ 0 (δ 1 , δ 2 , v 1 )| ≤ k 2 ( γ 1 -δ 1 L 2 (Ξ) + γ 2 -δ 2 L 2 (Ξ) ) v 1 -v 2 U , (20) if
(γ n 1 , γ n 2 ) ∈ (L 2 + (Ξ T )) 2 for all n ∈ N and (γ n 1 , γ n 2 ) ⇀ (γ 1 , γ 2 ) in (L 2 (0, T ; L 2 (Ξ))) 2 , then T 0 φ 0 (γ n 1 , γ n 2 , v) ds → T 0 φ 0 (γ 1 , γ 2 , v) ds ∀ v ∈ L 2 (0, T ; V 0 ). ( 21
)
Remark 3.1. i) Since by [START_REF] Kikuchi | Contact Problems in Elasticity : A Study of Variational Inequalities and Finite Element Methods[END_REF]

φ 0 (•, •, 0) = 0, from (20), for v 2 = 0, v 1 = v, we have ∀ γ 1 , γ 2 , δ 1 , δ 2 ∈ L 2 + (Ξ), ∀ v ∈ V 0 , |φ 0 (γ 1 , γ 2 , v) -φ 0 (δ 1 , δ 2 , v)| ≤ k 2 ( γ 1 -δ 1 L 2 (Ξ) + γ 2 -δ 2 L 2 (Ξ) ) v U . ( 22
)
ii) From ( 18) and ( 20), for δ 1 = δ 2 = 0, we derive

∀ γ 1 , γ 2 ∈ L 2 + (Ξ), ∀ v 1 , v 2 ∈ V 0 , |φ 0 (γ 1 , γ 2 , v 1 ) -φ 0 (γ 1 , γ 2 , v 2 )| ≤ k 2 ( γ 1 L 2 (Ξ) + γ 2 L 2 (Ξ) ) v 1 -v 2 U . ( 23
) iii) If (γ n 1 , γ n 2 ) ∈ (L 2 + (Ξ T )) 2 , for all n ∈ N, (γ n 1 , γ n 2 ) ⇀ (γ 1 , γ 2 ) in (L 2 (0, T ; L 2 (Ξ))) 2 , and v m → v in L 2 (0, T ; U ), then lim n,m→∞ T 0 φ 0 (γ n 1 , γ n 2 , v m ) ds → T 0 φ 0 (γ 1 , γ 2 , v) ds, ( 24
)
which can be proved by taking into account [START_REF] Migorski | A unified approach to dynamic contact problems in viscoelasticity[END_REF] in the following relations:

| T 0 {φ 0 (γ n 1 , γ n 2 , v m ) -φ 0 (γ 1 , γ 2 , v)} ds| ≤ T 0 |φ 0 (γ n 1 , γ n 2 , v m ) -φ 0 (γ n 1 , γ n 2 , v)| ds + | T 0 {φ 0 (γ n 1 , γ n 2 , v) -φ 0 (γ 1 , γ 2 , v)} ds| ≤ T 0 k 2 ( γ n 1 L 2 (Ξ) + γ n 2 L 2 (Ξ) ) v m -v U ds +| T 0 {φ 0 (γ n 1 , γ n 2 , v) -φ 0 (γ 1 , γ 2 , v)} ds|,
and passing to limits by using [START_REF] Kuttler | Existence and regularity for dynamic viscoelastic adhesive contact with damage[END_REF] and that (γ n 1,2 ) n are bounded in L 2 (0, T ; L 2 (Ξ)). Assume that f 0 ∈ W 1,∞ (0, T ; V 0 ), u 0 , u 1 ∈ V 0 are given, and that the following compatibility condition holds: κ(l(u 0 )) = 0 and ∃ p 0 ∈ H 0 such that (p 0 , w) + a 0 (u 0 , w)

+ b 0 (u 1 , w) = f 0 (0), w ∀ w ∈ V 0 . ( 25
)
Consider the following problem.

Problem Q 1 : Find u ∈ W 0 , λ ∈ L 2 (0, T ; L 2 (Ξ)) such that u(0) = u 0 , u(0) = u 1 , (λ + , λ -) ∈ Λ + (l(u)) × Λ -(l(u)) and u(T ), v(T ) -u(T ) U ′ × U -(u 1 , v(0) -u 0 ) + T 0 {-( u, v -u) + a 0 (u, v -u) + b 0 ( u, v -u)} dt (26)
+ T 0 {φ 0 (λ + , λ -, v + k u -u) -φ 0 (λ + , λ -, k u)} dt ≥ T 0 f 0 , v -u dt ∀ v ∈ L ∞ (0, T ; V 0 ) ∩ W 1,2 (0, T ; H 0 ),
where

W 0 := C 1 ([0, T ]; U ′ ) ∩ W 1,2 (0, T ; V 0 ). The sets Λ + (ζ), Λ -(ζ)
and Λ(ζ) have the following useful properties, see also [START_REF] Rabier | Fixed points of multi-valued maps and static Coulomb friction problems[END_REF].

Lemma 3.1. Let ζ ∈ L 2 (0, T ; L 2 (Ξ)) and (η 1 , η 2 ) ∈ Λ + (ζ) × Λ -(ζ). Then η 1 η 2 = 0 a.e. in Ξ T and there exists η ∈ Λ(ζ) such that η + = η 1 , η -= η 2 a.e. in Ξ T . Proof. If κ + • ζ ≤ η 1 ≤ κ + • ζ and κ -• ζ ≤ η 2 ≤ κ -• ζ a.e. in Ξ T , then (κ + • ζ) (κ -• ζ) ≤ η 1 η 2 ≤ (κ + • ζ) (κ -• ζ) a.e. in Ξ T . (27)
Since by (1) 0 / ∈ (κ(ζ(ξ, t)), κ(ζ(ξ, t))), it follows that for almost all (ξ, t) ∈ Ξ T the terms κ(ζ(ξ, t)) and κ(ζ(ξ, t)) have the same sign, or at least one term is equal to zero. Thus, (κ

+ • ζ) (κ -• ζ) = (κ + • ζ) (κ -• ζ) = 0 a.e. in Ξ T ,
so that by [START_REF] Rabier | Fixed points of multi-valued maps and static Coulomb friction problems[END_REF] one obtains η 1 η 2 = 0 a.e. in Ξ T .

To complete the proof, it suffices to take η = η 1η 2 and, using the relations η 1 ≥ 0, η 2 ≥ 0 and η 1 η 2 = 0 a.e. in Ξ T , to see that

η + = η 1 , η -= η 2 a.e. in Ξ T .
Based on the previous lemma, consider the following problem, which has the same solution u as the problem Q 1 , and the solutions λ 1 , λ 2 satisfy the relation

λ = λ 1 -λ 2 , where λ is a solution of Q 1 . Problem Q 2 : Find u ∈ W 0 , λ 1 , λ 2 ∈ L 2 (0, T ; L 2 (Ξ)) such that u(0) = u 0 , u(0) = u 1 , (λ 1 , λ 2 ) ∈ Λ + (l(u)) × Λ -(l(u)) and u(T ), v(T ) -u(T ) U ′ × U -(u 1 , v(0) -u 0 ) + T 0 {-( u, v -u) + a 0 (u, v -u) + b 0 ( u, v -u)} dt (28)
+ T 0 {φ 0 (λ 1 , λ 2 , v + k u -u) -φ 0 (λ 1 , λ 2 , k u)} dt ≥ T 0 f 0 , v -u dt ∀ v ∈ L ∞ (0, T ; V 0 ) ∩ W 1,2 (0, T ; H 0 ).

Some auxiliary existence results

For the convenience of the reader, an existence and uniqueness result proved in [START_REF] Cocou | Analysis of a dynamic unilateral contact problem for a cracked viscoelastic body[END_REF] will be restated here, under an adapted form. Let β : V 0 → R and φ 1 : [0, T ] × V 3 0 → R be two sequentially weakly continuous mappings such that

β(0) = 0 and φ 1 (t, z, v, w 1 + w 2 ) ≤ φ 1 (t, z, v, w 1 ) + φ 1 (t, z, v, w 2 ), (29)
φ 1 (t, z, v, θw) = θ φ 1 (t, z, v, w), (30)
φ 1 (0, 0, 0, w) = 0 ∀ t ∈ [0, T ], ∀ z, v, w, w 1,2 ∈ V 0 , ∀ θ ≥ 0, (31)
∃ k 3 > 0 such that ∀ t 1,2 ∈ [0, T ], ∀ u 1,2 , v 1,2 , w ∈ V 0 , |φ 1 (t 1 , u 1 , v 1 , w) -φ 1 (t 2 , u 2 , v 2 , w)| ≤ k 3 (|t 1 -t 2 | + |β(u 1 -u 2 )| + |v 1 -v 2 |) w , (32)
∃ k 4 > 0 such that ∀ t 1,2 ∈ [0, T ], ∀ u 1,2 , v 1,2 , w 1,2 ∈ V 0 , |φ 1 (t 1 , u 1 , v 1 , w 1 ) -φ 1 (t 1 , u 1 , v 1 , w 2 ) + φ 1 (t 2 , u 2 , v 2 , w 2 ) -φ 1 (t 2 , u 2 , v 2 , w 1 )| ≤ k 4 ( |t 1 -t 2 | + u 1 -u 2 + |v 1 -v 2 |) w 1 -w 2 . ( 33
)
Let L ∈ W 1,∞ (0, T ; V 0 ) and assume the following compatibility condition on the initial data:

∃ p 1 ∈ H 0 such that (p 1 , w) + a 0 (u 0 , w) + b 0 (u 1 , w) + φ 1 (0, u 0 , u 1 , w) = L(0), w ∀ w ∈ V 0 . (34)
Consider the following problem. Problem Q 3 : Find u ∈ W 2,2 (0, T ; H 0 ) ∩ W 1,2 (0, T ; V 0 ) such that u(0) = u 0 , u(0) = u 1 , and for almost all t ∈ (0, T )

(ü, v -u) + a 0 (u, v -u) + b 0 ( u, v -u) +φ 1 (t, u, u, v) -φ 1 (t, u, u, u) ≥ L, v -u ∀ v ∈ V 0 . ( 35
)
Under the assumptions ( 13), ( 14), ( 29), ( 30), ( 32)-(34) and the stronger condition

φ 1 (t, 0, 0, w) = 0 ∀ t ∈ [0, T ], ∀ w ∈ V 0 , (36)
an existence and uniqueness result for the problem Q 3 was proved in [START_REF] Cocou | Analysis of a dynamic unilateral contact problem for a cracked viscoelastic body[END_REF] but in its proof the relation (36) was only used to verify that the relation [START_REF] Sofonea | Analysis and Approximation of Contact Problems with Adhesion or Damage[END_REF] implies that φ 1 (t, z, v, •) is Lipschitz continuous on V 0 . Since ( 31) and ( 33) also imply that φ 1 (t, z, v, •) is Lipschitz continuous, we clearly have the following existence and uniqueness result.

Theorem 3.1. Under the assumptions ( 13), ( 14), ( 29)-( 34), there exists a unique solution to the problem Q 3 .

Lemma 3.2. Assume that ( 13), ( 14), ( 16)-( 18), [START_REF] Kuttler | Dynamic contact with Signorini's condition and slip rate depending friction[END_REF], and (25) hold. Then, for each (γ 1 , γ 2 ) ∈ (W 1,∞ (0, T ; L 2 (Ξ))) 2 ∩ (L 2 + (Ξ T )) 2 with γ 1 (0) = γ 2 (0) = 0, there exists a unique solution u = u (γ 1 ,γ 2 ) of the following evolution variational inequality: find u ∈ W 2,2 (0, T ; H 0 )∩W 1,2 (0, T ; V 0 ) such that u(0) = u 0 , u(0) = u 1 , and for almost all t ∈ (0, T )

(ü, v -u) + a 0 (u, v -u) + b 0 ( u, v -u) +φ 0 (γ 1 , γ 2 , v) -φ 0 (γ 1 , γ 2 , u) ≥ f 0 , v -u ∀ v ∈ V 0 . ( 37
)
Proof. We apply Theorem 3.1 to β = 0, L = f 0 and

φ 1 (t, z, v, w) = φ 0 (γ 1 (t), γ 2 (t), w) ∀ t ∈ [0, T ], ∀ z, v, w ∈ V 0 .
Since φ 0 satisfies ( 16)-( 18) one can easily verify the properties ( 29)- [START_REF] Shillor | Models and Analysis of Quasistatic Contact[END_REF]. Also, ( 25) and ( 31) imply the condition (34). Using [START_REF] Kuttler | Dynamic contact with Signorini's condition and slip rate depending friction[END_REF], we have

∀ t 1,2 ∈ [0, T ], ∀ u 1,2 , v 1,2 , w 1,2 ∈ V 0 , |φ 1 (t 1 , u 1 , v 1 , w 1 ) -φ 1 (t 1 , u 1 , v 1 , w 2 ) + φ 1 (t 2 , u 2 , v 2 , w 2 ) -φ 1 (t 2 , u 2 , v 2 , w 1 )| = |φ 0 (γ 1 (t 1 ), γ 2 (t 1 ), w 1 ) -φ 0 (γ 1 (t 1 ), γ 2 (t 1 ), w 2 )| +φ 0 (γ 1 (t 2 ), γ 2 (t 2 ), w 2 ) -φ 0 (γ 1 (t 2 ), γ 2 (t 2 ), w 1 )| ≤ k 2 ( γ 1 (t 1 ) -γ 1 (t 2 ) L 2 (Ξ) + γ 2 (t 1 ) -γ 2 (t 2 ) L 2 (Ξ) ) w 1 -w 2 U ≤ k 2 (C γ 1 + C γ 2 )|t 1 -t 2 | w 1 -w 2 U ≤ k 5 |t 1 -t 2 | w 1 -w 2 U ,
where C γ 1 , C γ 2 denote the Lipschitz constants of γ 1 , γ 2 , respectively, and

k 5 = k 2 (C γ 1 + C γ 2 ). Thus, |φ 1 (t 1 , u 1 , v 1 , w 1 ) -φ 1 (t 1 , u 1 , v 1 , w 2 ) + φ 1 (t 2 , u 2 , v 2 , w 2 ) -φ 1 (t 2 , u 2 , v 2 , w 1 )| ≤ k 5 |t 1 -t 2 | w 1 -w 2 U ∀ t 1,2 ∈ [0, T ], ∀ u 1,2 , v 1,2 , w 1,2 ∈ V 0 , (38)
and, since by the continuous embedding V 0 ⊂ U there exists

C U > 0 such that w U ≤ C U w ∀ w ∈ V 0 , it follows that φ 1 satisfies (33) with k 4 = k 5 C U .
Taking in (38) w 1 = w, w 2 = 0, by [START_REF] Rocca | Numerical analysis of quasistatic unilateral contact problems with local friction[END_REF] with θ = 0, we obtain

|φ 1 (t 1 , u 1 , v 1 , w) -φ 1 (t 2 , u 2 , v 2 , w)| ≤ k 5 |t 1 -t 2 | w U ∀ t 1,2 ∈ [0, T ], ∀ u 1,2 , v 1,2 , w ∈ V 0 , (39)
and using the continuous embedding V 0 ⊂ U , it follows that φ 1 satisfies [START_REF] Simon | Compact sets in the space L p (0, T ; B)[END_REF] with

k 3 = k 5 C U . Now, taking in (38) t 1 = t, t 2 = 0, u 1 = z, v 1 = v, u 2 = v 2 = 0, by (31) we have |φ 1 (t, z, v, w 1 ) -φ 1 (t, z, v, w 2 )| ≤ k 5 t w 1 -w 2 U ∀ t ∈ [0, T ], ∀ z, v, w 1,2 ∈ V 0 . (40)
As the embedding from V 0 into U is compact, from (39) and (40) it is easily seen that φ 1 , which is depending only on t and w, is weakly sequentially continuous. By Theorem 3.1 there exists a unique solution u = u (γ 1 ,γ 2 ) of the variational inequality (37).

Lemma 3.3. Let (γ 1 , γ 2 ), (δ 1 , δ 2 ) ∈ (W 1,∞ (0, T ; L 2 (Ξ))) 2 ∩ (L 2 + (Ξ T )) 2 such that γ 1 (0) = γ 2 (0) = δ 1 (0) = δ 2 (0) = 0 and let u (γ 1 ,γ 2 ) , u (δ 1 ,δ 2 )
be the corresponding solutions of (37). Then there exists a constant C 0 > 0, independent of (γ 1 , γ 2 ), (δ 1 , δ 2 ), such that for all t ∈ [0, T ]

| u(γ 1 ,γ 2 ) (t) -u(δ 1 ,δ 2 ) (t)| 2 + u (γ 1 ,γ 2 ) (t) -u (δ 1 ,δ 2 ) (t) 2 + t 0 u(γ 1 ,γ 2 ) -u(δ 1 ,δ 2 ) 2 ds ≤ C 0 t 0 {φ 0 (γ 1 , γ 2 , u(δ 1 ,δ 2 ) ) -φ 0 (γ 1 , γ 2 , u(γ 1 ,γ 2 ) ) +φ 0 (δ 1 , δ 2 , u(γ 1 ,γ 2 ) ) -φ 0 (δ 1 , δ 2 , u(δ 1 ,δ 2 ) )} ds. (41)
Proof. Let (γ 1 , γ 2 ), (δ 1 , δ 2 ) ∈ (W 1,∞ (0, T ; L 2 (Ξ))) 2 ∩ (L 2 + (Ξ T )) 2 and u 1 := u (γ 1 ,γ 2 ) , u 2 := u (δ 1 ,δ 2 ) be the corresponding solutions of (37), for which the existence and uniqueness are insured by Lemma 3.2. Taking in each inequality v = u2 and v = u1 , respectively, for a.e. s ∈ (0, T ) we have

(ü 1 -ü2 , u1 -u2 ) + a 0 (u 1 -u 2 , u1 -u2 ) + b 0 ( u1 -u2 , u1 -u2 )
≤ φ 0 (γ 1 , γ 2 , u2 )φ 0 (γ 1 , γ 2 , u1 ) + φ 0 (δ 1 , δ 2 , u1 )φ 0 (δ 1 , δ 2 , u2 ).

As the solutions u 1 , u 2 verify the same initial conditions and a 0 is symmetric, by integrating over (0, t) it follows that for all t ∈ [0, T ] Using the V 0 -ellipticity conditions [START_REF] Fan | Fixed points and minimax theorems in locally convex topological linear spaces[END_REF], the estimate (41) follows.

and so for almost every t ∈ (0, T ) we have By the relations ( 14), ( 23) for v 2 = 0, and (13), we obtain Since the sequence (γ n 1 , γ n 2 ) n is bounded in (L 2 (0, T ; L 2 (Ξ))) 2 , by Young's inequality it follows that there exists a positive constant C 1 , depending only on a 0 , b 0 , f 0 , u 0 , u 1 , the bound of (γ n 1 , γ n 2 ) n , k 2 and C U , such that the following estimates hold: This relation and the estimates (44) imply that there exists a positive constant C 2 having the same properties as C 1 and satisfying the estimate

∀ n ∈ N, | un (t)| ≤ C 1 , u n (t) ≤ C
∀ n ∈ N, ün L 2 (0,T ;U ′ 0 ) ≤ C 2 . (45)
From ( 44), (45), it follows that there exist a subsequence (u n k ) k and u such that

un k ⇀ * u in L ∞ (0, T ; H 0 ), u n k ⇀ * u in L ∞ (0, T ; V 0 ), un k ⇀ u in L 2 (0, T ; V 0 ), ün k ⇀ ü in L 2 (0, T ; U ′ 0 ). ( 46
)
According to Theorem 2.1 with

F = ( un k ) k , X = H 0 , Û = U ′ , Ŷ = U ′ 0 , r = 2, F = (u n k ) k , X = V 0 , Û = U, Ŷ = H 0 , r = 2, F = ( un k ) k , X = V 0 , Û = U, Ŷ = U ′ 0 , p = 2,

For each ζ ∈ L 2

 2 (0, T ; L 2 (Ξ)) the sets Λ(ζ), Λ + (ζ) and Λ -(ζ) are clearly nonempty, because the bounding functions belong to the respective set, closed and convex.

1 2 | 0 b 0 0 {φ 0 (γ 1 , γ 2 ,

 2000012 u1 (t) -u2 (t)| 2 + 1 2 a 0 (u 1 (t)u 2 (t), u 1 (t)u 2 (t)) + t ( u1 -u2 , u1 -u2 ) ds ≤ t u2 )φ 0 (γ 1 , γ 2 , u1 ) + φ 0 (δ 1 , δ 2 , u1 )φ 0 (δ 1 , δ 2 , u2 )} ds.

  u 0 , u 0 ).

  1 a.e. t ∈ (0, T ), un L 2 (0,T ;V 0 ) ≤ C 1 . (44) Using (43) for w = un ± ψ and (19), we see that for all ψ ∈ L 2 (0, T ; U 0 ), T 0 (ü n , ψ) ds + T 0 a 0 (u n , ψ) ds + T 0 b 0 ( un , ψ) ds = T 0 f 0 , ψ ds.

  ( γ n 1 L 2 (Ξ) + γ n 2 L 2 (Ξ) ) un U ds +

	≤	0	t	1 2 k 2 t | un (t)| 2 + m a u n (t) 2 + m b 2 0 f 0 un ds + t un 2 ds 0	1 2	|u 1 | 2 +	M a 2	u 0 2 .

Acknowledgments

The partial support of this work by "Laboratoire Européen Associé CNRS Franco-Roumain de Mathématiques et Modélisation" is gratefully acknowledged.

A fixed point problem formulation

Since D(0, T ; L 2 (Ξ)) is dense in L 2 (0, T ; L 2 (Ξ)), which is classically proved by using the convolution product with suitable mollifiers, it follows that for every γ ∈ L 2 + (Ξ T ), there exists a sequence (γ n ) n in W 1,∞ (0, T ; L 2 (Ξ))∩L 2 + (Ξ T ) such that γ n (0) = 0, for all n ∈ N, and γ n → γ in L 2 (0, T ; L 2 (Ξ)).

Theorem 3.2. Assume that [START_REF] Eck | Unilateral Contact Problems -Variational Methods and Existence Theorems[END_REF], [START_REF] Fan | Fixed points and minimax theorems in locally convex topological linear spaces[END_REF], ( 16)- [START_REF] Kuttler | Existence and regularity for dynamic viscoelastic adhesive contact with damage[END_REF], and [START_REF] Naniewicz | Mathematical Theory of Hemivariational Inequalities and Applications[END_REF] hold. For each (γ 1 , γ 2 ) ∈ (L 2 + (Ξ T )) 2 , let (γ n 1 , γ n 2 ) n be a sequence in (W 1,∞ (0, T ; L 2 (Ξ))) 2 ∩ (L 2 + (Ξ T )) 2 such that (γ n 1 , γ n 2 ) ⇀ (γ 1 , γ 2 ) in (L 2 (0, T ; L 2 (Ξ))) 2 , γ n 1 (0) = γ n 2 (0) = 0, and let u (γ n 1 ,γ n 2 ) be the solution of (37) corresponding to (γ n 1 , γ n 2 ), for every n ∈ N. Then (u (γ n 1 ,γ n 2 ) ) n is strongly convergent in W 0 , its limit, denoted by u = u (γ 1 ,γ 2 ) , is independent of the chosen sequence converging to (γ 1 , γ 2 ) with the same properties as (γ n 1 , γ n 2 ) n and is a solution of the following evolution variational inequality: 2 . Then, by Lemma 3.2, for every n ∈ N there exists a unique solution of the following variational inequality: find

From (43), for w = 0, w = 2 un , and integrating over (0, t) with t ∈ (0, T ), we derive

(47) By Lemma 3.3, for all k, l ∈ N we have

and passing to limits by using [START_REF] Migórski | Nonlinear Inclusions and Hemivariational Inequalities[END_REF], we find that ( un k ) k is a Cauchy sequence in L 2 (0, T ; V 0 ). Thus, ( un k ) k is strongly convergent to u in this space and since

we deduce

The limit u is the same for all the convergent subsequences, satisfying convergence properties similar to (47), corresponding to every sequence approximating (γ 1 , γ 2 ), as can be readily seen by passing to limits in the following relation, obtained from (41) for γ 1,2 = γ n 1,2 , δ 1,2 = δ n 1,2 and for all n ∈ N:

where

, and so integrating over (0, T ) yields

and integrating by parts the first term in (51) implies

Using e.g. ( 47), ( 13) and [START_REF] Migórski | Nonlinear Inclusions and Hemivariational Inequalities[END_REF], it is clear that we can pass to the limit in each term of (52) and so we obtain that u = u (γ 1 ,γ 2 ) is a solution of (42).

where u (γ 1 ,γ 2 ) is the solution of the variational inequality (42) which corresponds to (γ 1 , γ 2 ) by the procedure described in Theorem 3.2.

We shall consider a new problem, which consists in finding a fixed point of the set-valued mapping Φ, called also multivalued function or multifunction, which will provide a solution of Problem Q 1 .

Existence of a fixed point

We shall prove the existence of a fixed point of the multifunction Φ by using a corollary of the Ky Fan's fixed point theorem [START_REF] Fan | Fixed points and minimax theorems in locally convex topological linear spaces[END_REF], proved in [START_REF] Rabier | Fixed points of multi-valued maps and static Coulomb friction problems[END_REF] in the particular case of a reflexive Banach space. Note that since Y is a reflexive Banach space and D is convex, closed and bounded, there is no assumption that Y is separable, see [START_REF] Rabier | Fixed points of multi-valued maps and static Coulomb friction problems[END_REF][START_REF] Browder | Nonlinear operators and nonlinear equations of eyolution in Banach spaces[END_REF]. Theorem 3.3. Assume that ( 1), ( 2), ( 13)-( 21) and ( 25) hold. Then there exists

We apply Proposition 3.

The set D ⊂ (L 2 (0, T ; L 2 (Ξ))) 2 is clearly convex, closed and bounded. Since for each ζ ∈ L 2 (0, T ; L 2 (Ξ)) the sets Λ + (ζ) and Λ -(ζ) are nonempty, convex, closed, and bounded by R 0 , it follows that Φ(γ 1 , γ 2 ) is a nonempty, convex and closed subset of D for every (γ 1 , γ 2 ) ∈ D.

In order to prove that the multifunction Φ is sequentially weakly upper semicontinuous, let (

) and let us verify that (η 1 , η 2 ) ∈ Φ(γ 1 , γ 2 ). Using the Theorem 3.2 for each n ∈ N, it follows that there exists a sequence

where

is the solution of (37

is the solution of (42) corresponding to (γ n 1 , γ n 2 ) and to the procedure that enables to define Φ

and by (54) and the triangle inequality, we obtain

Now, by Lemma 3.1, the relation (

which may be rewritten as

for all n ∈ N, where l n := l(u (γ n

). The relations (57) are equivalent to

for every measurable subset ω ⊂ Ξ T and for all n ∈ N. Using (55), [START_REF] Han | Quasistatic Contact Problems in Viscoelasticity and Viscoplasticity[END_REF], the semi-continuity of κ and κ, the relation ( 2), the convergence property

, and passing to limits according to Fatou's lemma (see also [START_REF] Rabier | Fixed points of multi-valued maps and static Coulomb friction problems[END_REF]), we obtain

for every measurable subset ω ⊂ Ξ T , which implies (η 1 , η 2 ) ∈ Φ(γ 1 , γ 2 ).

4 Existence of a solution to the contact problem Theorem 4.1. Under the assumptions of Section 2 there exists a solution of the Problem P 1 v .

Proof. We shall prove that there exists at least a solution (u, λ + , λ -) of the Problem P 2 v which will provide a solution (u, λ) of the Problem P 1 v with λ = λ +λ -.

We apply Theorem 3.3 to

, we obtain [START_REF] Eck | Unilateral Contact Problems -Variational Methods and Existence Theorems[END_REF].

The condition meas(Γ α U ) > 0, the ellipticity properties of the coefficients A α ijkl , B α ijkl and the Korn's inequality imply that there exist m α a , m α b > 0 such that

and we obtain

where

b ). Also, the properties ( 15)-( 19), ( 21) and ( 25) can be easily verified. Now, let C tr be a positive constant such that v (L 2 (Ξ)) d ≤ C tr v H ι for all v ∈ H ι . Using [START_REF] Cocou | Analysis of a dynamic unilateral contact problem for a cracked viscoelastic body[END_REF], the following estimates hold:

and so ( 20) is satisfied with k 2 = C tr (1 + µ L ∞ (Ξ) ).