Gilles De Rapper

CROSSING BORDERS: MIGRATION IN AN ALBANIAN FRONTIER ZONE

Introduction

Looking at international borders with the tools of ethnographic observation allows us to understand general phenomena that appear in a particular light in a border context. This is particularly true in the Balkans where most international borders date back to the beginning of the 20th century only and are seen as unsatisfactory by all the states. Emigration is one of these questions as it is obviously connected to the crossing of borders. Moreover, the end of the Cold War saw the resurgence of migration in this area, after an interruption of several decades. Albania alone offers a great variety of situations, starting with the highly conflictual border with Serbia. Notwithstanding the territorial issue of Northern Epirus, the southern border is dominated by the question of Albanian emigration to Greece, which started as early as 1991, and has become a major problem between the two states, as most of this emigration is illegal. In the border area itself however emigration is not only a question of crossing the border, as it has to do with the very existence of local communities which have to deal with both states. This chapter looks at the way Albanian emigration, both in practice and discourse, is shaped by the vicinity of an international border.

Devoll is a small district of south eastern Albania, on the Greek border and close to the Macedonian border. It has approximately 43 000 inhabitants (1998) and had been for a long time a part of the larger district of Korçë when it gained its "independence" in 1992 by becoming a district (rreth) by itself. It is mostly rural and the population -except one Macedonian village -is all Albanian, though divided in two religious communities, Muslim and Orthodox Christian.

The profits generated by the border post of Kapshticë -known as Kristalopigi on the Greek side -played a great part in the decision of separating the Devoll from Korçë, as people from Devoll are now privileged in getting jobs and bribes at the border post. Kapshticë is one of the two main crossing points between Albania and Greece, the other being Kakavijë, in Epirus. Compared to the latter, the border post of Kapshticë looks less busy and somehow quieter. No gate of any kind has to keep the Albanian crowd inside Albania, as is the case in Kakavijë, and the lorries are usually less numerous. Kapshticë remains nonetheless the gateway to north eastern Greece, including the large urban centre of Thessaloniki and the agricultural regions of Macedonia and Thessalia, all very attractive to the Albanian workers. Devoll is thus at the same time a crossing point for migrants and one of the places where Greek army and police bring back the clandestines they have caught in Greece.

Life on the border

The town of Bilisht (8 000 inhabitants), Devoll's administrative centre, is obviously turned towards the border. At any time of the year, and even more between March and October, the town is used as a gathering place by many would-be migrants from most of Albania. Special bus lines wait for the travellers at the Pogradec train station, 65 km away, and regular ones bring many of them from Korçë bus station -Korçë being the main urban centre of south eastern Albania. A few hotels have opened in Bilisht which hosts migrants for a few days before their departure for the borderline. But the number of rooms is still very low and most of the migrants spend the night in the outside, sleeping in parks and orchards. This is especially true in summer time, when the number of candidates becomes very high and the warm weather allows them to sleep in the outside. The situation is the same, though at a smaller scale, in the border villages, higher in the mountains, which are used as illegal crossing points, some of them being less than half an hour walk away from the border.

People waiting to cross the border are of two kinds: some of them have just arrived from their villages in northern or central Albania, others have been brought back from Greece by the Greek police or army and wait a few days before trying again.

Migrants are a source of profit for the local population. First of all, as they usually stay a few days in Bilisht before leaving, they have to eat and drink. If they generally avoid the bars and restaurants, they rely on the contrary on the small food shops and stands which have opened on the main road, and can be seen everyday on the market, buying big loaves of bread, bottles of soda, fruit and biscuits. Most of them do not have any other luggage than the food they take on their trip towards the border, and they carry it in a white bag of the kind used to store flour or dry beans. Many shopkeepers now display such bags in front of their shops, as they are sure to sell them in a very short time. The same bags have thus come to be the "emblem" of the refugees, and anyone unknown who is seen carrying such a bag on its shoulder is always classified as "refugee".

Second, migrants always have to change money, when they leave as well as when they come back. The number of "exchange offices" in Bilisht is excessively high compared to the needs of the local population and is only explained by the coming and going of numerous refugees. "Exchange office" is a rather pretentious word, as many of them do not have any office at all but work on the central square and the bars around, waiting for the customer in front of a coffee and a glass of raki. There is much cheating on their part when exchanging the refugees' money -as many of the refugees coming from the North are not aware of the exchange rate of the Greek drachma -and there is a game going on among them, the winner being the one who earns more in one single operation with a refugee. The situation is even easier for them in the remote border villages where there is hardly no concurrence and where the refugees have to change money or starve in Greece (if the Greek drachma is usually accepted for payment on the Albanian side, this is of course not true of the Albanian lek on the Greek side).

The central square in Bilisht is also the place where many refugees look for the man who will help them in crossing the border and finding transportation once on the Greek side. These people are called "guide" (kullauz) but the phrase "the ones who deal with the refugees" (ata që merren me refugjatë) is heard more often, as this new "profession" is generally considered badly and people do not like to give it a name. The "guides" are necessary to most of the refugees as they offer services that seem a condition sine qua non for a successful stay in Greece: they guide them by night through the hills and mountains, know how to avoid Greek army patrols and usually work with Greek associates who drive the refugees to the very place where they want to go in Greece. These services are very expensive -many refugees have to borrow money before they leave their family in order to pay for the crossing. They then have to work between one and three months in Greece to pay back their debts, before even earning money for themselves. Most of them nonetheless believe it is worth it, since it avoids them a dangerous journey in an unknown country and allows them to work as soon as they are in Greece, without any lost of time. In the remote villages, the same man can be a shopkeeper, a guide and change money, dealing with the refugees in whatever need they have. This is for instance the case of Namik, owner of one the café-groceries in a mountain village, who never sleeps deeply between March and October, as he wants to hear any band of refugees coming to the village from either side of the border. He then runs out to meet them, houses them in his shop, leaving one of his son to watch on the goods, and offers his services. If they want to cross immediately, he calls for a young man who will guide them through the hills to the nearest Greek village, but he likes better keep them a day or two, pretending troubles on the border. He then can offer to feed them and he knows they will spend the time playing billiards -and paying for it. Everywhere in Bilisht and the border villages, people "who deal with the refugees" are looked at by the rest of the population with both envy -their pockets are full of bank notes when they come back from the border -and disgust: they do a bad job and live at the expense of people who are much poorer then they are.

The passing by of refugees is not the only way in which local life is concerned by emigration and border crossings. Local people do also go and work in Greece on a seasonal basis. This is especially true of the villages, where no work can be found apart from domestic agriculture, while the higher level of employment in Bilisht seems to prevent people from leaving. Being a commercial and administrative centre close to the border provides the town with very interesting job opportunities, and one can see Albanian workers in Greece coming back quickly to Bilisht when a place becomes vacant at the customs, where profits are even higher than in Greece. The villages on the contrary are not attractive at all and they seem to be emptied of their male population by the end of spring until September. Village men do the hard work on their lands before going to Greece, and leave their wife, old parents and children to care for the fields during the summer. According to the place they work in Greece, they can even come back to harvest and go again for a month or two. Some of them indeed are employed in the nearest Greek villages and cross the border daily. Among them, many take advantage of this situation to smuggle with the Greek villages, but this is another story.

Two kinds of migrants

According to local representations, migrants who cross the border with Greece are of two kinds: the first consists of migrants from the district itself; the second is made of migrants from other parts of Albania. There is no special name for the first category: people just refer to local migrants by saying: "they are in Greece, he works in Greece" (janë Greqi, punon Greqi). In contrast, migrants from other parts of the country are always referred to by the name "refugee" (refugjat), which is the Italian word, entered in Albania at the time of the "exodus" towards Italy at the beginning of the '90s. In the Devoll, it has lost its original meaning and come to mean "foreigner" or "traveller". In other words, these Albanian migrants are called refugee even before they leave Albania.

There are many ways in which this distinction is made obvious to the migrants themselves. The two categories rarely mix together. In the town of Bilisht, as well as in the border villages which are used as meeting and departure points for crossing the borderline, refugees usually stay aside from the local population: they sleep in the fields and the woods around the villages, they do not go to the bars and shops -except to buy food for their trip -and when they play billiards, they play among themselves, never with the local villagers. The separation and absence of communication can take extreme forms, since nobody seems to be willing to have contacts: arriving one morning in a village to visit a friend, I am told by his wife that he is at work in the orchard, just in the outskirts of the village. As I arrive on the spot I find my friend Ferdinant ploughing the soil in between the apple and plum trees. Under one of the trees however lies a group of 6 to 8 refugees, most of them sleeping or looking at the sky, ignoring both Ferdinant and me. Ferdinant goes on ploughing, turning around the refugees without asking them to move in order to let him finish his work. Everything appeared as if nobody was ready to establish contact, as if Ferdinant did not want to help these men nor even learn where they were from, and as if they knew Ferdinant would be of no help to them. The relationships between the migrants and the local population are generally of the same kind: they ignore each other as long as they do not have personal interest in dealing with each other. In Bilisht, where the refugees are many because the ones who are brought back to the border from Greece gather in town before trying again, a restaurant is named "the bar of the refugees" (klubi i refugjatëve) and most of its customers are indeed refugees. They nonetheless stay apart from the local customers and seem to avoid any clash with them. Moreover, like many travelling people, the refugees are considered to be a threat to the local population: villagers explain that they lock their door because of the refugees, and women confess that they do not travel alone on the roads any more, because, as one of them says, "many people travel by, which we do not know, and we wonder who they are". In the border villages, mothers warn their children saying: "be quiet, or the refugees will take you to Greece". In reality, because of what appears to be fear of contact with the locals, refugees bring little trouble in the area; and when they do, it has to do with their own stealing and quarrelling with each other, and not with the local population. Nevertheless, they are commonly considered as a source of insecurity.

The distinction between local migrants and others makes more sense when it is connected to the local conception of the national space of Albania. The opposition between North and South has come to be a part of common knowledge about Albania with its "ethnic" (Ghegs against Tosks) and political (anticommunist against communist) dimensions. It should nonetheless be noticed that the words "gheg" and "tosk" belong to a somehow high-educated language and are very rarely used by local populations, among which the sense of belonging and community appear at a smaller scale, usually that of the district. From the local point of view, the North-against-South opposition is rather used as a way of talking about "we" and "the others": seen from the South, the North represents everything people of the South deny for themselves. In that meaning there is no geographical border between North and South, and the river Shkumbin, traditionally referred to as the limit between Gheg and Tosk areas, is of no importance. For the Devoll population for instance, the town of Gramsh is already associated with the Norththough it is situated south of the Shkumbin -as it is considered to be a remote, backward and desolated area compared to the Devoll.

Seen from the Devoll, all refugees are "northerners" (nga veriu) or "mountaineers" (malok). Indeed, many of them come from the centre and the north of Albania, while migrants from the south usually cross the Greek border further in the west and do not have to travel through the Devoll district. Anyway, the point to be noticed here is that in local representations, people from the north have a strong and bad reputation. This reputation includes poverty, violence and cultural backwardness. In contrast, the population of south eastern Albania is proud of its own wealth as well as of its cultureculture (kulturë) is to be understood here as a mixture of development and personal education. The fact that most refugees are actually poor people tends to consolidate this negative representation.

As far as emigration is concerned, the two categories are also opposed on the basis of their behaviour in Greece. People from the North are said to go there for the money, and only for the money: they are no hard workers and they tend to steal and rob if their jobs do not please them. Local migrants on the other hand are said to be hard workers and loyal to the Greeks who give them jobs; they explain that they have too much pride and self-respect (sedër) to act badly towards the ones who help them. In other words, they can be trusted, while the others cannot. And, because of this self-pride, people from the Devoll tend also to pretend that they do not go that much to work in Greece, while the others, the northerners, have to go because of their extreme poverty. In the first group, the feeling of humiliation connected to emigration is said to be stronger than the actual need to work abroad, while in the second group the need is stronger than anything. The spatial opposition between North and South is thus used here as if it was expressing the feelings of the local population towards emigration: emigration is a bad thing, they say, we should not have to leave the country; so we leave it to those backward northerners. In this respect, local representations of migrants is a good example of the mirror effect of the opposition between North and South: in fact, people from the South do not differ very much from those from the North when it comes to the necessity and will to go and work in Greece, but since emigration is felt as something negative in the definition of one's identity -it means poverty, exploitation by a Greek boss, leaving one's family alone, etc. -it is projected on the Northerners alone.

Moreover, that same opposition between local and distant migrants tends to create a sense of local identity based on the familiarity with the border area. Refugees from the north always travel in bands of ten to twenty and more people, and they need the help of a local guide (kullauz) to cross the border safely (specially in winter time when the mountain tracks are covered with snow or disappear in the mist). In contrast, local migrants usually cross the border in small groups (up to five people), and they can rely on their personal knowledge and contacts on the other side of the border, among the Greek villagers. They were also the first ones, when the borders opened, to cross over and visit the Greek villages, before migrants from other parts of the country began to arrive. In that way, people from Devoll believe that they have special and stronger ties with the Greek population, and they sometimes feel closer to Greece than to the rest of Albania, especially when they compare themselves to the Northerners. Before the parliamentary elections of May 1996, Albanian Christians who live close to the border with Greece were sure that all the Devoll, Christians and Muslims together, would vote for the Human Rights Party, close to Greek interests in Albania, because of their links with Greece. "Just look at them, says one, they have all been to Greece at least once, soon, they will all speak the language. The Greek claims over Northern Epirus do not bother them; they will stop being Albanian and they do not care. They think it is better to change the borderline and to live like Greeks rather than to survive like Albanians." Later that same year, a refugee from Shkodër arrived from Greece and spent one night in the Muslim village I was staying at, one hour from the border. In the evening, as we were all at home watching television, the landlord kept playing with the remote control of the television, insisting on the wide range of Greek channels he could receive -he had never been to Greece and could not understand a word of Greek. As the landlord expected, the Shkodran was much impressed by all these Greek channels. Half smiling and pointing to the direction of the border, the landlord answered: "Ah! Greece is just there, we are in Greece here!" Needless to say that the same man was never the last one in the village to say how badly he considered the Greeks -except when he had to show the rest of Albania how privileged the Devoll is.

The distinction between the local migrants and the others is not made by the Devoll population alone. On the other side of the border, the Greeks tend to operate the same distinction between local and distant migrants. The first ones are often called "our Albanians". Most of them have been crossing the border since the beginning and they are quite well-known in the Greek villages. They are more likely to find jobs in these Greek border villages, while the others are always sent away in the cities. By calling them "our Albanians", the Greeks do not mean they consider them as good friends or loyal workers, and they actually do not trust them very much. But, as they put it, "at least, if one of our Albanians does something wrong, we know where to find him," meaning that they can easily cross the border to the Albanian villages and get back what was stolen. Nevertheless, there exists a sense of community among Greeks and Albanians in the border area, which does not exist at all with the other Albanian migrants. Very few Greeks have been to central and northern Albania -while they are much more interested in visiting Northern Epirusand they tend to consider the population of these distant areas as dangerous murderers and Muslim fanatics. People from Bilisht pretend the Greeks never go further than Korçë because they are afraid of what could happen in more distant places where all Albanians are mafiosi and carry weapons. Altogether, the local conceptions of the Albanian migrants on both the Albanian side and the Greek side of the border are quite similar. It is difficult to say how much what the local Greeks think and say is influenced by what the local Albanians tell them about themselves, but the fact that the Greek population agrees on the same distinctive opposition is taken by the Devoll population as evidence of their own identity and superiority. The sense of community that exists between both sides of the border does not mean real knowledge of one another, as many Greeks have never been to the Albanian villages, but feel anyway a kind of proximity with Albania. "What would like to know about Albania? asks a Greek villager. Just ask me." "Have you ever been there?" "Oh, no! Why should I go? It is so dangerous, they can rob you of everything and leave you naked on the road."

The sense of local identity was given a strong emphasis in 1992, when the Devoll became a district by itself, while it had been so far a part of the district of Korçë. This decision is said to have been taken by the local population in order to "regain" its independence (the Devoll had been a separate district until 1958). The fact that the new district can keep a part of the incomes from the border post of Kapshticë probably played a role in this decision. It also gives the population the opportunity to create its own links with the neighbouring Greek districts of Florina and Kastoria. Officials from the Devoll regularly cross the border to meet their Greek counterparts and discuss common projects or take part in religious celebrations. Greek officials from the border villages recognize that they have many projects concerning the cooperation with their Albanian neighbours -at least before the events of 1997. This situation should be compared to the former one, when the border with Greece was tightly locked and when no contact of any kind could be established across the border. The Devoll was then at the same time a peripheral area, a place where no one should go to, and a place with strong ties with the centre of the state. As a state border area it was indeed highly watched by the army and police forces, most of which were not recruited among the local population, but came from other parts of Albania. Today the army has disappeared from the border, the policemen are mainly local people, and the Devoll seems to look towards Greece more than towards Tiranë and Korçë.

Migration and local society

Emigration is not only a journey into space, it is also a journey through time, in which Albania represents the past and Greece the future. Migrants are expected to bring a piece of that future into Albania when they come back. In this respect emigration is not only an answer to the lack of work and money, but is also an answer to the lack of modernity. Apart from this, the temporal dimension of emigration appears in two different cyclic conceptions, a long-term one, and a short-term one.

In the long-term cyclic conception, the present emigration phase is seen as the repetition of past migrations. Before World War II, a lot of people from the Devoll, especially among the Christians, left their country to go to America or Australia. This period of time is still remembered as a time of progress and innovations for the whole district. Migrants would send money home and come back after five years abroad, bringing modern clothes, modern equipment and modern mentality to their villages.

The interesting point is that this migration was mainly a Christian phenomenon and can be seen as one of the causes of the cultural superiority of Christians over Muslims in today's Devoll. It is indeed commonly assumed by local people, both Muslims and Christians, that Christians enjoy a higher level of culture: Christian villages were the first ones to open schools in the first decades of 20th century, affording their inhabitants to become civil servants, intellectuals and entrepreneurs instead of peasants. They were also more inclined to go abroad, especially to the USA, from where they used to send money back to the village. Today, Christian villages are still known in the Devoll to be better equipped than Muslim ones: they've got water adduction, paved street and large stone houses, giving them an urban character which contrasts with the very rural Muslim villages. All this, including the Chritians' identification to "Western" way of life (dressing, religion, attitudes towards women), is called "culture" (kulturë) in the Devoll, a word which could also be translated by "civilization" in this context. In local discourse, the higher cultural level of Christians is due to their contacts with the foreign world through emigration and trade; "they have seen the world", as people say. Today, as Muslims seem to be more involved in emigration than Christians, who prefer to move to Albanian cities rather than to Greece, the situation is opposite. It is a fact that, in the Devoll alone, emigration towards Greece seems to be more a Muslim than a Christian phenomenon. First of all, when the borders opened, Christians seem to have reactivated the old American network, which allowed them to migrate in a safe way to a very attractive country (this is also true, though on a lesser scale, of Muslims who had family in America -USA or Argentina -and Australia). They also tended to consider it better to be a rich and influent trader in an Albanian city rather than a second-class worker in Greece. That is why many of them, after having raised funding in Greece as migrants in the first two or three years, settled back in Bilisht, Korçë or Tiranë as traders, usually taking advantage of the links they had formed in Greece. For instance, Kujtim, a Christian from Bilisht, came back after two years in Greece, working in an orchard, bringing to Albania an old lorry his Greek boss had sold him. He now imports fruit and vegetables from Northern Greece to Korçë. Second, though they are Orthodox, the Devoll's Christians insist on the fact they are Albanian and not Greek, and they regularly accuse Greeks to consider them as Greeks (or Northern Epirotes). This does not mean there is no sympathy at all with the Greeks, but rather a fear of becoming an "ethnic" minority in a largely Muslim area: the Albanian Christians intend to remain Albanian, as long as they live in an Albanian district. As one of them complains: "At the time of the cooperative, the Muslims used to insult us as « Greeks of the devil », but when the border opened they all went to Greece, while we Christians stayed in Albania. Who's Greek in this story?" If it might be a good strategy for Albanians to claim Greek national and religious backgrounds, it is less so in a border area, where the proximity of the borderline helps to maintain a strong feeling of difference between Albanians and Greeks, especially when they share the same religion.

From the Muslim point of view, the present period of emigration is thus seen as a kind of revenge over the Christians: contact with the West and its civilization is not anymore a privilege of the Christians; Muslims too have now an access to modernity. Where as from the Christian point of view, Muslims are seen as traitors who leave their country and their religion when the country is faced with difficulties.

A second comparison is made by local people on emigration. Before and during the invasion of Greece by Italy, at the beginning of the forties, the economic situation in the Greek villages was worse than it was in the Albanian villages of Devoll. As long as the border was open, many Greeks would cross over to the Albanian villages and beg for food and work. In 1991 and 1992, when Albanians began to cross the border, begging for food, clothes and work, some of them were welcomed by Greek families who remembered their own situation during the war. To face the humiliation of being dependant on Greeks for food and clothing, many Albanians pretend that it is just reciprocity and they blame the Greeks who do not act good to them by saying: "those Greeks, they feel superior to us because we need them to survive, but you should have seen them in the past, when they were begging for bread!"

As can be seen from the spatial and temporal conceptions linked with emigration, the self, the human being is transformed by the crossing of the border. Muslim migrants usually change their names for Christian names as soon as they enter Greece. Some of them pretend to be of another religion (Christian instead of Muslim) or of another nationality (Vorio Epirote instead of Albanian). Family and relatives, who count for a great part in the definition of the self, have to be left in Albania. While an Albanian can always rely on his relatives to support him and avoid him any humiliation and social judgements, migrants in Greece are alone and have to manage for everything by themselves. National pride has also to be forgotten, as Albanians in Greece are often considered as second class workers and hardly as human beings. As one of them complains: "We always live in fear and danger. We have to sleep in the woods, just like wild animals."

As a result of these personal changes, emigration also generates new discourses on society and national identity. For example, there is a strong emphasis on family and kinship relationships which tend to consider the family as one of the bases of national identity. The Albanians, both migrants and non migrants, insist on the fact that family solidarity is much more developed among them than anywhere else, especially more than in Greece. Everyone can tell a story about Greek parents and grand-parents left by their children or send to an asylum where they are left to die alone. In contrast, they say, such things never happen in Albania. People are poor, and their only wealth is their family; they have to stick together to survive because nobody else will care for them. They cannot rely on society, or on the state, to keep them away from poverty: social organisations, such as cooperatives, which used to offer work and services to the whole village do not exist any more and have not been replaced yet. Traditional social groups, such as religious communities, have also been destroyed and, as they reappear today, they are not ready yet to offer all their members opportunities of socialisation. Family is thus seen as the only strong basis of society.

This tendency is somehow encouraged by the state and political parties, and gives family a religious or sacred dimension. "Fatherland begins at home" (atdheu fillon tek shtëpia jote), said the Republican party in the 1996 electoral campaign, as if the country has to be reconstructed around the last piece of national identity left by the communist regime and transition phase: family.

There would be much more to say about the use of kinship and family in today's Albania but, as far as emigration is concerned, two points can be noticed here. First, the discourse on family organisation is based on the opposition between an imagined Albanian family, on the one hand, and an imagined Greek or Western family, on the other. This opposition is mainly expressed in terms of solidarity and love, as opposed to individualism and distance. Even if the disintegration of family in the near future is sometimes feared as a consequence of the modernisation of Albania, people more usually consider that their family is the only place where Albanian identity can be preserved.

Second, that same discourse on Albanian family is rarely experienced in reality, as many families are dispersed by emigration, either inside or outside Albania. Familial solidarity does not function as it is said to function. It acts mainly in people's mind, as a response to social disintegration. It is frequent to hear people say that, "in the past, familial solidarity was even stronger than it is today, and it would extend to a man's distant cousins". Today, solidarity is usually limited to rather small nuclear families. A man leaves for Greece while his parents and his wife stay at home to care for the children, the garden and the animals. When he is back, the money he has earned is spent on a strict domestic basis: it is used to repair the house or build a new one, to buy food in winter time. In that way, familial relationships are as a much shaped by emigration than by traditional models. The importance of family is thus felt by people because emigration tends to weaken more traditional forms of kinship relationships. In other words, emigration often contributes to question cultural and social behaviour.

Family is not the only part of society to be questioned. As mentioned above, emigration brings to Albania many new behaviours which question unconscious as well as conscious tradition. These have to do with money and wealth, with the relationships between parents and children, or between men and women. In general, all these are expressed in terms of "good" and "bad". Among the new behaviours, some are seen as "good" ones (for example private enterprises are generally seen as a good thing), while others are seen as "bad" things (for example, spending one's money to buy a car when agricultural material would be more necessary).

It is interesting to notice that in this religiously mixed area, where Christians are a minority, many of these social changes tend to be organised around a religious opposition. First, emigration gives meaning and value to the already existing religious opposition between Muslims and Christians. It is commonly assumed by people that, since Greece is a Christian country, Albanians who want to find jobs should be Christians or at least pretend to be Christians. It thus gives rise to a religious differentiation inside Albania, between Christians who seem to be privileged, and Muslims whose religious identity has to be questioned. Second, many innovations tend to be associated with the Christian minority, as if this minority was acting as a carrier of modernity. For example, Muslims who want to look modern keep their shoes on as they enter their house, as Christians do; they also prefer Christian plum raki to Muslim grape raki, and so on.

To be Christian or Muslim, to act as a Christian or a Muslim, is thus very important in the construction of the self. It should be remembered that this fact is in contradiction with the state's official position on religion. While in national conceptions one is supposed to be Albanian before being Christian or Muslim, on the local level it seems that one has to be Christian or Muslim in order to be Albanian. Emigration to a Christian country seems to reactivate the already existing opposition between local Albanian Christians and Muslims, and to give it a new relevance in the definition of the self. The representations of space, time and self are influenced by the fact that the Devoll is a border area: it creates a specific sense of identity, made of distance from the centre of the state and community with the other side of the border. But this does not mean that the Albanian state does not exists any more in that area, or that the border has already changed and people consider themselves to be Greek. All seasonal migrants from Albania live in two states, according to the time of the year. This phenomenon is more emphasized in the border area where people live in two different states all along the year. Border people are much aware of what the Greek authorities decide relating to their status in Greece and to the border formalities. They have Greek friends and visit them; they take part in Greek religious feasts. They are looking for a form of integration inside Greek society, for example by bringing their family with them. Their children go to Greek schools, and their results in Greek history or language is a source of pride not only for their parents but also for the rest of the family in Albania. A Muslim woman from a border village was proud to tell me that her brother had been in Greece for such a long time that he found it difficult to speak Albanian. His children, she said, were first at the Greek school. It is interesting to notice that the Albanian state has started to react to this "hellenisation" of Albanian migrants: in March 1998, a Greek delegation was asked by the Albanian parliament to respect the right of Albanians for education in their language when they live in Greece. "Albania has always respected the rights to education of its Greek minority, declared the Albanian part. Greece should act the same. There are 30 000 ethnic Albanians around Kastoria and Florina [Northern Greece] and between 250 000 and 300 000 Albanian migrants live in Greece" (Rilindja,31.03.98). The two parts agreed on meeting every three months to discuss economic and political cooperation between the two states. In this kind of negotiations, it should also be noticed that the Albanian migrants are always compared to the Greek minority in Albania, which is a way of saying that, if they still belong to the Albanian nation, they also belong to the Greek state: they have come to form a new Albanian minority in Greece (the old one, the Arvanites, dating back to the 15th century, being now fully assimilated in the Greek nation-state). The ones who have brought their family to Greece do not intend to leave the country but rather to settle and look for complete regularisation through Greek citizenship. The decision taken in 1997 by the Greek authorities to regularise many of the illegal migrants (Albanian and others) goes in the same direction: Albanians are not anymore refugees that should be helped and send back to their country.

The search for integration within Greece includes also official cooperation between the local communities on both sides of the border. The district of Devoll has thus started to establish regular cooperation with its Greek neighbour of Kozani. This cooperation includes cross border tourist projects (originating from the Greek side, more aware of some modern trends in tourism -trekking, green and cultural tourism), border crossing agreement which would allow border people (within 20 km from the border) to cross freely, as well as material assistance in education and agriculture. All these projects are however very dependant on the general situation in Albania, and each crisis (winter 1996(winter -1997(winter , September 1998) ) seem to slow them down as insecurity on the Albanian side grows up. On the Greek part, local officials also complain of not being allowed to carry negotiations as they want to, due to the centralisation of the state: the border crossing agreement for instance, in project for one and a half year, still waits for the Greek parliament's vote.

Since they keep an eye on both sides of the border, border people are prompt to criticize and to react to what is happening in one or the other of the two states. They also feel easily abandoned by their own state, as long as they feel attracted by the other state (the situation is probably different on the northern border, where Yugoslavia is not attractive to Albanians). In that meaning, the state is contested here as in other parts of Albania because it is not present enough: the state should care for the roads and schools, for the water and electricity supply, for security and employment, but people complain because nothing is done. On the other hand, border people probably feel better than the others the need for the state: they exist as nation, society, culture and as individuals as long as the state maintains its presence at the border. But the state is not the only institution to be questioned by border people. Nation, religion, culture and, in one word, identity are continuously questioned and modified by border communities. As seen above, border people can pretend to be Greek, not only to find a job in Greece, but also to mark their difference with northern Albanians. They are also bringing home cultural models and behaviours from Greece, to mark their new social status inside Albania. This does not mean they can easily cross the boundaries of national identities, as Albanian society seems to have its own protections. Here is an example: an old Muslim from the village of Sinicë arrived one day in the village of Miras, to meet the mayor. One of the communal workers, who regularly goes to Greece, explains him where and when he can meet the mayor. He ends his explanation by asking the old man, in Greek: endaksi? (all right?), as it is usual among the migrants to use Greek words in the conversation. But the old man reacted with violence to this word: "What is this endaksi, he said, can't you say mirë? You are Albanian, you are not Greek! The Greeks destroyed our villages, they killed the Muslims! Never forget it!" The old man came from a village which was burnt by the Greeks during the first world war, as many other Muslim villages of the area. In the Devoll, where the national and state borders approximately coincide with a religious boundary between Muslims and Orthodox, it is not so easy to cross the national boundary, even if it is only by using Greek words.

Conclusion

The influence of Greek culture and economy in Albania is now a general phenomenon, at least in Southern Albania, since Albanian migrants who come and go between Greece and Albania come from almost all parts of the country. But it gets most obvious in border areas, where it started earlier and in a more intensive manner. The observation of international borders is also of interest to the anthropologist. Because it deals with people involved in different states, cultures and identities, it helps to question the anthropological concepts of 'society', 'space' or 'culture', which are usually taken for granted by the anthropologist as the 'natural' boundaries of the communities he studies. International borders remind us that every society or culture has to be constructed by its members, and that this is done as much at the periphery as at the centre of the so-called 'society'.

Both practices and representations of emigration in this border area appear to be embedded in a much wider context. Emigration is not only an economical necessity, nor a "natural" trend of Albanians, even if the present day migration phase reminds of earlier migrations. Being migrant or refugee (in the specific meaning, not the Albanian one) implies the crossing of borders and a new definition of the self. The borders the migrants have to cross are not only international borders separating different states; they are also the borders of collective identities and as such connect emigration to several oppositions and symbolic systems which question the national, religious and personal dimensions of one's identity: north and south, Muslim and Christian, civilization and barbary, past and present.


Guillaume Haiat

Etienne Barthel

An approximate model for the adhesive contact of rough viscoelastic surfaces

Surface roughness is known to easily suppress the adhesion of elastic surfaces. Here a simple model for the contact of viscoelastic rough surfaces with significant levels of adhesion is presented. This approach is derived from our previous model [E. Barthel and G. Haiat Langmuir, 18 9362 2002] for the adhesive contact of viscoelastic spheres. For simplicity a simple loading/unloading history (infinitely fast loading and constant pull-out velocity) is assumed. The model provides approximate analytical expressions for the asperity response and exhibits the full viscoelastic adhesive contact phenomenology such as stress relaxation inside the contact zone and creep at the contact edges. Combining this model with a Greenwood-Williamson statistical modeling of rough surfaces, we propose a quantitative assessment of the adhesion to rough viscoelastic surfaces. We show that moderate viscoelasticity efficiently restores adhesion on rough surfaces over a wide dynamic range.

Introduction

Surface roughness is a major parameter for the control of the adhesion of viscoelastic materials and as such is central to many technological processes. As an example, in the automotive industry, when assembling a polymer interlayer with two glass sheets to form a laminated windshield, a periodic roughness is intentionally generated on the polymer to reduce the adhesion and facilitate positioning before the final heat treatment [1]. Roughness also plays a role in the self-adhesion mechanism (or tack) of soft polymers [2,3] like repositionable adhesives. In a similar way, in the glass industry, glass molding is performed at elevated temperatures for easy glass flow but at the risk of adhesion. Indeed adhesion of hot glass to the mold halts production until the mold has been cleaned of its adherents. The roughness of the mold strongly impacts the adhesion of the viscoelastic glass gob [4] and a stochastic but carefully tailored roughness, obtained through shot peening of the steel surfaces for instance, will promote an easy molding process.

The adhesive contact to rough viscoelastic surfaces is a complex problem however because it couples the statistical approach of the adhesive contact to rough surfaces with the difficulties involved in the viscoelastic adhesive contact. For elastic solids, the adhesive contact to rough surfaces is reasonably well understood. Building upon the Greenwood-Williamson independent asperity approach, Fuller and Tabor [5] have given a useful description of the strong reduction of the adhesive force incurred when roughness increases. They introduce an adhesion parameter σ s /δ c where σ s is the standard deviation of the summit height distribution and δ c is the maximum single asperity extension before rupture. This adhesion parameter is a measure of the energy balance between stretched and compressed asperities. If the adhesion parameter is large, the compression of the upper asperities dominates the traction of the adhesive lower asperities which significantly depresses the adhesion.

The calculations are relatively straightforward for elastic contacts which are essentially reversible. For viscoelastic contacts, adhesion depends upon the full contact history so that the calculations are more involved. In the absence of adhesion, the viscoelastic contact problem was first solved by Ting [6]. Subsequently, important results for the viscoelastic crack problem were obtained by Schapery [7,8] and Greenwood and Johnson [9,10]. Hui and coworkers [11,12,13] have tackled the problem of viscoelastic adhesive contacts. We recently proposed a model for the full viscoelastic contact of adhesive bodies [14,15,16].

The present paper inquires into the application of this model to the adhesive contact to rough surfaces. To model the response of a distribution of asperities a simplified description of the viscoelastic contact is needed and the first part of the paper develops such a minimal model. It is devised so that the two main phenomena characteristic for viscoelastic adhesive contact -stress relaxation under the contact and creep induced by the adhesive interactions at the contact edges -are preserved while keeping calculations to a minimum. The model exhibits typical viscoelastic adhesive contact phenomenology, and in particular the time lag between indenter retraction and the actual contact radius recession, also called "stick period" [14]. In the second part of the paper examples of the use of this approximate model for rough surfaces are presented. The impact of the pull-out velocity on the pull-out force for typical rough surfaces is calculated. The results emphasize how a moderately viscoelastic material effectively restores the adhesion lost through roughness.

Model

Single asperity contact

Modeling the viscoelastic contact of an axisymmetric asperity in the presence of adhesion is complex because, as in all viscoelastic problems, the solution depends upon the full contact history.

In addition, for contact problems, one must deal with mixed boundary conditions: the surface normal displacement u(r) is prescribed inside the contact zone whereas the normal surface stress σ(r) due to the adhesive interactions is specified inside the so-called cohesive zone (Figure 1), outside the contact zone. Moreover the boundary between these two regions is not fixed but moves during the contact history. For a viscoelastic material, the solution is reasonably simple for growing contact radius [11,17]. However, during pull-out (i.e. decreasing contact area), the residual deformation has to be taken into account. This in turn depends upon the full stress history, which leads to a complex situation, both for adhesionless [6] and adhesive [11,12,13,14,15] contacts.

Resolution method

Under the assumption of axial symmetry, we resort to two auxiliary functions g(r) and θ(r), which are the following Abel transforms of respectively the distributions of normal surface stress σ(r) and normal surface displacement u(r):

g(r) = - +∞ r sσ(s) √ s 2 -r 2 ds, (1)
θ(r) = d dr r 0 su(s) √ r 2 -s 2 ds, (2)

Boundary conditions

A first benefit of these transforms is their pertinence for axisymmetric boundary conditions. In particular, inside the contact zone (r < a), the normal surface displacement is known through the contact condition

u(r) = δ -h(r) (3)
where δ is the penetration and h(r) the shape of the indenter (Figure 1). Then, using Eq. 2, inside the contact zone

θ(r) = δ -δ 0 (r) (4)
where

δ 0 (r) = d dr r 0 ds sh(s) √ r 2 -s 2 • (5)
It turns out that δ 0 (r) is the Hertz penetration for a contact radius r as shown in the next section. Note that this function δ 0 (r) is determined by the shape of the indenter only. It is equal to r 2 /R for a sphere of radius R and to π 2 r/ tan β for a cone of apical angle β.

Conversely, as mentioned above, the adhesive stress distribution σ(r) outside the contact zone is known. More precisely, in the present case, we will show in section 2.2.3 that it can be self-consistently determined through an independent set of equations, under the assumption of small cohesive zone size. Therefore the g function is known outside the contact zone (r > a).

Mechanical Equilibrium -Viscoelastic materials

The second benefit of these auxiliary functions is that they naturally handle the long range nature of the elastic field: for instance, under the conditions of linear elastic behaviour (Young's modulus E, Poisson ratio ν) and absence of shear stresses at the contact, mechanical equilibrium leads to

g(r) = E ⋆ 2 θ(r) (6)
where

E ⋆ = E (1 -ν 2 ) (7)
The contact problem is solved under the assumption of continuity of the stress distribution at a. The penetration is then directly obtained by [14] g

(a) = E ⋆ 2 θ(a) (8)
For example for an adhesionless contact, σ(r) = 0 for r ≥ a so that g(a) = 0 and δ = δ 0 (a) as mentioned in section 2.2.1. If adhesion is present, g(a) is not zero and a more complex situation arises.

For a linear viscoelastic material, under the approximation of constant Poisson ratio [18] we introduce the creep and the relaxation functions

φ(t) = 2 E * φ(t) (9)
ψ(t) = E * 2 ψ(t) (10)
with

φ(t) = 1 + (1 -k) k (1 -e -t µ ) (11)
ψ(t) = k + (1 -k)e -t kµ ( 12
)
where µ is the creep time. The parameter k lies between 1 (elastic) and 0 (Maxwell). Due to the decoupling between spatial and temporal responses, the equilibrium equation (Eq. 6) will now be

g(r, t) = t 0 dτ ψ(t -τ ) d dτ θ(r, τ ) (13)
or its inverse θ(r, t)

= t 0 dτ φ(t -τ ) d dτ g(r, τ ) (14)
where the relaxation function ψ(t) and the creep function φ(t) are inverse for this product of convolution. In the case of increasing contact radius, Eqs. 4 and 14 determine the penetration by

δ(t) = δ 0 (a(t)) + t 0 dτ φ(t -τ ) ∂ ∂τ g(a(t), τ ) ( 15
)
This result is equivalent to previous formulations [12,17]. The decreasing contact radius case is more intricate. Taking into account the domains where g and θ are known respectively, we have to resort to Eqs. 4, 13 and 14 to determine the penetration by the integral equation

g(a(t), t) = t t a-(a(t)) dτ ψ(t -τ ) ∂ ∂τ {δ(τ ) -δ 0 (a(t))} + ḡ(a(t), t) (16)
Here t a-(a(t)) is the time at which the contact radius was first equal to a(t), during the loading phase. The corrective term

ḡ(a(t), t) = t a-(a(t)) 0 dτ ψ(t -τ ) ∂ ∂τ τ 0 dτ ′ φ(τ -τ ′ ) ∂ ∂τ ′ g(a(t), τ ′ ) ( 17
)
is unessential, except for the cross-over between increasing and decreasing contact radius regimes (a = a max , t = t max ).

Then for a given history of the penetration, one can calculate the history of the contact radius and subsequently the force [14,15,16]. Under the assumption of small cohesive zone size (see section 2.2.3 for the cohesive zone and section 6 for details of the calculation) the force is given by Eq. 19 in [14]:

P (t) = 4 t 0 dτ ψ(t -τ ) d dτ min(a(t),a(τ )) 0 dr (δ(τ ) -δ 0 (r)) (18)
Note that this equation is valid for both inward and outward runs. Eqs. 16 and 18 involve stress relaxation inside the contact zone but also crack tip viscoelasticity through g(a(t), t), as discussed in more details in the next paragraph.

Cohesive Zone

For any adhesive contact, in the context of the present (Sneddon) method, the quantity g(a), which is the function g evaluated at the edge of the contact zone, plays a central role. For the contact zone, it determines Eqs. 15, 16 and 18 between macroscopic contact variables. But we have previously shown that, for the cohesive zone, g(a) determines the stress intensity factor [19] characteristic of the local deformation process at the crack tip.

Here we dicuss, for the specific case of a viscoelastic adhesive contact, how g(a(t), t) is determined by the cohesive zone parameters. This results in an independent relation between g(a) and the contact radius velocity ȧ through Eqs. 20, 21, 22 and 23. This problem is similar to a viscoelastic crack [7,8,9,10] and calls for a consistent treatment of an adhesive-stress induced creep inside a cohesive zone of finite size (c(t) ≥ r ≥ a(t), Figure 1). The equivalence of various methods for the description of the cohesive zone has been demonstrated earlier [20]. In brief, the details of the description of the stresses inside the cohesive zone is unimportant as long as the effective range of these interactions is preserved. Greenwood and Johnson [21] have proposed a description (the so-called "Double-Hertz") which is analytically simple and which we have retained for our viscoelastic models [14]. Following these authors, we assume a specific stress distribution which results in

g(r) = π 4 σ 0 r 2 -c 2 √ c 2 -a 2 . ( 19
)
Assuming that the size of the cohesive zone is small compared to the contact zone (a ≫ c -a ≡ ǫ) and taking the limit when r → a gives:

g(a) = - π 4 σ 0 √ 2aǫ (20)
For a decreasing contact radius the self-consistent equation coupling the total adhesion energy and the deformation of the contact zone writes (Eq. 23 of Ref. [14]):

w = π 8 σ 2 0 ǫφ 1 (t r ) ( 21
)
where w is the adhesion energy,

φ 1 (t) = 2 t 2 t 0 dτ τ φ(t -τ ) (22)
and t r is a fracture dwell time. For non vanishing contact radius velocity, this is the time it takes for the crack tip to move a distance equal to the cohesive zone size. It is given by:

t r (t) = -ǫ(t) ȧ(t) (23)
The simultaneous resolution of Eqs. 20-23 determines the velocity dependence of g(a). It is typical of viscoelastic crack problems : compare our set of Eqs. 20, 21 and 23 with, for instance, Eqs. 1, 47 and 48 in [7]. As a result, g(a) increases with ȧ as displayed on Figure 2: again compare e. g. with Fig. 6 b in [10].

General Model -stick period

For the viscoelastic adhesive contact, the main phenomenon we have identified [14,15] is the "pinning" of the contact upon unloading (Figure 3). More precisely, for such viscoelastic contacts, there is a typical time lag between the retraction of the indenter and the actual contact radius recession. We have called "stick period" this period over which the contact radius stays pinned, before contact edge retraction sets in, finally leading to contact rupture.

The physical interpretation of the stick period phenomenon has already been given in [14]. Relaxation of the stresses inside the contact zone reduces the stress intensity factor and hence g(a). This results in a slow crack motion (Figure 2) and the contact radius practically comes to a halt. When pulling the indenter out, enough tensile stresses must be re-built inside the contact zone before g(a) is large enough for the contact edge to be restored into propagating condition (Figure 2), and this depends upon the competition between stress relaxation and pull out velocity as discussed in [16].

Approximate Model

We need a simple description of the viscoelastic adhesive contact for rough surface calculations. We restrict the developments to the spherical asperity (radius R) usual in rough surface models although extension to other asperity shapes should be straightformard. We also introduce two major simplifications to the full model. One is an assumption on the type of loading/unloading. The other one is an approximation for the stick period.

Simplifying assumption -Direct unloading

We consider fixed-grip loading, i.e. a loading where the penetration is prescribed. In addition, the loading phase is considered to be infinitely fast and the pull-out velocity δ a (negative) constant (Figure 3), so that the penetration history simply writes:

δ(t) = (δ 0 + t δ)Θ(t) (24)
where δ 0 is the initial penetration1 , taken positive when the two surfaces are pushed one into another. Θ is the Heaviside step function. Due to the infinitely fast loading phase the quantity t a-(a(t)) defined in the unloading phase (section 2.2.2) is equal to zero.

Stick period approximation

This approximation stems from the observation that for large penetrations, the contact radius upon loading is primarily determined by the instantaneous elastic response and that crack tip creep during loading is not a dominant process [14].

As a result, in the following, the stick period is approximately modeled by a period of constant contact radius prescribed by the instantaneous modulus. This will be a good approximation except for small initial contact radii where the adhesive term is dominant.

In summary the approximate model is as follows:

1. calculate the initial contact radius from an elastic model with the instantaneous Young's modulus; assume this contact radius stays constant to the end of the stick period. The main viscoelastic process at work here is the stress relaxation inside the contact zone, which directly impacts the force (section 2.3.3); 2. determine if there is a stick period and if so calculate ending time t f (section 2.3.4);

3. calculate the unloading phase (section 2.3.5) and in particular the adhesion force.

Stick Period -Force

The force in the stick period is obtained from Eq. 18 taking into account the discontinuity of the penetration at τ = 0. This leads to (section 6.1):

P 1 (t) = 4ψ(t)   2a 3 0 3R - σ 0 πa 3/2 0 √ ǫ 0 √ 2E *   + 4a 0 δ t 0 dτ ψ(t -τ ) ( 25
)
which can be rewritten as:

P 1 (t) = ψ(t)P 0 + 4a 0 δ t 0 dτ ψ(t -τ ). ( 26
)
where the normalized form of ψ (Eq. 12) has been used. P 0 is the initial force and ǫ 0 the cohesive zone size at t = 0 which can be calculated from Eqs. 20-21.

The first term results from the relaxation of the initial force and the second one is due to the flat punch viscoelastic response and is proportional to the (constant) pull-out velocity.

Approximate determination of the end of the stick period

At the end of the stick period a(t f ) = a 0 because the contact radius increase has been neglected by assumption. However we do take into account the crack tip evolution which is reflected in a cohesive zone size ǫ(t f ) smaller than ǫ 0 . Since the contact radius is a constant, the crack dwell time t r (section 2.2.3) is approximated as the crap tip age, which is also the contact age. Therefore, when the contact radius starts decreasing at t = t f , the dwell time t r is taken equal to t f . Inserting Eq. 21 into Eq. 20 provides an approximate criterion for the begining of the contact radius decrease:

g (a(t f ), t f ) = - wπa 0 φ 1 (t f ) (27)
When the contact radius decreases, Eq. 16 becomes

g(a(t), t) = δ t 0 dτ ψ(t -τ ) + ψ(t) δ 0 - a(t) 2 R ( 28
)
Then t f is solution to the following implicit equation:

ψ(t f ) δ 0 - a 2 0 R + δ t f 0 dτ ψ(t f -τ ) + wπa 0 φ 1 (t f ) = 0 (29)
Note that the purely elastic case is recovered for t = 0. It is also recovered for constant ψ(t) (elastic material) because δ(t) = δ 0 + δt, and for long waiting times where ψ(t) ≃ ψ(+∞) and t 0 ψ(τ )dτ ≃ tψ(+∞). Injecting the initial elastic solution, t f is identified as the non zero root of function f , with

f (t) = δ t 0 dτ ψ(t -τ ) + √ wπa 0   1 φ 1 (t) -ψ(t) 2 E *   (30)
Time t f only depends on the mechanical properties of the material and on the history of the penetration (δ 0 and δ). Considering a given contact experiment on a rough surface, the mechanical and adhesive properties as well as the pull-out velocity are identical for all asperities. Therefore, t f will only be a function of the initial penetration δ 0 which is depends on the initial height of a given asperity.

The numerical calculation of t f (δ 0 ) is achieved using a bisection method. From Eq. 30, and knowing that a 0 increases with δ 0 , t f (δ 0 ) is a strictly increasing function of δ 0 and can thus be numerically inverted to obtain the function z f (t), representing the initial penetration necessary for a given asperity to reach the end of the stick period at time t. This function will be useful for the computation of the total force contribution (section 2.5) from the asperity height distribution.

Decreasing contact radius

The computation of the load P 2 (t) in the case of a decreasing contact radius is performed using Eq. 18 which leads (section 6.2) to:

P 2 (t) = 4a(t)ψ(t) δ 0 - a(t) 2 3R + 4a(t) δ t 0 dτ ψ(t -τ ) (31)
The two terms are respectively the elastic force for a spherical asperity with the relaxed modulus at time t in presence of adhesion and the response of the viscoelastic medium to a flat punch displacement (contact radius a(t)). Note that the purely elastic case is recovered for elastic materials because δ(t) = δ 0 + δt. Eqs. 21, 23, 28 and 31 completely describe the decreasing contact radius phase.

Single Asperity -Normalization

For a spherical asperity, following Maugis [14,22], the normalization of the contact variables is performed by introducing the following quantities:

P = P πwR , (32)
A = a 3πwR 2 4E ⋆ 1/3 , (33) δ
= δ 9π 2 w 2 R 16E ⋆2 1/3 , (34)
λ = 2σ 0 πw16E ⋆2 9R 1/3 , (35)
G(A) = - π 3 λ √ 2Aǫ. ( 36
)
were ǫ is the normalized size of the cohesive zone (same length scale as contact radius a). In normalized form, g(a) is given by:

G(A 0 ) = - π 3 λ 2A 0 ǫ0 (37)
In normalized form, the self-consistent equation 21 is given by:

1 = π 2 12 λ 2 ǫ(t) φ1 (tr(t)) ( 38
)
where φ1 is given by: φ1

(t r ) = 2 t r 2 tr 0 dτ (τ ) φ(τ ) (39)
Consequently, Eq. 29 used for the determination of the end of the first regime now writes:

ψ(t) δ(0) -A 2 0 + δ t f 0 dτ ψ(t f -τ ) + 2 2A 3 φ1 (t f ) = 0 (40)
In the stick period, the force for one isolated asperity (Eq. 18) is now given by:

P1 (t) = ψ(t) A 3 0 (t) -A 1.5 0 √ 6 + 3 2 A 0 δ t 0 dτ ψ(t -τ ) (41)
When the contact radius starts decreasing, G(A(t)) (Eq. 28) writes as follow :

G(A(t)) = δ t 0 dτ ψ(t -τ ) + ψ(t) δ(0) -A 2 (t) (42)
and the force in the second regime is expressed through:

P2 (t) = ψ(t) 3 2 A(t) δ0 - A 3 (t) 2 + 3 2 A(t) δ t 0 dτ ψ(t -τ ) (43)

Accounting for a rough surface

We follow the roughness description proposed by Greenwood and Willamson [23] in their adhesionless elastic contact study. Namely, a surface of nominal area A 0 with N asperities is in contact with a rigid flat plane. All asperities are assumed to have the same radius of curvature R and a Gaussian height distribution of standard deviation σ s :

χ(z) = 1 σ s √ 2π exp - z 2 2σ 2 s ( 44
)
where χ(z) is the probability that the summit of an asperity stands between z and z + dz. For a penetration of the rigid flat δ 0 , the number of asperities in contact is given by: n = N ∞ δ 0 dzχ(z) and all contact variables can be obtained with the same integration. Similarly, the calculation of the load at time t is performed by integrating the viscoelastic response of each asperity initially in contact:

P (t) = N ∞ 0 dzχ(z + δ 0 )P a (z, t) (45)
where P a (z, t) is the load at time t of an asperity undergoing an initial penetration of z -δ 0 .

3 Results and discussion

Single asperity -Impact of pull-out velocity

Force vs penetration curves (force plots) for a single asperity were calculated with λ=5 -a value consistent with our assumption of small but finite cohesive zone size -and for a moderately viscoelastic material with k = 0.5. Figures 4 and5 display results obtained for δ0 = 5 for pull-out velocities δ ranging from -0.01 to -50.

For very low pull-out velocity values (| δ| ≤ 0.1), there is an initial rapid decrease of the force, followed by a linear variation of the force as a function of penetration. This is due to the fast relaxation of the (predominantly compressive) stresses. The stress field is then controlled by the relaxed modulus which leads to constant contact stiffness and a linear force-distance curve segment, before the contact radius starts decreasing. When the penetration decreases further, the force becomes tensile (negative), displays an incurvation, levels off, finally increases and jumps to zero: this evolution of the force results from contact edge recession and asperity snap-off. The absolute value of the maximum tensile force defines the pull-out force F pullout , and the (positive) effective adhesion w ef f through w ef f = F pullout /(3/2 πR).

The very low velocity case is an elastic adhesive contact with an effective modulus equal to the relaxed modulus and a pull-out force equal to 3/2 which is the JKR value. At slightly larger pull-out velocities, we observe that the pull out force increases with pull-out velocity from this 3/2 value: higher pull-out velocity leads to increased contact radius velocity. Enhancement of the viscoelastic crack tip dissipation and effective adhesion follow as amply demonstrated in the literature [7,8,9,11,12,17,24]. It results from the combination of a fully relaxed contact zone and a less relaxed cohesive zone.

In contrast, for large pull-out velocities (| δ| ≥ 1), the initial precipitous drop of the force is no longer present and the pull-out force decreases with pull-out velocity. Eventually for very large pull-out velocity (e.g. δ = -50) the behavior tends toward an elastic behavior and the pull out force reverts to 3/2. Indeed, for large pull-out velocities, the relaxation of the compressive stresses in the contact zone is far from complete during the experimental time. The pull-out force is lower than for low pull-out velocity because the compressive contact stresses favor contact rupture.

As a consequence, there exists a transition with an optimum pull-out velocity (here of around δ = -1) for which the pull-out force is maximum. This result specifically originates from the consistent inclusion of both viscoelastic crack tip dissipation and contact stress relaxation.

Rough surfaces

As an example of results for rough surfaces, the same material parameters λ = 5 and k = 0.5 are chosen. The initial penetration is δ0 = 5.

Force distribution

Normalizing the roughness as the penetration (Eq. 34), we compute the force curve for a rough surface with a normalized standard deviation σs = 2. Computing the contributions of asperities displaying each regime (stick period vs. decreasing contact radius) to the total force for pull-out velocity δ = -1 (Figure 6), we observe that the proportion of asperities initially in contact is almost 100%, as expected for such a ratio of initial penetration δ0 to roughness σs . This type of contact is relevant to glass molding where a rather homogeneous distribution of asperities, with narrow asperity height and radius distributions, is generated on the mold surface. Note that the Greenwood-Williamson model is particularly apt at treating such a surface morphology. It would be less adequate for a less regular surface with a wide distribution of asperity height and/or a wide distribution of asperity radius.

The force is initially dominated by asperities pinned in the stick period (black dashed line). A gradual transition where the contact zones of the lower asperities de-pin and start to recede is observed: the contribution of pinned asperities drops to zero. Then the lowest asperities start to undergo contact rupture. It is in the middle of this transition that the maximum tensile force (pull-out force) is recorded.

Impact of roughness -Constant pull-out velocity

Force plots with increasing roughness (from σs = 0.1 to σs = 5) are displayed on Figure 7. For small σs , the results track the single asperity limit: all the asperities in the population exhibit an identical behaviour, having nearly the same initial penetration. For larger σs , the initial contact force increases as expected from the elastic (JKR) limit: for a single asperity the increase of the force with initial penetration is more rapid than linear. Simultaneously, the duration of the contact increases and the pull-out force decreases with roughness as in the elastic case [5]. Indeed, for one isolated asperity, the pull-out time increases with initial penetration. Therefore, when σs increases, the distribution of the pull-out times spreads out and the different asperities contribute less constructively to the total pull-out force, which therefore decreases.

Impact of pull-out velocity -Constant roughness

Figures 8 and9 display force plots at different pull-out velocity values, from δ = -0.01 to δ = -10 for a normalized standard deviation σs = 2. The overall picture is similar to the single asperity case (section 3.1) and the same transition from low to high velocity regimes is observed. At low pull-out velocities, a fast decrease of the force due to stress relaxation is followed by a quasi elastic behavior while at large velocities, the system is purely elastic. The pull-out forces in these two limit regimes are differentiated by the roughness. This is because the elastic moduli involved are different: indeed, the relaxed modulus is pertinent for the very low pull-out velocity regime, and the instantaneous modulus for the very high velocity regime. A larger elastic modulus will result in a larger adhesion parameter and a lower adhesion force [5,25]. A pull-out force enhancement is recorded between these two limit regimes because of the combination of partial contact stress relaxation and crack tip dissipation.

It is interesting to assess the details of the transition between these two regimes. Calculation of pull-out forces as a function of pull-out velocities were performed for a rough surface with a roughness of σs = 2, a cohesive zone parameter λ = 5 and a moderately viscoelastic material k = 0.5. The pull-out force as a function of pull-out velocity has been normalized to the rough surface limit in the high velocity elastic regime (instantaneous modulus). The results are displayed on Fig. 10 where the isolated asperity result is also shown for comparison. As expected both exhibit enhancement of the pull-out force in the intermediate regime where contact stress relaxation and crack tip creep occur at the same time. Figure 10 clearly evidences that the enhancement of adhesion is much more pronounced on rough viscoelastic surfaces than for an isolated asperity. Indeed for elastic materials a roughness as large as σs = 2 severely depresses the pull-out force due to the stored elastic energy of the higher asperities. However for a viscoelastic material these elastic stresses rapidly relax while the single asperity adhesive force is enhanced through crack tip dissipation. As a result, even a very moderately viscoelastic material is sufficient to enhance the effective adhesion on rough surfaces by an order of magnitude over a large range of pull out forces (Figure 10).

Conclusion

A simple model for the adhesive contact of rough viscoelastic surfaces has been developed based on a Greenwood-Williamson roughness distribution. For that purpose a simplified model has been proposed for the adhesive contact of viscoelastic asperities, assuming instantaneous loading and constant pull-out velocity. It retains the full phenomenology of the viscoelastic contact including stress relaxation inside the contact zone and creep in the cohesive zone.

For elastic surfaces, it is well known that increasing roughness rapidly suppresses adhesion. Our calculations give a quantitative assessment of how efficiently viscoelasticity restores adhesion on rough surfaces through the interplay between contact stress relaxation and cohesive zone creep. The phenomenon operates in a wide range of pull-out velocities.

Acknowledgements

We thank I. Sharma for numerous suggestions.

Appendix -Computation of the total force

The total force writes as [15]:

P (t) = 4 +∞ 0 g(r, t)dr (46)
The contribution of the cohesive zone is of higher order in ǫ (cohesive zone size) and is neglected, which is typical for large λ. Then Eq. 46 becomes P (t) = 4 a(t) 0 g(r, t)dr and Eq. 13 becomes

g(r, t) = t t a-(a(t)) dτ ψ(t -τ ) d dτ θ(r, τ ) (47)
with θ(r, τ ) = δ(τ ) -δ 0 (r) which is Eq. 4. Inserting Eq. 47 into 46 and inverting the order of integration (cf also Ref. [15], section 3.2.4) results in Eq. 18.

During the stick period

During the stick period by assumption a(t) = a 0 is constant and min(a(t), a(τ )) = a 0 . In addition for a sphere of radius R loaded as specified in section 2.2.1, combining Eqs. 4 and 24

θ(r, τ ) = (δ 0 + τ δ -r 2 /R)Θ(τ ) (48)
Then Eq. 18 gives:

P 1 (t) = 4ψ(t) a 0 0 dr(δ 0 - r 2 R ) + 4a 0 δ t 0 dτ ψ(t -τ ) (49) P 1 (t) = 4ψ(t)(a 0 δ 0 - a 3 0 3R ) + 4a 0 δ t 0 dτ ψ(t -τ ) (50)
where P 1 denotes the force during the stick period. The initial penetration δ 0 is given by the double Hertz model and writes:

δ 0 = a 2 0 R - σ 0 π √ a 0 ǫ 0 √ 2E * . ( 51
)
This last equation, in combination with the previous one, gives Eq. 25.

[24] Greenwood, J. A.; Johnson, K. L. J. Colloid Interface Sci 2006, 296, 284.

[25] Maugis, D. J. Adhes. Sci. Technol. 1996, 10, 161.

Captions and Figures

Fig. 1: Schematics of the viscoelastic adhesive contact showing the gap between the surfaces and the normal surface stress distribution. Special emphasis is given to the cohesive zone where the attractive interactions operate across the gap between surfaces. Fig. 2: Schematics of the relation between g(a) -proportional to the stress intensity factor -and the crack velocity (after [14]).

Fig. 3: Prescribed penetration as a function of time (δ(t)) and resulting contact radius history (a(t)). The constant radius approximation proposed in the present model is also shown as a appro (t). The end of the stick period is t f . Fig. 4: Single asperity force plot for different values of the pull-out velocity (from δ = -0.01 to δ = -1) and for the elastic limit (JKR) with the relaxed modulus.

Fig. 5: Single asperity force plot for different values of the pull-out velocity (from δ = -1 to δ = -50) and for the elastic limit (JKR) with the instantaneous modulus.

Fig. 6: Force distribution during adhesion rupture on a rough surface showing the contributions from asperities in the stick period and asperities with receding contact radius. The proportion of asperities in contact is shown on the right hand axis. Fig. 7: Force plots for rough surfaces with increasing roughness. Here δ 0 = 5 and δ = -1.

Fig. 8: Force plots for a rough surface with σ s = 2 for increasing pull-out velocity at low velocities. Also shown is the elastic limit for the relaxed modulus. Fig. 9: Force plots for a rough surface with σ s = 2 for increasing pull-out velocity at high velocities. Also shown is the elastic limit for the instantaneous modulus.

Fig. 10: Pull-out force enhancement over elastic case as a function of pull-out velocity for an isolated asperity and for a rough surface with σ s = 2. . δ=-0.01

. δ=-0.1 . δ=-1

Figure 1 :

 1 Figure 1:

Figure 2 : 20

 220 Figure 2:

Figure 3

 3 Figure 3:

Figure 4

 4 Figure 4:

Figure 6

 6 Figure 6:

Figure 7

 7 Figure 7:

Figure 8 :

 8 Figure 8:

This initial penetration cannot be confused with the δ 0 (r) function introduced earlier and which will now be substituted by its specific value for a sphere r

/R in the rest of the paper.

Decreasing contact radius

By assumption the contact radius is constant or monotonously decreasing from τ = 0 so that min(a(t), a(τ )) = a(t) and Eq. 18 directly results in Eq. 31. . δ=-50

. δ=-10

. δ=-1


Matthieu Tourbier
email: matthieu.tourbier@ec-nantes.fr

Bernard Peseux
email: bernard.peseux@ec-nantes.fr

Bundi Donguy
email: bundi.donguy@ec-nantes.fr

Laurent Gornet
email: laurent.gornet@ec-nantes.fr

MODELLING AND SIMULATION OF THE THREE-DIMENSIONAL HYDRODYAMIC PROBLEM

This paper deals with slamming phenomenon (impact between bow ship and water free surface). Slamming loads on ship may be sufficiently important so as to create plastic deformations of the hull external structure. In extreme cases, they have been recognised for being responsible for the loss of ships.

The problem to solve is transient and highly non-linear due to the character of the flow. In the present paper, the three-dimensional Wagner problem is solved numerically using a variational formulation together with a Finite Element Method. Threedimensional results for simple rigid bodies such as a cone and an ellipsoid are successfully compared with analytical results. Results for deformable structure will be presented.

INTRODUCTION

In rough seas where large relative amplitude ship motions occur, the bow may emerge out of the water. In these conditions of the state of the sea, the ship hull can experience impact loads as the bow reenters the incoming wave. This phenomenon is commonly referred as slamming. Despite the pressure field due to the impact remains localized in space and time, slamming loads can locally create plastic deformations to the hull external structure. In extreme cases, they have been recognized for being responsible for the loss of ships. The purpose of this paper is to give some insight about this subject.

There are several approaches to study such a problem. On one hand, the Navier Stokes equations can be used to solve in velocity formulation, and on the other hand a velocity potential theory is introduced. In this paper, the second approach is going to be applied. The velocity potential equations are studied by an asymptotic analysis. The two dimensional problem has been widely studied and the theoretical analysis is now well established in case of a rigid body. First order analytical solutions have been found for a wedge with small deadrise angles [START_REF] Wagner | fdber Stoss und Gleitvorgfinge an der Oberflfiche von Flttssigkeiten[END_REF], a cylinder [START_REF] Cointe | L'impact hydrodynamique en deux dimensions[END_REF], and arbitrary two dimensional blunt body shapes [START_REF] Cointe | Two dimensional water solid impact[END_REF], Howinson, et al 1991). The latest consider an axisymetric body yet the results are closed to two-dimensional ones. In above mentioned studies, the body is assumed to be rigid.

The two dimensional asymptotic solutions have been well compared to drop tests of different shapes [START_REF] Cointe | Hydrodynamic impact analysis of a cylinder[END_REF][START_REF] Fontaine | Asymptotic theory of water entry[END_REF][START_REF] Zhao | Water entry of arbitrary two-dimensional sections with and without flow separation[END_REF][START_REF] Magee | A coupled approach for the evaluation of slamming loads on ships[END_REF] and for rigid cone in three-dimensional [START_REF] Faltinsen | Water entry of ship sections and axisymmetric bodies", High Speed Body Motion in Water[END_REF]. For three-dimensional case, analytical solutions are far from possible expect in case of particular geometry [START_REF] Scolan | Three-dimensional theory of water impact. Part 1. Inverse Wagner problem[END_REF]. [START_REF] Korobkin | Wave impact on the bow end of a catamaran wetdeck[END_REF], [START_REF] Kvalsvold | Hydroelastic modeling of wet deck slaming on multihull vessels[END_REF][START_REF] Faltinsen | Water entry of ship sections and axisymmetric bodies", High Speed Body Motion in Water[END_REF], studied ship bow behaviour modelling the structure as a beam so that the coupled fluid-structure mathematical problem can be solved simultaneously. It results that hydroelasticity have not to be neglected. More recently, Donguy, Peseux and Fontalne (2000) have shown that structural deformations present an important threedimensional character. So three-dimensional effects have to be taken into account.

This study takes place within this framework. We try to develop a purely numerical approach to solve general three dimensional problem of slamming, for rigid and deformable body, considering fully coupled fluid-structure interaction problem.

In a first part, the three dimensional governing equations for the fluid are presented. The setting equation relies on the asymptotic theory. For any geometry of body, the velocity potential formulation can not be solved analytically. So, a numerical resolution is performed by the finite element method. In a second part, the structure problem is considered and is classically solved by the finite element method. Then, we will focus on the coupled problem. The fluid structure interaction introduces a coupling matrix that we validate in two or three dimensional cases. Finally, we present solution of the coupled problem, which is iteratively solved by the Newmark time integration scheme. We consider the problem of the impact of a three-dimensional body on the free surface (Fig. 1). The water entry problem is expressed within the framework of potential flow theory. Indeed, the fluid is assumed to be perfect and incompressible and the flow is irrotational. The velocity field can therefore be evaluated according to ~ = gr?tdO, where ~=~x,y,z,t) is the velocity potential. Under these assumptions, the velocity potential must satisfy the following boundary value problem:

A¢ =0 in t2 z (1)
-~n = u, .n on the body (2) 9¢p 1 _ 2 + P + g z =0 on the free surface (3) "~t +-2 ( grad~ ) 9

Oh 94 "7-+ g rdd~,g rddh -7 -= 0 dt OZ on the free surface (4)

where the displacement of the body is given by its ff,(x,y,t) and the free surface elevation is written as z = h(x,y,t).

The body boundary condition, Eq. 2, expresses the continuity of the normal velocity component on the fluid-structure interface. The kinematic and dynamic free surface conditions, Eq. 3 and Eq. 4 respectively, state that the free surface is a material surface along which the pressure is constant. Moreover, the fluid is assumed to be initially at rest:

h(x,y,O) = 0 ~(x,y,z,O) = 0 (5)
and the flow to be unperturbed far from the body :

[grad(p I --> 0 when (x2+y2+z2)m ~ oo (6)
Once the velocity potential is known, the pressure on the body is calculated by Bemoulli's equation:

p Ot z (7)

2.__22 Asymptotic formulation

The above mentioned boundaries value problem cannot easily be solved in 3D, even numerically, due to the local and moving flow character. Classically, the hydrodynamic problem is simplified into:

A(~ = 0 in ~Qy (8) 9¢ -_ On tq.n on FB (9) = 0 on FL (10) Oh 00 Ot = 9-"z on FL ( 11
)
where FB is the projection on z = 0 of the wetted body surface and FL the position of the undisturbed free surface. Of represent the linearized fluid domain (see Fig. 1). [START_REF] Cointe | Two dimensional water solid impact[END_REF], [START_REF] Wilson | The mathematics of ship slamming[END_REF] used the asymptotic expansion method to justify the mixte Neuman-Dirichlet problem. The perturbation procedure strongly relies on the blunt body assumption since the small parameter used in the asymptotic expansion is the ratio between the immersion parameter and the characteristic length scale of the body wetted width. Within a far field point of view, the body boundary condition, Eq. 2, can be written on z = 0 without introducing significant error, therefore justifying Eq. 9. The exact dynamic condition, Eq. 3, has been replaced by a Dirichlet condition for the potential on the undisturbed position of the free surface, Eq. 10. Physically the acceleration in the fluid is assumed to be large compared to gravity which can be neglected during the first instants. Finally, the simplified kinematical free surface condition, Eq. 11, states that the vertical displacement of the free surface is equal to the fluid vertical motion, evaluated on the undisturbed position of the free surface. The quadratic terms have to remain small compared to the linear ones in order to keep valid this approximation. The resulting problem, Eqs. 8-11 is often referred to the generalised [START_REF] Wagner | fdber Stoss und Gleitvorgfinge an der Oberflfiche von Flttssigkeiten[END_REF] problem. In the present formulation, three-dimensional effects are retained through the Laplace's equation, and structural deformations of the body surface are taken into account in the body boundary condition.

Numerical resolution

The resolution procedure for the velocity and displacement potentials is based on the Finite Element Method. The weighted residual method is applied considering the weighting functions ~0 which verifies the boundary condition ~o = 0 on FL • The problem comes to minimize the integral quantity:

W ((o) = In, (oR((o)dD (12)
where the residue R(~) is set equal to Aq~. Then, applying Green's identity and taking into account of boundary conditions, Eq. 13, leads to the weak formulation:

In, g r~a(p.grffdO . dD = It. <p(u,.fi). dS (13)
The potential ¢ and weight functions q~ are approximated using Galerkin's method leading to:

q) = <N]>lq)] e ¢ = <Ny>[¢] e (14)
where the shape functions ivy depend on the types of elements, and (0} e denotes the nodal potential vector of the finite element (e).

Finally, the following linear system is deduced from the discretization of the bilinear forms, Eq. 13:

[HI{c} = {G, }

H,; = Ia" Nij N j,flD

(16) [G) ] e = Iced [N f ](@~)dS (17)
To evaluate the right hand side of eq. ( 17), it is necessary to know the wetted portion of the body Fn or, equivalently, the position of the contact line between the body and the fluid. Due to the deformations of the free surface, the wetted surface is part of the unknown of the problem. An additional equation is therefore needed to close the problem. Physically, the contact line is determined by imposing to the solution to satisfy volume of the fluid conservation [START_REF] Wilson | The mathematics of ship slamming[END_REF][START_REF] Fontaine | A second-order solution for the wedge entry with small deadrise angle[END_REF] or equivalently, by imposing the existence of an intersection point between the elevation of the free surface and the body as intuitively done by [START_REF] Wagner | fdber Stoss und Gleitvorgfinge an der Oberflfiche von Flttssigkeiten[END_REF]. Volume conservation should be automatically satisfied since Laplace's equation is solved, the wetted surface beeing determined through an iterative procedure. [START_REF] Donguy | On the ship structural response due to slamming loads[END_REF] Once convergence has been reached, the position of the contact line is known. The linear system for the velocity potential is then solved. As a result, a simple first order backward finite difference scheme gives an accurate estimation of the pressure. This last solution represents only the fluid flow far from the body. Consequently, pressure distribution is singular close to the boundary wetted surface. This singularity is corrected by matching numerical solution with the analytical solution of the flow near the body.

HYDROELASTICITY

In this section, the previous analysis is extended to treat the case of deformable body. The equations governing the structural deformations are first recalled. A coupled method to solve the fluidstructure interaction problem is then proposed.

3.__! Governincj structural equations

The usual assumptions of small perturbations is made. The equilibrium equation reads:

Ot-'q =d'[v,t-~+] in g2, (18)
where X denotes the stress tensor, ff, corresponds to the structural displacement, and f corresponds to volume force associated for example with the gravity field. If gravity effects do not contribute to the local deformations here, they nevertheless influence the global motion of the body before impact. Boundary condition on the structure is:

~.~ =~,~ on re (19)
where the subscript f and s refer to the fluid and the structure, respectively, The structural modeling is performed within the framework of linear elasticity. Hooke's law is used to express the relation between the stresses and the=strains which are given by the linearised Green-Lagrange tensor E : E,,j =/(u,,j + uj,~) (20)

3.__22 Coupled problem

The structural problem is also solved using the Finite Element Method. The principle of virtual works is applied to Eq. 18 together with boundary condition, Eq. 19, leading classically to the discretised form:

+[Kl{U}={r) (21) {F} ' -- (22)
where the mass and stiffness matrices, respectively [MA and [K],

together with the generalized load vector {F}, are obtained by assembly of the elementary terms calculated on each finite element (e). The nodal displacements [U} is linked to the interpolation of the displacement field through:

{u, }= [N~ ]{U}, (23)
By definition, hydroelastic effects arise as a result of a strong interaction between the fluid flow and the structural response. From Eq. 22, it can be seen that the pressure imposed by the fluid will drive the structural deformations.The structural deformations influence the pressure field through the body boundary condition as shown by Eq. 17. In order to obtain a robust method to solve the coupled fluidstructure interaction problem, all the evolution equations are integrated simultaneously. The structural deformations are included in the fluid flow discrete formulation, while the pressure is expressed as the time derivative of the velocity potential in the discrete structural analysis. The following problem is then obtained:

[FS } [FSf = ~r$ {N I }{n}r[Ns ]dS (24) (25) (26)
where the coupling terms [FS] representing the fluid -structure interaction appear explicitly. Eq. 26 shows that the coupling term involves only the values of the structural displacement g, and the velocity potential ~ on the wetted surface of the body, FB.

3.__33 Validation of the couplinq matrix in 2D

The numerical resolution of Eq. 24 or Eq. 25 is performed using commercial Finite Element Method numerical code Castem. An external procedure is implemented for the evaluation of the coupling matrix [FS]. The correct implementation of the approach is checked for the test case problem of sloshing in a rectangular tank including an elastic beam in its center, as shown in Figure 2 for twodimensional case. This problem has been studied experimentally by Cha'f (Cha~, 1996).

For this problem, gravity cannot be neglected and so the formulation introduces a mass matrix. We obtained the following problem [START_REF] Morand | Interactions Fluides-Structures[END_REF]): ( 27)

[FSff = Ires {N/]{n }r [N s Iris (29)
The mechanical excitation consists in a harmonic load applied to the beam or the plate extremity. A frequency scanning is performed, see Fig. 4, to evaluate the eigenfrequencies which correspond to the peaks of the beam displacement. The free surface elevation for the first four modes of the coupled problem is shown in Fig. 3, together with the modal shape of the elastic beam. The numerical values of the eigenfrequencies are presented in table 1, column C1. They are compared to the results by [START_REF] Peseux | Dynamic response of partially filled tanks -Explicite time integration method[END_REF], column entitled "Ref'. A toolbox that allows the direct computation of the resonant frequencies was also applied, see column C2, All the results are relatively close to each other, especially, the two last columns. The computation by Peseux was performed using another f.e.m, code, which may explains the small differences in the results. The agreement between the three methods is very satisfactory. To validate the coupling matrix in three-dimensional case, we are interested in the test case problem of sloshing in a rectangular tank including an elastic plate in its center, as shown in Figure 5a. The figure 5b shows the finite elements mesh used, which are hexaedron elements for the fluid and quadrilateral elements for the structure. The equations of the problem are the same as the Eq. 27 -Eq. 29.

Once again, the mechanical excitation consists in a harmonic load applied to the plate extremity. A frequency scanning is performed, see Fig. 6, to evaluate the eigenfrequencies which correspond to the peaks of the plate displacement. The free surface elevation with the modal shape of the plate beam for the first four modes of the coupled problem is shown in Fig. 7. In table 2, we reported the numerical values of the eigenfrequencies are presented in table 2. Morever, experimental results for plate vibration [START_REF] Peseux | Contribution ?t l'dtude de structure partiellement immergdes en matdriau homogbne ou en composite[END_REF] are given in the column "Exp". Comparison of frequencies leads to good agreement, which give confidence into the external numerical procedure that has been developed in the present study. Developing this procedure, and controlling precisely the coupling terms resulting from the fluid -structure interaction is necessary to treat the water impact problem of an elastic body. (31) where:

The fluid -structure interaction problem, Eq. 31, is then integrated starting from the initial conditions: W(t=O)=W, 02) 6qt=oo,=b+() (33) in a numerical temporal explicit or implicit scheme with a time step dr. Nevertheless, studying for rigid body slamming, it appears that time step dt for the fluid problem has to be less than the fiftieth part of the instant to calculate accurately pressure distribution. Choosing the same time step for the coupled problem simulation leads to generate a high costly time procedure.

Consequently, we solved the fluid-structure interaction problem with an iterative procedure. The fluid problem is first solved with a step time S t. After determination of the force due to the pressure, the structural resolution is within the Newmark temporal integrated sheme with the step time At [START_REF] Donguy | Etude de l'int6raction fluide-structure lors de l'impact hydrodynamique[END_REF]. The numerical results have been compared to the experimental ones obtained for the impact of cone with thickness 1.5mm and deadrise angle 14". The figure 9 represent the pressure history at two points of the structure. The structural deformation present a strongly axisymetric character as shown in the figure 10. In the present paper, a numerical method is proposed to solve the three-dimensional hydrodynamic impact problem for blunt bodies.

In a first part, we have presented the fluid problem formulated in velocity potential and relying on asymptotic theory. The equations deduced from such formulation have been solved through the FEM. In a second part, we have examined the structural governing equations, which have been classically resolved by FEM as well. Finally the two previous methods had been used to treat the case of deformable bodies. Yet this formulation introduces a coupling matrix. The correct evaluation of the terms has been checked for the sloshing problem of a tank with an elastic beam in two-dimensional case and elastic plate in three-dimensional case. The coupling problem has been integrated in time by a Newmark schema with an iterative coupling. The results that we have obtained under these assumptions are rather good.

This formulation entails the boundary conditions projection. To avoid this assumption, and follow the evolution of the free surface, other method can be used like the ALE (Arbitrary Lagrangian Eulerian) method.

  Figure 1. Geometrical definitions

  Figure 2. Elastic beam in a heavy fluid problem

  Figure 3. Eigemnodes shapes

  Figure 5b. Mesh of the elastic plate problem

Figure 8 .

 8 Figure 8. Iterative scheme used to solved the coupled problem Experimental drop tests of elastic cones have been realized at the Mechanic and Material Laboratory of the Ecole Centrale of Names[START_REF] Donguy | Etude de l'int6raction fluide-structure lors de l'impact hydrodynamique[END_REF]. The numerical results have been compared to the experimental ones obtained for the impact of cone with thickness 1.5mm and deadrise angle 14". The figure9represent the pressure history at two points of the structure. The structural deformation present a strongly axisymetric character as shown in the figure10.

Figure 9 .Figure 10 .

 910 Figure 9. Comparison of pressure history for the impact of a cone with thickness 1,5mm (deadrise angle ZJ"). In dot line : numerical solution -In straight line : experimental values