senior lecturer, John Mullen

War Memory in British Soldier Songs of the First World War La mémoire de guerre dans les chansons de soldat britanniques de la Grande Guerre

Keywords: War memory, popular song, popular culture, First World War, appropriation Résumé Première Guerre Mondiale, chanson, culture populaire, appropriation

dans les chansons populaires britanniques de 1880 à 1920.

John Mullen

In what is no doubt the biggest wave of commemoration events in British history, the myriad activities organized for the centenary of the outbreak of the Great War have made it clear to more people than ever that "remembering" is not simply recalling the past, but seeing the past, in the words of Paul Fussell, "remembered, conventionalized and mythologized". [START_REF] Fussell | What a Lovely War: British Soldiers' Songs 3 F.T. Nettleingham, Tommy's Tunes, 1917 and More Tommy's Tunes[END_REF] In this contribution I would like to look at how one particular cultural objectthe soldiers' song has been used over the last century to revive a memory of the war, for particular and varying objectives.

The major characteristics of soldiers' songs

Soldiers' songs are generally anonymous, invented by troops to sing in groups, whether on the march or at rest behind the lines. They are sometimes referred to as "trench songs", though in front-line trenches they may not have been much used. They were not written to be sold, and so are distinct from commercial music-hall songs performed at a theatre, such as "It's a Long Way to Tipperary" or "Oh, What a Lovely War!" They are normally only a few lines long, and are set to well-known tunes, since their inventors were rarely trained musicians. The tunes come from music hall hits or from hymns, and the songs often exist in a series of variants, since each group of singers might add or change a verse or two.

Soldiers' songs have existed since well before 1914, 2 but the fact that First World War armies were exceptionally large, and immobile for long periods, made for a much richer production. There was even, during the war, a certain popularization of some soldiers' songs among civilians via the music hall: a few singers, such as male impersonator Hetty King, produced on-stage medleys of "songs the soldiers sing".

The First World War soldier songs are available to us because of collections published either at the time, often by officers (since they had more access to publishing) 3 or by veterans after the war, who collected them through associations such as the British Legion, or through appeals in the press. [START_REF] Brophy | The Long Trail 5 For example M. Arthur, When This Bloody War Is Over: Soldiers' Songs of the First World War, and M. Pegler, Soldiers' Songs and Slang of the Great War 6 Burnley News[END_REF] Sometimes collections of "songs the soldiers sang" do not distinguish between music hall songs and soldiers' songs. 5 This is unfortunate, because the filters the two genres needed to pass through to become popular were quite different.

I have previously worked on a corpus of 160 soldiers' songs. Most of these were in published collections, although some were quoted in the press during the war or appeared in personal diaries. The repertoire for such an ephemeral type of singing was varied and unstable. A local newspaper, The Burnley News, declared in 1919: nearly every unit in the British army has songs in its repertoire that are their own peculiar treasure and are not sung by anyone else. These are usually composed by the company 'funny man'... These songs however, have often only quite a short reign of popularity and are soon discarded and never sung again. 6 The British soldiers' song repertoire is widely recognized as dissenting in character, and in this very different from commercial music hall songs. 7 Dissenting music hall songs were rare because of the centrality of singalong to the music hall experience. The singer was expected to persuade almost the whole of the audience to sing along with the chorus. If half refused to sing, the turn would be deemed a failure, and the singer might well find themselves without further bookings. Thus anti-war songs are not sung, since they would neither achieve consensus among the audience nor be favoured by generally conservative theatre managers anxious to gain valued respectability by showcasing their patriotism. 8 The soldiers' songs, if they were to catch on, had also to be consensual, but within a much more restricted social group: the lower ranks of the army, solidly made up of working class men. Superior officers were likely to be absent during the singing, or "turning a deaf ear". It is very significant, then, that the soldiers' songs are uncompromisingly dissenting and reject warmongering, hatred of Germans and glory of war discourses. They almost never go so far, however, as to be pacifistic or mutinous, for the same reason: the need to achieve consensus within the immediate groups who are being expected to sing along.

It is essential to remember that the soldier songs do not represent the only groupsinging activity of the lower ranks. They also sang, in unison, music hall hits and religious hymns. The soldiers' songs constitute in fact a supplementary repertoire, used to express sentiments which are not provided for by the other repertoires. Thus, none of the soldiers' songs are love songs, because recruits' knowledge of music hall songs was sufficient to provide any love song required: there was no need to invent any. Similarly, the sentiment of trusting resignation in God or destiny, well-provided for in popular hymns, is not found in the soldiers' songs. The two main types of sentiment not provided by other repertoires are the vulgar and the dissenting, as briefly illustrated by these two extracts from songs:

Do your balls hang low

Do they dangle to and fro

Can you tie them in a knot

Can you tie them in a bow? Can you sling them on your shoulder Like a lousy fucking soldier Do your balls hang low? 9

Greeting to the Sergeant

You've got a kind face you old bastard You ought to be bloody well shot You ought to be tied to a gun wheel And left there to bloody well rot1

"Patriotic" War memory and the soldiers' songs

The repertoire of soldier songs becomes, after the war, a part of war memory which can be either forgotten or (selectively) revived and used for different ideological purposes.

There had been collections of soldiers' songs published while the war was still going on, but two ex-soldiers decided in 1930 they wanted a more comprehensive collection. This was the now classic publication edited by John Brophy and Eric Partridge: The Long Trail: What the British soldier sang and said in 1914-1918. Brophy had joined the army as an under-age volunteer at the age of fourteen in 1914, and later in his life became a journalist on the Daily Telegraph. Partridge, who was somewhat older, had been a schoolteacher before the Great War, served in the Australian army, and would later become known as a writer, producing, in particular, a series of dictionaries of slang. Indeed, the 1930 volume contained, in addition to a few dozen songs, a dictionary of soldier slang, and a substantial explanatory introduction.

The songs were sorted into categories "Songs sung on the March" "Songs sung either on the march or in the billets" "Songs rarely if ever sung on the march". The categorization shows the almost anthropological interest of the editors in transmitting some of the texture of the ordinary frontline soldiers' lives. It also suggests the nostalgic revival which was part of the writers' objective. Sadly, it is not the case that a dozen different ex-soldiers collected songs and published their views about the meanings of the songs. We have just the one book, with just the one viewpoint. It is valuable and interesting, but we must take into account that an exsoldier who becomes a journalist with the Daily Telegraph, a British newspaper with a very precise political orientation, may well not be representative. The question of why only one such book should appear is likely to be linked to the fact that, before the Second World War, publishing interest in the first war focussed on general questions of strategy and battle tactics, or on diplomatic and political analyses: the experience of the ordinary soldier was thought an unsuitable subject for study. It is also well-documented that a large proportion of ordinary soldiers preferred not to discuss their war experience, often not even with close family members.

Brophy and Partridge's book was well received: the work was republished, in 1930, 1931, 1965, 1969, 1972 and 2008. The collection did not only circulate the memory of the songs, but also a discourse on the nature of the repertoire. This discourse insisted firstly on the authenticity of these "grass roots" songs, representing the war memory of the ordinary soldier, very distant from either official anthems or high culture musical productions such as Benjamin Britten's War Requiem or Ravel's La Valse. As the Evening Telegraph declared, these pieces "come from the ranks, especially from the private soldiers without ambition to bear office or special responsibility…" 10 Brophy's introduction to a later edition also stressed authenticity:

They are the songs of homeless men, evoked by exceptional and distressing circumstances; the songs of an itinerant community, continually altering itself under the incidence of death and mutilation. [START_REF]From Brophy's introduction to the[END_REF] Secondly, the dissenting tone of much of the repertoire was underlined:

One of the major themes of the soldiers' songs was the theme of satire, on the war, and on all kind of mock heroics. They blatantly denied the sugary patriotism, the 'let me like a soldier fall' attitude, and brought out all the bitterness, the disillusionment, in a spirit of jocular irony. 12 Indeed, sentiments such as the following are quite typical: What did you join the Army for? Why did you join the Army? What did you join the Army for? You must have been bloody well barmy! Of the song "I Want to Go Home", 13 the 1930 reviewer commented: I think there is infinite pathos in that song. It marks the first disillusionment of the volunteer when he discovered that war is not romantic, and that the Germans were not subhuman. 14 Establishment commentators have regularly seemed ill-at-ease with the dissenting content, and attempted to find ways to reduce its impact. A Scottish journalist in September 1939 (no doubt a time when the eventuality of cynical soldiers was a sensitive subject) insisted:

Tommy loved 'safety valve' songs. Anyone unaccustomed to the workings of the British mind might have thought melancholy ditties were an indication of a weakening morale. But it was only the British way of keeping ourselves cheery by exercising our traditional right to grumble. 15 This term "grumbling" will be heard right up to the present day from those who wish to insist that the dissenting content of soldier songs is not to be taken seriously. The term, however, is problematic: to grumble is, according to the New Oxford Dictionary 16 "to complain or protest about something in a bad-tempered but typically muted way". Can the bitter and even murderous tone of some of these songs 17 really be reduced to being "badtempered", a question of individual mood? And is public group singing a "muted" form of protest? This definition of soldiers' songs as "grumbling" is an example of the conventionalizing and mythologizing of which Fussel speaks. The tremendous popularity, once the war was over, of a song like "Pop Goes the Major", a 1920s music hall song whose subject is the joy of finding one's superior officer from wartime and burning him alive, [START_REF] Adams | Pop Goes the Major[END_REF] suggests a more solid content to the dissent in soldiers songs, as does the tremendous force within war memories of a "Lions led by donkeys" view of hierarchical relations inside the army.

Along with dismissal as "grumbling", a more sophisticated opinion frequently encountered is that which sees dissenting singing as a deliberate way of exorcising one's own cynicism and demoralization. This is suggested already in the 1930 review: But, although [the soldier song repertoire] expresses the soldiers' war weariness, it also conquers it with ridicule, and that was how they kept going [… ] Messrs Brophy and Partridge have produced a most unusual book that brings back the spirit of the war, and the spirit that won the war. 19 Malcolm Brown, author of a dozen titles on the First World War, writes in the introduction to the 2000 edition of Brophy and Partridge's collection a similar view: Oh yes, there's anger and cynicism, and a world-weary longing for the whole damned thing to be over, and couldn't we all go back to dear old Blighty? But there's also a strong sense that by slagging off the war, and finding ways of laughing, even jeering, at it, the men deputed to fight it could find a courage that gave them the determination to win through. [START_REF] Brophy | Daily Telegraph Dictionary of Tommies' Songs and Slang[END_REF] John Brophy, in his own introduction to the 1965 edition, seems to go even further: … when the romantic conception of war proved false, out of date, useless, the man in the line was helped in his daily endurances if he could ridicule all heroics and sing, with apparent shamelessness "I Don't Want to Be a Soldier" or "Far Far from Ypres I Long to Be". These songs satirized more than war: they poked fun at the soldier's own desire for peace and rest , and so prevented it from overwhelming his will to go on doing his duty. They were not symptoms of defeatism, but strong bulwarks against it. [START_REF] Brophy | [END_REF] So the memory of war presented in these largely dissenting songs is reinterpreted in a way that the traditional narrative of determined troops who do not doubt can be maintained. Although it is obviously true that the songs helped the soldiers survive, it by no means follows that their main effect was to increase the determination to fight of the troops in question. As Tony Ashworth's work comprehensively shows, [START_REF] Ashworth | Trench Warfare 1914-1918, The Live and Let Live System[END_REF] soldiers frequently had other priorities, such as working out how to avoid fighting without being caught by superior officers. Even if, as ex-soldiers themselves, Brophy and Partridge deserve their voice to be heard, does this discourse not represent a case of dominant ideology reinterpreting the voice of the dominated? Through their songs, the soldiers seem to be showing their hatred of their superior officers and their cynicism about official patriotism, but we are told that this only appears to be the case. Popular grassroots expression is reduced by post-hoc cultural revisionism to a simple means of psychological relief aiming at reinforcing fighting spirit. My analysis does not of course imply that the dissenting tone of the British soldier songs shows a structured pacifist ideology or a reasoned political opposition to Empire war aims. In a situation where practically all the available political leadershipunion leaders, feminist leaders and influential intellectualscampaign in favour of the war, a politicized opposition was simply not available to most rank and file soldiers.

"Antimilitarist" war memory and the soldiers' songs

After this patriotic/ nostalgic configuration, the second major way in which these songs have been used to revive and create memories is by antimilitarist groups, singers or writers. By far the most important example of this has been the series of artistic productions of which the musical show Oh What a Lovely War, first produced in 1963, is the most wellknown. [START_REF] Littlewood | Oh What a Lovely War. 24 Chilton had already made radio documentaries on other subjects[END_REF] This production was inspired by a radio documentary, The Long Long Trail. Its creator, Charles Chilton, 24 whose father had died at 19 years old, a private in the First World War, began searching for more information about his father's experience. He said he was looking for "the true songs of the trenches", which would show, he believed, an authentic expression of grass roots experience. The resulting programme was broadcast on the BBC in 1961, and contained a mix of historical information, music hall songs, and soldiers' songs.

The tone was one of disgust with the war. 25 After hearing The Long Long Trail, Joan Littlewood and her theatre workshop worked with Chilton and others to produce their innovative musical production. Littlewood was an influential experimental theatre director, best known for her productions of Brecht's plays, and for that of A taste of Honey by Shelagh Delaney. In Oh What a Lovely war!, the Great War is presented as an Edwardian end-of-the-pier Pierrot show. In the first act the audience is encouraged to sing along with jingoistic music hall songs from the first few months of the conflict, such as "We Don't Want to Lose you, but We Think You Ought to Go". In the second act, they sing along mostly with the soldiers' songs which represent the disillusion and anger of the ordinary soldiers crushed by a military machine serving elite interests. The bitterly sarcastic presentation of the war as a pierrot show is intended to echo the tone of the dissenting soldiers' songs, though unlike the soldier song repertoire, the show communicates a coherent political critique of the war as the result of a society divided between ordinary people, just good for cannon fodder and an extremely rich elite who profit from the division and re-division of the world between competing empires.

From the point of view of war memory, it is important to note that Littlewood does not only propose to audiences a particular discourse on the war, presenting soldier songs as particularly suitable vehicles to transmit a truth she considers hidden by official discourses. She also proposes a war memory activity : that of singalong. Singing in unison has been defined by musicologist Philip Tagg as "a rather obvious aural icon of individuals sharing (supposedly or actually) a common cause, or identity, or set of values and beliefs". [START_REF] Tagg | Music's Meanings[END_REF] That is to say that, as the soldiers' songs were in their original contexts, the songs of Oh What A Lovely War are proposed in the theatre for group singing. In each situation, the attempt to build a public emotional expression of anti-war consensus is the aim.

The theatre production, which was a tremendous critical success, soon moved to the West End. [START_REF] Goorney | The Theatre Workshop Story[END_REF] Littlewood was keen that the anti-war message should be the most prominent element: a scoreboard on stage showed the rising death toll throughout the war. On the US stage it was replaced by a count of US citizens who became millionaires by selling arms during the war, an addition which caused frequent walkouts from the more expensive seats. [START_REF] Sweet | Oh What A Lovely Warwhy the battle still rages[END_REF] In the French production in Paris in 1963, the well-known antimilitarist song from 1917, the Chanson de Craonne was sung on stage, although at the time the song was still banned.

Those who have the dough, they'll be coming back, 'Cause it's for them that we're dying. But it's all over now, 'cause all of the grunts Are going to go on strike.

It'll be your turn, all you rich and powerful gentlemen,

To go up onto the plateau.

And if you want to make war, Then pay for it with your own skins.

There is conventionalizing and mythologizing in Oh What a Lovely War! too of course, notably in the presentation of the soldiers songs as a relatively pure authentic voice of the lower ranks to be contrasted with the jingoistic music hall. In fact a mix of music hall songs and soldier songs is used. The title song, "Oh What a Lovely War" is not a soldier song, but a music hall hit, and its existence underlines the fact that the dichotomy of music hall songs/ soldier songs has been much exaggerated. If there are practically no jingoistic soldier songs, there is significant dissent expressed in the music hall repertoire. 29 Despite Littlewood's determination that the radical message not be lost, not all antimilitarists were happy with the production. Some were very conscious of the complexity of presenting an anti-war message through singalong: Littlewood's husband, communist singer and actor Ewan Maccoll accused the show of leaving audiences "feeling nice and comfy, in a rosy glow of nostalgia". [START_REF] Goorney | The Theatre Workshop Story[END_REF] Littlewood, for her part, comments on the effects of her presentation of war memory:

Oh What a Lovely War awakened race memory in our audiences. At the end of each performance people would come on stage bringing memories and mementoes, even lines of dialogue which sometimes turned up in the show. [START_REF] Littlewood | Joan's book[END_REF] Littlewood's claim is that her production revived and revealed important truths about the war at this time when veterans were still common. This claim has recently been vigorously contested by important commentators, such as the present British Education secretary Michael Gove, who claimed :

The conflict has, for many, been seen through the fictional prism of dramas such as Oh! What a Lovely War, The Monocled Mutineer and Blackadder, as a misbegotten shamblesa series of catastrophic mistakes perpetrated by an out-of-touch elite. Even to this day there are Left-wing academics all too happy to feed those myths. 32 Mr Gove is not alone: the play has become a frequent target for historians who have chosen the label "revisionist" and who wish to defend different aspects of empire policy in 1914 to 1918. Brian Bond in his 2002 book writes: "For a new generation in the 1960s the play and film of Oh What a Lovely War had a dramatic effect … Until … recently, historians had either reinforced the myths, or had failed to counter them." [START_REF] Bond | The Unquiet Western Front[END_REF] Others go further: "Who will not have seen the scurrilous 1960s film Oh What a Lovely War! or the equally reprehensible and more recent BBC production Blackadder" [START_REF] Davies | Bloody Red Tabs[END_REF] writes Frank Davies, while Wolfson history prize winner, Andrew Roberts, in his A History of the English-Speaking Peoples since 1900, also denounces its "outrageous lampooning".

The stage show was succeeded by a film version, directed by Richard Attenborough and starring Susannah York, John Gielgud, Laurence Olivier, Vanessa Redgrave and Dirk Bogarde. Richard Attenborough was able to get the funding to produce the film by impressing the financiers with a star studded cast. The film was the 16th most popular movie at the UK box office in 1969. 35 Given the fact that the film's Box Office results were not those of a smash hit, and also the fact that the theatre show was not shown professionally in the theatre for over twenty years, and the first DVD release of the film was in 2006, ten years after DVD film formats became common [START_REF] Newman | Another opening, another show: Kat and the Kings and Oh what a lovely war![END_REF] it seems highly unlikely that the musical had the decisive effect on popular views of the war which some have claimed. Its lasting influence in some circles, though, is attested by its prominent presence in 2014 on the BBC History website. 37

The songs in the 21 st century and tensions within discourses

We have seen, then, the soldiers' songs used by two politically dissimilar traditions.

The question of why this is possible will be dealt with in the conclusions, but first I would like to present some very recent uses of the songs, which sometimes reveal tensions inside the two discourses. The songs fit most readily into musical genres which have been constituted as dissenting, in particular folk genres and punk rock. Frank McConnell produced in 2007 a folk version (few instruments, regional accent) of "Hanging on the Old Barbed Wire" accompanied by two other somewhat suggestive soldiers songs: "That's the Wrong Way to Tickle Marie" and "Mademoiselle from Armentiers". [START_REF] Arthur | When This Bloody War is Over…[END_REF] The CD also contains readings of a series of war poems of which the dominant theme is antimilitarist. Peter Rothstein produced in 2007 a theatre show All is Calmthe Christmas Truce of 1914, a recounting of the 1914 truce, with several soldiers' songs, including "We're Here Because We're Here", "I Want to Go Home" and "Hanging on the Old Barbed Wire", as well as a series of carols and quotations from interviews with soldiers. The show went on stage in the United States between 2007 and 2014, and was also produced for the radio. In the United Kingdom, Folk at the Oak, a series of local folk concerts in Crick Northamptonshire, used both "Hanging on the Old Barbed Wire" and "I Don't Want to Be a Soldier". 41 David Olney, a folk singer, in concert in 2011, sang an a Capella version of "Hanging on the Old Barbed Wire". The US punk group "Sucker Punched by Christ" produces an accelerated version with prominent drums, and angry punk voicing, the singer stripped to the waist, a style perhaps particularly suited to the bitter tone of the song. 42 Not all uses of the soldiers' songs in recent years are in antimilitarist or antiestablishment contexts, however. The version by The Spinning Jennys, 43 whose band name evokes "the olden days" and who sing dressed in Edwardian clothing, with a Union Jack draped on their piano, is followed by singing of music hall songs such as "A Long Way to Tipperary" and "Pack Up Your Troubles". The general tone of the concert seems to be nostalgia, so the cynical soldiers' song might seem out of place.

The development of the internet and particularly of

More distant still from antimilitarism is the case of the Military Wives choir. This singing group is a pro-military initiative aiming at raising public support for British soldiers and ex-soldiers, particularly in the context of the first ten years of the twenty first century when mass public opposition to British military interventions, actual or potential, in Iraq, Afghanistan, Iran and Syria has been so vocal. 44 The choir sang the following soldiers' song at a number of their concerts.

The Bells of Hell go ting-a-ling-a-ling For you but not for me:

For me the angels sing-a-ling-a-ling, They've got the goods for me.

Oh! Death, where is thy sting-a-ling-a-ling? Oh! Grave, thy victory?

The Bells of Hell go ting-a-ling-a-ling For you but not for me. This soldiers' song, initially reported as being sung sardonically by soldiers leaving the trenches to their replacement regiments, was part of the Military Wives Choir concert in York in October 2013. 45 The singers sang in evening dress, in a traditional choral style These examples bring up the question of appropriation, of who the soldiers' songs might be seen to belong to. As a number of commentators have pointed out, military policy in contemporary society requires ideological campaigns at home. "Given that wars tend to be unpopular, the government and military try to recast appeals for war as support for individual troops." writes Lindsay German of the Stop the War Campaign. 46 The use of soldiers' songs to showcase concern for the lives of individual soldiers can be part of attempts to defend British military policies. For antimilitarists, on the contrary, the songs are presented as belonging to the tradition of opposing elite plans which are against the interests of the mass of the people.

Tensions in antimilitarist uses of the soldiers' songs can also appear. In Derby, for the centenary of the outbreak of the First World War, the musical play "Oh What a Lovely War!" was produced. This might appear to be a clearly antimilitarist choice. Nevertheless, one sees the following note: "Tickets are priced £6, with £1 from each sale being donated between The British Legion and Help for Heroes". The choice of charities, in particular "Help for heroes" participating in the opposite discourse, since a "hero" is necessarily someone wounded or killed in a worthwhile cause (brave serial killers are not called heroes). The "Help for heroes" website is clearly not antimilitarist, the founder making it clear "I loved my time in the army" 47 . The choice of these charities is likely to be a response to common opinions of which tradition the songs should belong to, and the naming of two charities is probably a sign that there was some disagreement involved. Another production of the show, in late 2014, has announced that profits will be given to a hospital for military veterans. 48

Conclusion

One of the reasons the songs speak to us today is because of what they say about human beings faced with extreme, life-threatening situations. This is, perhaps, just one more aspect of a more general characteristic of cultural production: the preference of cinema, television and even visual art for extreme situationsmurders, kidnappings, catastrophesis well-known. For creating and communicating war memories, we have shown that soldiers' songs are highly flexible objects. The French historian, Nicolas Offenstadt, categorized discourses on war memory as made up of three major strands: genealogical or family discourses, local pride discourses, and activist discourses. [START_REF] Offenstadt | [END_REF] Soldiers' songs can be seen as belonging to two separate activist discourses. Antimilitarists who stage or support the production, in 2014, of the musical comedy Oh What a Lovely War, are hoping it will encourage anti-war feeling and activism, which, in the Britain of the last fifteen years has been at its highest point for several decades, in reaction to British interventions in Iraq and Afghanistan. For the Military Wives Choir or the Daily Telegraph, the celebration of these songs of "modest heroes" fits with campaigns of solidarity with "our soldiers" in these twenty first century wars. This celebration is part of a right wing populist discourse which takes for granted the general correctness of the military policy of the British state, and campaigns for more recognition of the cost to ordinary soldiers; such discourses also hope to mute opposition to British foreign policy.

How can many of the same songs be used in such radically different ways? In fact, the anti-militarists and the military populists have some things in common. Both share, along with a large section of the population, a desire to explore close-up the texture and feelings of life in the trenches. This experiential approach to the war has been dominant in the centenary, as witness the new galleries at the Imperial War Museum, or their campaign "Lives of World War One". 50 Both groups want to privilege the voice of the ordinary soldier. For the military populists, the heroes who fought for the Empire deserve listening to and having their life in wartime vividly recounted, despite the dissenting tone of much of the soldier song repertoire.

At the end of the day, these soldiers won, and clear victories for Britain have been much rarer these last decades, leading to an increased concentration in conservative circles on past glories. The fact that the dissenting tone of the songs had often been toned down even in the trenches, in order to please a mixed group of other ranks makes this process of reappropriation easier.

For the anti-militarists, the victims of a bloodbath which in no way served their interests deserve to have their voices heard, in particular in order to underline the huge gap between their viewpoint and that of the military authorities. The fact that the soldiers sing about suffering and sometimes about dying, but not about killing, facilitates both the initial popularity of the songs among frontline soldiers, and the incorporation of the songs into each of these two discourses

The importance of popular song in war memories, I think, will bear further exploration, which to be fruitful needs to bear in mind that popular music must be analysed as a mass activity and not just as a corpus of texts and melodies. For a particular repertoire, such as the soldier song, understanding the songs also means understanding the selection mechanisms which allow a song to join the repertoire. There is also a need for further research on the "nostalgia" aspect of the use of these songs. Nostalgia was considered a danger for some of the team producing Oh, What a Lovely War, and is often a sentiment difficult to identify and analyse. A number of researchers have worked on nostalgia and popular music, and the application of some of their conclusions to the case of soldiers' songs would certainly be productive. [START_REF] Barrett | Music-evoked nostalgia[END_REF]

Works cited

Arthur, Max. When This Bloody War Is Over: Soldiers' Songs of the First World War.

  His rhetoric seems effective, and allows conservative opinion to embrace the soldiers' songs in war memory. It is notable that the newspaper which republished recently in 2008 the Brophy and Partridge collection should be the Daily Telegraph, generally recognized as the most conservative of the serious daily newspapers. It was not The Guardian nor The Independent, who might be expected to be more open to antimilitarist sentiment.

  YouTube has made visible uses of the soldiers' songs in local or marginal events which would previously have been invisible. I have collected a few examples, which involve some of the most well-known of these songs. The one most frequently sung is "Hanging on the Old Barbed Wire", one of the most bitter of the pieces, which includes the following lyrics: 38 If you want to find the Sergeant I know where he is (x3) He's lying on the canteen floor I saw him, I saw him Lying on the canteen floor I saw him, Lying on the Canteen floor The brass hats...Drinking claret at Brigade HQ. The general: pinning another medal on his chest. The politicians....Drinking brandy at the House of Commons bar. If you want the old battalion I know where they are (x3) They're hanging on the old barbed wire. I saw them, I saw them Hanging on the old barbed wire I saw them Hanging on the old barbed wire. Cover versions of such songs constitute a commemorative activity, and in most cases it is part of an antimilitarist and anti-establishment activist discourse. The highly successful Leeds rock group Chumbawumba included the song in 2008 on their album English Rebel Songs 1381-1914, alongside songs about the fourteenth century peasants' revolt, radical Diggers during the English civil war, and the Chartist movement of the mid-nineteenth century. 39

  accompanied by a grand piano, and the song was inserted into a medley with "The Bells are ringing for me and my gal" a 1917 wedding song used in the 1942 musical comedy of the same name. Another use of a soldiers' song by institutions close to the military establishment is contained in the poetry reading organized in April 2014 by the Royal United Services Institute conflict war and culture event, in partnership with the Josephine Hart poetry foundation. The Royal United Services Institute presents itself as "an independent think tank engaged in cutting edge defence and security research. A unique institution, founded in 1831 by the Duke of Wellington." Other recent RUSI events included "General Stanley McChrystal on Operational Leadership", the "Chief of Defence Staff Lecture" and "Anders Fogh Rasmussen on NATO and Missile Defence". At their event, "I Don't Want to Be a Soldier" was read in formal poetry reading style, by top British actor, Rupert Evans, complete with cravate and leather-backed chair. I don't want a bayonet up my arsehole I don't want my bollocks shot away I'd rather stay in England In merry merry England And fuck my bloody life away.

London: Piatkus 2001 Ashworth,

 2001 Tony. Trench Warfare 1914-1918, The Live and Let Live System. London: Pan 1980 Barrett, Frederick S. et al. "Music-evoked nostalgia": Affect, memory, and personality", Emotion, Vol 10(3), June 2010. Bond, Brian. The Unquiet Western Front. Cambridge: Cambridge University Press 2002

J.Brophy and E. Partridge, The Long Trail, p.67


Matthieu Génévriez

Kevin M Dunseath

Mariko Terao-Dunseath

Xavier Urbain
email: xavier.urbain@uclouvain.be

Sequential double photodetachment of He -in elliptically polarized laser fields

Four-photon double detachment of the helium negative ion is investigated experimentally and theoretically for photon energies where the transient helium atom is in the 1s2s 3 S or 1s2p 3 P o states, which subsequently ionize by absorption of 3 photons. Ionization is enhanced by intermediate resonances, giving rise to series of peaks in the He + spectrum, which we study in detail. The He + yield is measured in the wavelength ranges from 530 nm to 560 nm and from 685 nm to 730 nm and for various polarizations of the laser light. Double detachment is treated theoretically as a sequential process, within the framework of R-matrix theory for the first step and effective Hamiltonian theory for the second step. Experimental conditions are accurately modeled, and the measured and simulated yields are in good qualitative and, in some cases, quantitative agreement. Resonances in the double detachment spectra can be attributed to well defined Rydberg states of the transient atom. The double detachment yield exhibits a strong dependence on the laser polarization which can be related to the magnetic quantum number of the intermediate atomic state. We also investigate the possibility of non-sequential double detachment with a two-color experiment but observe no evidence for it.

I. INTRODUCTION

Double photodetachment, the process in which a negative ion absorbs one or several photons and ejects two electrons, has been much less studied than double photoionization, its counterpart for atoms and positive ions [1]. While the important structural differences between anions and atoms are expected to alter the dynamics of double electron ejection, experimental investigation however has been hampered by the difficulty of producing anions in sufficiently high density.

Early work on double photodetachment involved moderately intense fields (∼ 10 10 W•cm -2 ) and aimed at either studying excess photon detachment, i.e. the absorption by the system of more photons than is energetically required, or at performing spectroscopy of autoionizing states embedded in the continuum [2,3]. The detection of positive ions following photodetachment is also at the basis of resonant ionization spectroscopy, although in this case the atom is optically excited to a Rydberg state and ionized by a static electric field [4]. In these studies, the production of positive ions is a means to study single photodetachment of the negative ion, and the ionization dynamics of the second electron from the neutral atom are not considered in detail. More recently, a number of studies have been devoted to double detachment in an intense field (≥ 10 13 W•cm -2 ), where the absence of a long-range Coulomb potential and the lower binding energy compared to atoms are expected to significantly modify the dynamics of non-sequential processes [5][6][7][8][9][10].

The work reported here concerns double detachment of the 1s2s2p 4 P o metastable state of He -in moderately

σ α f L f M f (ǫ) = 4π 2 αa 2 0 ω 2L i + 1 Mi L,L ′ (2L + 1)(2L ′ + 1) ℓ f ,m f L f ℓ f L M f m f -M L f ℓ f L ′ M f m f -M × α f L f ℓ f L ′ M |D(ǫ)|α i L i M i * α f L f ℓ f LM |D(ǫ)|α i L i M i , (1)
which is derived from the general expression of the dipole matrix elements given by Burke [START_REF] Burke | R-Matrix Theory of Atomic Collisions[END_REF], and where α is the fine-structure constant, a 0 is the Bohr radius, ω is the photon angular frequency and D(ǫ) is the dipole length operator for a given polarization vector ǫ. In the "natural" reference frame of Tumaikin and Yudin [START_REF] Tumaikin | [END_REF]28], this is expressed as

ǫ = e 0 √ cos 2ε -e ±1 √ 2 sin ε (2)
where e 0,±1 are spherical unit vectors [START_REF] Edmonds | Angular Momentum in Quantum Me-1018 chanics[END_REF]

246

With the definition (2) for the polarization vector, the 247 dipole matrix elements appearing in (1) can be obtained

248 from the reduced matrix elements (α f L f ℓ f L||D||α i L i ) 249
using the Wigner-Eckart theorem [START_REF] Edmonds | Angular Momentum in Quantum Me-1018 chanics[END_REF]:

250 α f L f ℓ f LM | D(ǫ) |α i L i M i = (-1) -L-M L 1 L i -M 0 M i √ cos 2ε - L 1 L i -M ±1 M i √ 2 sin ε (α f L f ℓ f L||D||α i L i ), ( 3
)
where D is the tensor operator corresponding to D(ǫ).

251

Note that the reduced matrix element does not depend 252 on the polarization.

253

The cumulated photodetachment probability to a par-

P L f M f (t) = σ α f L f M f σ × 1 -e -t -∞ dT σφ(T ) , (4)
where φ(T ) is the instantaneous photon flux and the to-

H eff |ψ p = E |ψ p , (5)
where |ψ p is the model space wave function. Each eigen-

304

value can be written as E = E 0 + ∆ -i Γ 2 , the real part being the Stark-shifted energy of the field-dressed atomic state and Γ its total ionization width. The effective Hamiltonian valid up to the second order of perturbation theory is

H eff = P H 0 P + P V P + P S + Ω -i Γ 2 P ( 6
)
where V is the exact atom-field interaction operator and H 0 is the exact "free" Hamiltonian, containing the field-free atomic and light-field Hamiltonians. P is the Feshbach projection operator, projecting the wave function onto P-space states: i∈P |i i|. The operator S is a two-photon transition operator connecting the Pspace states via nonresonant bound states and the operators Ω and Γ represent two-photon couplings between model-space states through the ionization continua (see P1 for the detailed expression of these operators).

The eigenvalues of H 0 for bound states are

E i = E i -N ω
, where E i is the field-free energy of the bound state and N is the number of photons absorbed. Fieldfree energies for low-lying bound states (n ≤ 3) are taken from the NIST atomic database [START_REF] Kramida | NIST Atomic Spectra Database[END_REF]. The energies of Rydberg states are computed from their principal quantum number n and quantum defect δ nℓ . The latter is calculated using Ritz's expansion with coefficients given by Drake [START_REF] Drake | Springer Handbook of Atomic, Molecular, 1030 and Optical Physics[END_REF]. By convention, N = 0 for either of the 1s2s or 1s2p initial states, N = 1 for the 1s3s and 1s3d states where ǫ f is the photoelectron energy.

332

The one-and two-photon matrix elements (V ij , S ij ,

333

Ω ij and Γ ij ) coupling
356 |ψ p (t) = c 2s0 (t) |2s0 + 27 n=4 c ns0 (t) |ns0 + M c ndM (t) |ndM . (7)
The 1s orbital has been omitted in the |nℓM basis vectors for brevity. The total orbital angular momentum L has also been omitted and is equal to that of the outer electron (L = ℓ). The summation over the total magnetic quantum number M runs from -ℓ to +ℓ for each basis vector. For the 1s2p state, ionization proceeds through the (1+1+1) and (2+1) REMPI schemes shown in Fig. 1. The model space is thus spanned by one of the magnetic sublevels of the 1s2p state, defined by its magnetic quantum number M f , and the 1s3s, 1s3d, 1snp and 1snf states. The wave function is given by

|ψ p (t) = c 2pM f (t) |2pM f + c 3s0 (t) |3s0 + M c 3dM (t) |3dM + 45 n=6 [c npM (t) |npM + c nf M (t) |nf M ] . (8)
For both initial states, the Q-space is truncated to a ing the time-dependent Schrödinger equation [START_REF] Baker | [END_REF], whose 380 formal solution may be written as

381 |ψ p (t + ∆t) = e -iH eff (t)∆t |ψ p (t) . (9)
The wave function is iteratively propagated by computing numerically the matrix exponential e -iH∆t with the expokit computer package [38]. The time dependence of the effective Hamiltonian arises from the dependence of the couplings between model-space states on the intensity of the laser field, itself evolving in time as the atom travels through the laser focus. The wave function is propagated up to about t = 4 × 10 7 a.u. in steps of ∆t ≃ 10 4 a.u. The initial condition for propagation starting from the He(1s2s 3 S) state is

c 2s0 (0) 2 = P 2s0 (t → ∞), (10)
where P 2s,0 is the detachment probability, given by Eq. ( 4). All other coefficients are zero. Similarly, the initial condition for the He(1s2p 3 P o ) state is given by

c 2pM f (0) 2 = P 2pM f (t → ∞), (11)
and all other coefficients are zero. Such initial conditions imply that the photodetachment and ionization processes are sequential, and that the former occurs well before the latter, i.e. at much lower intensities. Due to the non-hermicity of the effective Hamiltonian, the norm of the wave function at the end of the propagation has decreased by an amount p. This quantity corresponds to the probability of ejecting 2 electrons from the He -ion via either the He(1s2s 3 S) state or one of the magnetic sublevels of the He(1s2p 3 P o ) state. The total ejection probability p tot is the sum over all channels. Note that for each photon energy, p tot is computed for a range of peak intensities and appropriate averaging over the ion beam section and integration over the pulse duration is subsequently performed in order to simulate experimental conditions (see paper P1). Simulated results are also multiplied by 0.56 to account for the MCP detection efficiency.

IV. RESULTS AND DISCUSSION

A. Single photodetachment

We have studied the one-photon, single detachment of He -under elliptical polarization and in the wavelength range from 685 nm to 730 nm (1.698 eV to 1.810 eV).

Branching ratios to He(1s2s 3 S) and He(1s2p 3 P o , M f = 0, ±1) final states were calculated using Eq. ( 1) with reduced dipole matrix elements obtained from the Rmatrix calculation described in the appendix. This calculation yielded an electron affinity of 75.5 meV, which compares favorably with the experimental value of 77.516 meV [39]. The total and partial cross sections are in good agreement with those of an earlier R-matrix calculation [11].

The branching ratios are in fact only weakly dependent on the wavelength. A representative set is shown The branching ratios to the various magnetic sublevels We note that in paper P1, a population distribution of (0.25, 0.5, 0.25) was assumed for the (-1, 0, 1) magnetic sublevels of the 1s2p state for linear polarization, as it provided the best fit to some of the experimental results.

The present calculation proves this assumption incorrect and, while it does not modify the conclusions drawn in P1, it changes to some extent the relative height of the 1snp and 1snf peaks in the simulated He + ion spectra.

The behavior of the branching ratios for magnetic sublevels is established here for photodetachment of He The evolution of double detachment as the polarization is tuned from linear to left circular is shown in Fig. 6.

Measurements were performed both for the 1s2s-1s9d and REMPI preferentially via a 1snf Rydberg state. In linear polarization, the magnetic sublevel of 1s2p with M f = 0 ionizes through the (1+1+1) scheme while those with M f = ±1, which cannot couple to the 1s3s state, preferentially do so through the (2+1) scheme.

In the detailed peak profiles presented in Fig. 8(a), the rightmost peak, corresponding to a 1snf resonance, exhibits the same spread towards higher photon energies observed for the He(1s2s 3 S) transient state, which can be attributed to the same dynamical processes. The leftmost peak, corresponding to 1snp resonances, spreads instead towards lower photon energies, indicating that the 1s2p state is shifted up in energy. The spread however changes direction towards higher photon energies below the 1s2p-1s3s resonance (see Fig. 7). This spreading and its reversal can be explained by the strong AC Stark shift of the 1s2p state due to the 1s3s state, which has opposite signs on either side of the resonance [18].

We next investigate the effect of polarization on the double detachment dynamics, and in particular on the respective contributions of the various magnetic sublevels 1s2p energy is smaller, the peaks will spread less towards 678 higher photon energies. However, above the 1s2p-1s3s resonance, the 1s2p state will be significantly shifted upward in energy due to its interaction with the 1s3s state.

The spread of the 1snp peak is therefore the result of a competition between the non-resonant and resonant AC Stark shifts, so that a smaller non-resonant contribution leads to an increased upward shift, and therefore an increased spread towards lower photon energies. In Fig. 8, the intensity of the 1snp peak is essentially the same for linear and circular polarization. Although the ionization rate of the 1snp states is 16% higher for circular polarization, this increase is compensated by the lower (19%) population of the M f = -1 sublevel for circular polarization compared to that of the M f = 0 sublevel for linear polarization.

Let us now consider the 1snf peak. and no evidence of non-sequential processes was found.

834

The study of double detachment of He -in more in- to assess the reliability of the reduced dipole matrix elements thus obtained.

In the R-matrix approach, configuration space is di- Since the initial He -anion has a 4 P o symmetry, after photodetachment the residual helium atom can only be left in a triplet state. The present calculation includes the five lowest triplet states of helium, whose wave functions were obtained using the CIV3 atomic structure computer code [44]. This code is based on a configuration interaction approach, in which the wavefunctions for a particular symmetry |LSπ are expressed as antisymmetric linear combinations of products of one-electron orbitals. In CIV3, the radial part of each orbital is written as a as a function of the photon wavelength. Full line: R-matrix calculation using the length form of the dipole matrix elements; dashed line: R-matrix calculation using the velocity form of the dipole matrix elements; broken line: reference [11]; circles: experimental results from reference [13].

In the R-matrix calculation, the inner region extends This compares favourably with the value of 77.518 meV obtained by a more extensive calculation [39], the experimental value of 77.516 meV [39], and is slightly better than that of an earlier R-matrix calculation [11], in which the computed electron affinity was then slightly adjusted to agree with the accurate value. While such small differences may be important close to threshold, they have little effect on the overall cross sections in the range of photon wavelengths (500-800 nm) considered in this study, and we do not perform such an adjustment here. As shown in figure 11, presenting the total cross section for photodetachment of He -, the agreement with the earlier work is very good [11].

FIG. 1 .

 1 FIG. 1. Schematic energy level diagram of He -and He. The dotted arrows show the pathway responsible for double detachment of He -via the 1s2s intermediate state and through (2+1) REMPI (Resonance Enhanced MultiPhoton Ionization) of the atom. The solid arrows show the pathway via the 1s2p state and through (1+1+1) and (2+1) REMPI of the atom.

  and the ellipticity angle ε can take the values -π/4 ≤ ε ≤ π/4. On 226 the right-hand side, the helicity of the vector e ±1 corre-227 sponds to the sign of ε. The polarization is linear when 228 ε = 0, left circular when ε = π/4 and right circular when 229 ε = -π/4. The choice of reference frame is arbitrary, and 230 a commonly used convention is to choose the quantiza-231 tion axis z along the electric field in the case of linear po-232 larization, and along the direction of light propagation in 233 the case of circular polarization. The natural frame pos-234 sesses the advantage of bridging these two conventions by 235 performing a continuous rotation of the reference frame 236 as the ellipticity angle evolves from 0 to ±π/4. The ori-237 entation of the reference frame in the laboratory frame 238 thus depends on the ellipticity angle, and it is identical 239 to standard conventions in the limiting cases of linear 240 and circular polarization. Any other frame would yield 241 identical results for the partial photodetachment cross 242 sections, keeping in mind that the M values are projec-243 tions of the angular momentum along the quantization 244 axis, and an appropriate rotation must therefore be per-245 formed in order to make meaningful comparisons.

329 and N = 2

 2 for the Rydberg states. The eigenvalues of 330 the continuum states of H 0 are, similarly, e = ǫ f -3ω, 331

  finite size and includes only the bound and continuum states involved in two-photon couplings between P-space states. It comprises bound states up to n = 70 and continuum states up to energies where bound-free couplings are negligible. For linear polarizaton, selection rules (∆M = 0) permit only those states with the same magnetic quantum number as the initial state to be populated. This in turn keeps the size of the effective Hamiltonian small, e.g. , 83 × 83 for the 1s2p initial state. In the general case of elliptic polarization, looser selection rules (∆M = 0, ±1) yield a significantly larger effective 374 Hamiltonian, with a size up to 409 × 409 for the 1s2p 375 state. 376 Once the effective Hamiltonian is constructed, it is 377 used to propagate the wave function along the atom's 378 trajectory through the laser pulse. This is done by solv-379

FIG. 3 .

 3 FIG.3. Branching ratios R2s,2p for the photodetachment of He -into the He(1s2s 3 S) and He(1s2p 3 P o ) states at λ = 690 nm. Full thin line: 1s2s state, dashed line: 1s2p state. Branching ratios to the various magnetic sublevels of the 1s2p state are also shown, and corresponding M f values are labeled on the right hand-side of the graph. Dotted line: M f = -1, full thick line: M f = 0, dash-dotted line: M f = +1.

  453

454

  of the 1s2p state depend on the choice of the reference 455 frame, since M f values represent the projection of the 456 orbital angular momentum onto the quantization axis z.457In the natural reference frame, the polarization vector 458 given by Eq. (2) rotates in the laboratory frame with 459 the ellipticity angle, so that magnetic quantum numbers 460 for different polarizations correspond to projections onto 461 quantization axes with different orientations. Such de-462 pendence however does not affect the final conclusions since all calculations to determine measurable quantities are performed within the same frame, and the final results are summed over all M f values, i.e. all orientations, before being compared to experiment. As noted in the previous section, for linear (ε = 0) and circular (ε = ±π/4) polarizations, the natural frame coincides with the standard choice of reference frame and comparisons with other data are straightforward. Moreover, calculations within other reference frames would yield results equivalent to the present ones and which can be compared to one another after appropriate rotation by means of Wigner D matrices [29].

FIG. 4 .FIG. 5 .

 45 FIG. 4. Experimental double detachment spectrum as a function of the photon energy. Top graph (a): linear polarization. Bottom graph (b): left circular polarization. Data shown is the number of He + ions produced by a 6 mJ laser pulse and for a He -beam of 1 nA .

604 2 .FIG. 7 .

 27 FIG. 7. Double detachment spectrum as a function of the photon energy. Top graph (a): linear polarization. Bottom graph (b): left circular polarization. The vertical dashed lines indicate the position of the 1s2p-1s3s resonance. Resonances with Rydberg states are observed up to n = 35. The laser pulse energy is 6 mJ.

FIG. 8 .

 8 FIG. 8. Details of the double detachment spectrum around the 1s2p-1s13p (left peak) and 1s2p-1s13f (right peak) resonances. Top graph (a): linear polarization; bottom graph (b): left circular polarization. Full circles: experimental data, dash-dot-dot line: simulated M f = -1 contribution, dash-dot line: simulated M f = 0 contribution, dotted line: simulated M f = +1 contribution. For linear polarization, M f = +1 and M f = -1 contributions are identical, therefore only twice the M f = -1 contribution is shown. All simulated contributions are scaled by 0.45.

  photodetachment in circular polarization. Such increases explain why the 1snf peak becomes larger and significantly broadens when switching from linear to circular polarization.Discrepancies between experimental and simulated yields, as observed in Fig.8, may be due to variations in the transmission of the He + ions to the detector, slightly imperfect circular polarization of the laser light, imperfect modeling of the experimental conditions and inaccuracies in the effective Hamiltonian matrix elements.Finally, the evolution of the double detachment yield as a function of the light polarization is shown in Fig.9.The experimental yield was measured for a photon energy of 1.767 eV, chosen to probe the minimum between the 1s2p-1s12f and 1s2p-1s13p resonances, where the influence of the polarization is most prominent. Simulations were also performed at the minimum, which in the calculations is located at a slightly lower photon energy (1.766 eV). The double detachment signal comes mainly from the 1s2p-1s13p resonance, with the contributions of the various magnetic sublevels evolving from predominantly M f = 0 to mostly M f = -1. The onset of the contribution from M f = +1 as the polarization becomes more circular is reminiscent of the broadening and increase in magnitude of the 1s2p-1snf peaks discussed above, with n = 12 here.

FIG. 9 . 3 .

 93 FIG.9. Experimental and simulated double detachment yield as a function of the ellipticity angle, at the photon energy corresponding to the minimum between the 1s2p-1s12f and 1s2p-1s13p resonances. Full circles: experimental yield, full line: simulated yield, dash-dot-dot line: M f = -1 contribution, dash-dot line: M f = 0 contribution, dotted line: M f = +1 contribution. The experimental yield is for a photon energy of 1.767 eV (701.72 nm) and the simulated one for 1.766 eV (702.05 nm). Simulated yields are scaled by 0.65 to match experimental values. The laser pulse energy is 6 mJ.

FIG. 10 . 8 ×

 108 FIG. 10. Double detachment spectrum around the 1s2p-1s3s resonance. Dot-dashed line: two-color, sequential measurement, multiplied by 35; full line: one-color measurement. The vertical dotted line indicates the exact position of the 1s2p-1s3s resonance. The laser pulse energy is 6 mJ.

  vided into two regions by a sphere encompassing the charge density of all states of the residual atom included in the calculation. Within this sphere, the states of the (N + 1)-electron system are represented by a discrete set of antisymmetrized basis functions built from linear combinations of the residual atomic states coupled with a set of continuum orbitals representing the ejected electron, supplemented by a number of bound or short-range correlation configurations. The (N + 1)-electron Hamiltonian is diagonalized in this basis, and the resulting eigenvalues and eigenvectors are used to construct the inverse logarithmic derivative matrix or R-matrix on the boundary of the inner region. In the outer region, the ejected electron moves far from the target while the other electrons remain bound. Exchange between the ejected and bound electrons can then be neglected, so that the wavefunctions for the full system can be represented by a standard close-coupling expansion involving products of the residual atomic states and a set of unknown functions representing the ejected electron. These unknown functions satisfy an infinite set of coupled second-order differential equations, with the appropriate asymptotic boundary conditions determining if the solutions represent a bound state of the initial anion or a continuum state of the atom plus ejected electron. The initial bound state and final continuum states are then determined by matching the solutions in the inner and outer regions at their common boundary. For the initial bound state, this matching can only be performed at discrete energies, which are found by an iterative search algorithm.

C

  jnℓ r I jnℓ exp(-ζ jnℓ r). (A.1) The coefficients C jnℓ are uniquely determined by or-910 thonormality constraints if k = n -ℓ, while the in-911 dices I jnℓ and the exponents ζ jnℓ are variational parame-912 ters chosen to minimize the energies of particular atomic 913 states. 914 The values of the parameters used in the current study 915 are given in table I. The 1s orbital is simply that for 916 the hydrogenic He + ion. The n = 2, 3 orbitals were all 917 optimized on the associated 1snℓ state of helium. The 918 pseudo-orbitals 4s and 5s were optimized on the 1s2s 3 S 919 state, 4d and 4f on the 1s2p 3 P o state. The 4p and 920 5p pseudo-orbitals were optimized on a linear combina-921 tion of the 1s2p 3 P o and 1s3p

out to 40 a 0

 0 , and 30 continuum orbitals per angular momentum ℓ are used to represent the ejected electron. The (N + 1)-electron Hamiltonian in the inner region is diagonalized for the initial 4 P o symmetry and the three final symmetries 4 S , 4 P, 4 D allowed by the dipole selection rules. In the outer region, imposing decaying boundary conditions on the solutions of the coupled second-order differential equations in the 4 P o symmetry yields an electron affinity of about 75.5 meV for the initial He -state.

  states of the model space are

334

expressed within the dipole and rotating-wave approx-335 imations

[START_REF] Shore | The Theory of Coherent Atomic Excitation[END_REF]

, justified by the moderate laser intensities 336 and the explicit treatment of only quasiresonant bound 337 states. Their calculation for all polarization states ǫ is 338 prohibitive, but can be greatly simplified, as for the pho-339 todetachment of He -, by virtue of the Wigner-Eckart 340 theorem. Reduced matrix elements, independent of ǫ, 341 are calculated only once using: (i) two-electron DVR ba-342 sis functions for couplings between low-lying states; (ii) 343 Quantum Defect wave functions for couplings amongst 344 Rydberg states and between Rydberg and continuum 345 states (see paper P1). Dipole matrix elements for any, 346 arbitrary ǫ are then readily calculated using Eq. (3) and 347 the effective Hamiltonian matrix is then constructed us-348 ing these elements. 349 We have built two effective Hamiltonians to describe 350 the REMPI of the 1s2s and 1s2p initial states respec-351 tively. For the 1s2s state, ionization proceeds via a 2-352 photon resonance with the Rydberg 1sns and 1snd states. 353 The model space includes the 1s2s, 1sns and 1snd states 354 with n in the range from 4 to 27, and the wave function 355 is expressed as

TABLE I .

 I Slater orbital parameters used to build the helium triplet state wavefunctions.

		C jnℓ I jnℓ	ζ jnℓ	C jnℓ I jnℓ	ζ jnℓ
	1s	5.65685 1 2.00000	2p 0.25282 2 0.54467
	2s	1.02331 1 1.57920	3p 0.14871 2 0.50810
		-0.33960 2 0.59932	-0.00928 3 0.31496
	3s	0.51628 1 1.53349	4p 5.69931 2 1.81486
		-0.18699 2 0.46220	-0.14792 3 0.88532
		0.02242 3 0.37501	0.00010 4 0.31256
	4s	3.72051 1 0.57067	5p 7.08316 2 2.03268
		-23.34422 2 1.97582	-1.97318 3 0.90825
		7.95138 3 1.78719	0.75867 4 1.01816
		-1.16998 4 1.23540	-0.00003 5 0.35646
	5s 11.03473 1 1.17185	3d 0.00904 3 0.33361
		-24.98834 2 1.09440
		15.71030 3 1.09465	4d 1.20358 3 1.35014
		-3.08663 4 0.98350	-0.00025 4 0.36349
		0.46320 5 1.03562
				4f 0.57024 4 1.43379

  3 P o states as it was found 922 that this gave the best energy separation between the two Total cross section for the photodetachment of He -

	923
	924	states.
	925	The energies and excitation thresholds thus obtained
		are presented in table II, where they are compared with

926

those of a more accurate calculation

[START_REF] Drake | Springer Handbook of Atomic, Molecular, 1030 and Optical Physics[END_REF]

and with the val-927 ues recommended by the National Institute of Standards 928 and Technology (NIST). Oscillator strengths and transi-929 tion probabilities are compared with the NIST values in 930 table III. 931 FIG. 11.

TABLE II .

 II Energies and excitation thresholds for the five lowest triplet states of helium. The theoretical values are compared with the accurate, non-relativistic energies taken from chapter 11 of[START_REF] Drake | Springer Handbook of Atomic, Molecular, 1030 and Optical Physics[END_REF] and the thresholds recommended by the National Institute of Standards and Technology (NIST)[START_REF] Kramida | NIST Atomic Spectra Database[END_REF].

			Present (au)	Accurate (au)	Present (eV)	NIST (eV)
	1s2s	2 3 S	-2.17513	-2.17523	0.0	0.0
	1s2p	2 3 P o	-2.13294	-2.13316	1.14805	1.14449
	1s3s	3 3 S	-2.06866	-2.06869	2.89720	2.89885
	1s3p	3 3 P o	-2.05798	-2.05808	3.18754	3.18746
	1s3d	3 3 D	-2.05562	-2.05564	3.25203	3.25404

TABLE III .

 III Oscillator strengths f and rates A in length (L) and velocity (V) forms for dipole allowed transitions involving the five lowest triplet states of helium, compared with the values recommended by NIST[START_REF] Kramida | NIST Atomic Spectra Database[END_REF]. The figures in parentheses are the powers of ten by which the preceeding number must be multiplied.

	Transitions		f		A (sec -1 )
			Present	NIST	Present	NIST
	1s2s 3 S -1s2p 3 P o	L	0.5421	0.5394	0.1033 (8)	0.1022 (8)
		V	0.5320		0.1014 (8)
	1s2s 3 S -1s3p 3 P o	L	0.5744 (-1)	0.6448 (-1)	0.8442 (7)	0.9475 (7)
		V	0.6378 (-1)		0.9373 (7)
	1s2p 3 P o -1s3s 3 S	L	0.7019 (-1)	0.6951 (-1)	0.2795 (8)	0.2785 (8)
		V	0.6786 (-1)		0.2703 (8)
	1s2p 3 P o -1s3d 3 D	L	0.6161	0.6102	0.7100 (8)	0.7070 (8)
		V	0.6085		0.7012 (8)
	1s3s 3 S -1s3p 3 P o	L	0.9126	0.8914	0.1114 (7)	0.1074 (7)
		V	0.8693		0.1061 (7)
	1s3p 3 P o -1s3d 3 D	L	0.1095	0.1120	0.1179 (5)	0.1292 (5)
		V	0.1055		0.1136 (5)


Christophe Dano
email: christophe.dano@ec-nantes.fr

Pierre-Yves Hicher

IDENTIFICATION OF THE PARAMETERS OF AN ELASTO-PLASTIC MODEL WITH STRAIN-SOFTENING BY INVERSE ANALYSIS OF PRESSUREMETER TESTS

Identification of the parameters of an elasto-plastic model with strain-softening by inverse analysis of pressuremeter tests

Christophe Dano, Pierre-Yves

-INTRODUCTION

The pressuremeter test is an in-place test widely used in soil mechanics in order to assess soil properties [START_REF] Baguelin | The pressuremeter and foundation engineering[END_REF][START_REF] Cassan | Essais in situ en mécanique des sols -Tome I : réalisation et interprétation[END_REF][START_REF] Wroth | The interpretation of in situ test[END_REF]. The pressuremeter device (Fig. 1) is composed of a radially expandable cylindrical probe either inserted into a pre-bored hole (Ménard pressuremeter test) or directly driven into the soil (Self-boring pressuremeter test) at the desired depth. Application of water pressure increments, monitored from the pressure -volume control unit located at the surface, causes the probe to be inflated. A standard procedure (French Standard NF P 94-110 or US Standard ASTM D 4719-87) then allows to deduce an experimental stress -deformation curve, named S exp . Its shape (Fig. 1) depends on the mechanical properties of the soil in which the probe has been inserted.

A theoretical pressuremeter curve S th can also be obtained by solving the mechanical boundary problem posed by the pressuremeter test, either analytically or by finite elements computations, depending on the degree of complexity of the constitutive model adopted for representing the behaviour of the tested soil. Then, it becomes possible to identify the parameters of the constitutive model by inverse analysis. The purpose of this mathematical tool is to fit the estimates of the theoretical pressuremeter curve S th to field data S exp by iterative computations on the values of the parameters of the constitutive model.

In this paper, we analyse some pressuremeter tests carried out in uncemented soils. The soil behaviour is represented using the linear isotropic elastic plastic model shown in Figure 2. The elastic part is defined by the elastic modulus E and the Poisson's ratio ν. The shear modulus G is equal to E/(2(1+ν)). The Mohr-Coulomb failure criterion (Eq. 1), characterised by the friction angle ϕ', determines the value of the peak strength :

( ) ( ) ( ) ( ) 0 ' sin k k , F r p d r = σ + σ × ϕ × ε + σ - σ = σ θ θ (Eq. 1)
where σ r and σ θ are respectively the radial stress and the orthoradial stress. Volumetric strains in the plastic domain are calculated assuming a non-standard flow rule (Eq. 2) :

( ) ( ) ( ) ( ) r r sin g G σ + σ × ψ χ + σ - σ = σ θ θ (Eq. 2)
where ψ is the dilation angle and χ a value between 0 and 1. ( [START_REF] Carter | Analysis of cylindrical cavity expansion in a strain weakening material[END_REF] have noted that the post-peak behaviour has a great effect on the stress distribution around the cavity, that is why we consider a post-peak strain softening law that depends on the deviatoric plastic strain ε d p such as :

( )

p d p d 1 k ε × α + = ε (Eq. 3) p p r p d θ ε - ε = ε (Eq. 4)
where α represents the strain softening rate. The function k is equal to 1 at the peak strength and is set to the value β < 1 at the critical state at which the strength remains constant. We model therefore a gradual decrease of the friction angle between the peak and the critical state. The semi-analytical expression of the pressuremeter curve developed from this constitutive model is then compared to relationships found in the literature [START_REF] Hughes | Pressuremeter tests in sands[END_REF][START_REF] Yu | Finite cavity expansion in dilatant soils : loading analysis[END_REF][START_REF] Monnet | Etude théorique de l'équilibre élasto-plastique d'un sol pulvérulent autour du pressiomètre[END_REF]. We show the effect of the different assumptions on the values of the friction angle obtained by optimisation computations.

-EXPRESSIONS OF THE PRESSUREMETER CURVE

-Resolution of the mechanical problem

The pressuremeter test is formally a problem of expansion of a cylindrical cavity, with an initial radius a, into an infinite medium (Fig. 2). We assume that : -the initial stress state is isotropic and is defined by the earth pressure at rest p 0 ; -the medium is homogeneous and isotropic ; -the problem is axisymmetric ; -the vertical stress is always the intermediate principal stress.

The assumption of plane strain is also adopted. It is justified by the use of guard cells placed on both sides of the probe. Finally, the elastic strains ε e and the plastic strains ε p are linked to the total strain ε by the partition rule (Eq. 5) :

p e ε + ε = ε
(Eq. 5) To determine the relation between the deformation u(a)/a and the pressure p applied by the probe at the cavity wall, we have to combine equations of the constitutive model, equilibrium equations in the horizontal plane (Eq. 6), compatibility equations (Eq. 7) that link the strains to the displacements, initial and boundary conditions (Eq. 8).

( ) ( ) ( ) 0 r r r dr r d r r = σ - σ - σ θ (Eq. 6) ( ) ( ) dr r du r r = ε ( ) ( ) r r u r = ε θ (Eq. 7) ( ) p a r r - = = σ ( ) 0 r p r - = ∞ → σ (Eq. 8)
The following notations are also adopted :

ψ χ + ψ χ - = χ sin 1 sin 1 n 1 1 n n = χ = G 2 ' sin p c k 0 2 3 ϕ = ' sin 1 ' sin 1 k p ϕ + ϕ - = 2 3 10 c k 2 1 k α + = 2 3 11 c k 2 k α - = ( ) [ ] n 1 1 1 11 lim lim k 1 1 b c R + - - β + =       = ( ) ( )( ) ' sin k 1 n 1 2 n 1 11 10 10 1 1 ' sin 1 X ' sin k ' sin k 1 X f ϕ + + +       ϕ + ϕ + ϕ + =

-Pressuremeter curve equation using the strain softening model

Initial pressure increments are small enough to cause small and reversible strains. We can therefore consider that the soil behaves elastically as long as p is less than p pl such as :

( )

' sin 1 p p 0 pl ϕ + × =
(Eq. 9)

The equation of the pressuremeter curve in the elastic domain is then :

( ) G 2 p p a a u 0 - = (Eq. 10)
As soon as p reaches the value p pl , a plastic annulus develops concentrically around the borehole. The soil between the radius a and the radius c (Fig. 2) undergoes strain softening. Then :

( ) p a c a c f ' sin 1 p ' sin k 1 ' sin k 2 0 10 10 =       ×       × ϕ + ϕ + ϕ (Eq. 11a) and ( ) 1 n 1 2 3 1 2 3 1 1 a c c k n 1 2 c k 1 n 1 n a a u +       + + + - = (Eq. 11b)
Equation ( 11a) is a polynomial function of the variable (c/a) raised to two distinct powers that depend on the values of the friction angle and the dilation angle.

The determination of this variable requires to calculate the deformation at the cavity wall (eq. 11b). It can be done by optimisation using a Gauss-Newton algorithm.

These equations (11a,b) are suitable as long as the pressure p into the probe is less than p pa for which the soil reaches the critical state. The pressure p pa mainly depends on the parameters α and β :

( ) ( ) ' sin k 1 ' sin k 2 lim lim pl pa 10 10 R R f p p ϕ + ϕ × × =
(Eq. 12)

When p > p pa , a third domain (a < r < b) coexists with the two previous zones (Fig. 2) : the elastic zone (r > c) and the strain softening zone (b < r < c). Assuming that the area of the strain softening zone is set when the critical state is reached, that is to say that the ratio (c/b) is set to the value R lim , the pressuremeter curve (Eq. 13) can be explicitly calculated :

( ) ( ) ( ) ( ) χ χ + χ χ θ + - + +                   + - + - ε = χ n 1 c k 1 n k a b n 1 c k 1 n k b a a u 2 3 1 n 1 2 3 1 (Eq. 13) with ( ) ( ) 1 n 1 lim 2 3 1 2 3 1 1 R c k n 1 2 c k 1 n 1 n b + θ + + + - = ε and ( ) ( ) ( ) ( )( ) ( ) 2 10 10 ' sin k 4 ' sin 1 ' sin k 1 lim ' sin 2 ' sin 1 lim 0 R R f ' sin 1 p p a b ϕ β ϕ β + ϕ + ϕ β ϕ β + ×       × ϕ + =

-Pressuremeter curve equation using the small strain assumption

The previous developments constitute a generalisation of the pressuremeter curves proposed in the literature. Indeed, if we assume : -a linear elastic perfectly plastic constitutive model ; -a small strain hypothesis ; -α = 0, β = 1, then p pl = p pa and (Eq. 13) becomes :

( ) ( ) ( )           ψ -         × ψ + ϕ = ψ + ϕ ϕ + sin p p sin 1 G 2 ' sin p a a u sin 1 ' sin ' sin 1 pl 0 (Eq. 14)
Equation ( 14) is similar to the relation proposed by [START_REF] Monnet | Etude théorique de l'équilibre élasto-plastique d'un sol pulvérulent autour du pressiomètre[END_REF] if the elastic strain is set to a constant value in the plastic domain. On the contrary, if the elastic strain is supposed negligible in the plastic domain, we obtain the relation proposed by [START_REF] Hughes | Pressuremeter tests in sands[END_REF] :

( ) ( )                   ϕ = ψ + ϕ ϕ + sin 1 ' sin ' sin 1 pl 0 p p G 2 ' sin p a a u (Eq. 15)
The equation ( 15) leads to a stiffer pressuremeter curve than the equation ( 14).

2.4 -Pressuremeter curve equation using the large strain assumption [START_REF] Yu | Finite cavity expansion in dilatant soils : loading analysis[END_REF]) have put the validity of the small strain hypothesis into question since it tends to give too high values of the limit pressure. With the linear elastic perfectly plastic model, they have developed an expression of the pressuremeter curve, using a large strain hypothesis and new definition of the compatibility rules (Eq. 16) :

( )         = ε 0 r dr dr Ln r ( )         = ε θ 0 r r Ln r
(Eq. 16) [START_REF] Yu | Finite cavity expansion in dilatant soils : loading analysis[END_REF] then stated that :

( ) ( ) ( ) 1 p ' sin 1 p a a u ' sin 1 2 ' sin 0 sin 1 2 -       ϕ + × Γ = ϕ + ϕ ψ + (Eq. 17)
where Γ is a function of the parameters of the constitutive model. Γ requires the calculation of a serie whose first five terms provide a good estimation.

-OPTIMISATION PROCEDURE

-Minimisation algorithm

The different equations of the pressuremeter curve (Eqs. 10, 11, 13, 14, 15, 17) have been implemented in a commercial software that also provides an optimisation algorithm, based on either the Gauss-Newton algorithm or the conjugate gradient method. The Gauss-Newton algorithm minimises the difference, raised to the power 2, between the experimental pressuremeter data S exp (p i ) and the theoretical one S th (p i , J) where p i is the pressure into the probe and J is the set of parameters :

( ) ( ) [ ] ∑ - = N 1 2 i th i exp J , p S p S L ( ) ( ) k k f J L Inf J L ℜ ∈ = (Eq. 18) ( ) ( )       ∂ ∂       ∂ ∂ + = - + k 1 k 2 2 k 1 k J J L J J L J J
However, the simultaneous optimisation of all the parameters by the Gauss-Newton algorithm leads to divergent computations or erroneous estimates because of the identical weight assigned to the parameters. This problem is emphasised by the coupling that links some of the parameters, the friction angle and the dilation angle for instance.

Therefore, in order to reduce the number of parameters that have to be simultaneously optimised, we establish an identification procedure based on a sensitivity study and on laboratory experiments.

-Identification procedure

The linear elastic perfectly plastic model is defined by 4 parameters : the earth pressure at rest p 0 , the shear modulus G, the friction angle ϕ' and the dilation angle ψ.

The value of the earth pressure at rest measured at the beginning of the pseudo-elastic stage of the pressuremeter curve, often integrates experimental errors (removal of the load during sounding, partial collapse of the borehole, …). So, without reliable geotechnical investigations or information about the historical record of the site to estimate the earth pressure, we calculate p 0 from the relation :

' K p v 0 0 σ = (Eq. 19)
where K 0 is the coefficient of earth pressure and σ v ' the vertical effective stress at the depth z at which the pressuremeter test is carried out (σ' v = γ h .z -u where γ h is the unit weight of the soil and u the pore pressure). The unknown disturbance due to the boring stage also affects the beginning of the pressuremeter curve. Consequently, the optimisation procedure is applied on the portion of the curve which corresponds to u(a)/a > 1,5 %, as proposed by [START_REF] Cambou | Utilisation de l'essai pressiométrique pour l'identification de paramètres intrinsèques du comportement d'un sol[END_REF]).

The shear modulus G is determined on an unload / reload cycle on which the effect of the initial disturbance is negligible :

( ) cycle cyc a a u 2 p G       ∆ ∆ = (Eq. 20)
But this value G cyc has to be corrected since the elastic properties of soils are highly dependent of the stress and strain states. [START_REF] Bellotti | Interpretation of moduli from self-boring pressuremeter tests in sands[END_REF] proposed the relation :

( ) 5 , 0 0 c 0 0 cyc ' p ' p 2 , 0 ' p ' p G G         - × + × = (Eq. 21)
where p c ' is the pressure into the probe at the time of unloading.

For natural uncemented sands, the dilation angle ψ is correlated to the friction angle ϕ' by : °-ϕ = ψ 30 ' (Eq. 22)

Consequently, only the friction angle ϕ' has to be optimised when considering the elastic perfectly plastic model.

The strain softening constitutive model adds 3 parameters : the softening rate α, the decrease of the friction angle β and the dilatancy characteristic in the critical state χ. The parameter α is optimised simultaneously with the friction angle ϕ'. The value of χ is set to 0 in the critical state since it is characterised by a constant volume deformation in the case of uncemented sands. Finally, the parameter β is the ratio between sin(ϕ cv ) and sin(ϕ') where ϕ cv and ϕ' are respectively the critical state friction angle and the peak friction angle. The two angles are highly correlated. The parameter β mainly depends on the density of the soil and on the mean stress. [START_REF] Bolton | The strength and dilatancy of sands[END_REF] proposed : ( )

' sin 8 , 0 ' sin ϕ ψ - ϕ = β (Eq. 23)
In order to avoid convergence towards values which do not present any physical meaning, we also add initial constraints on the parameters. For instance, the friction angle must be comprised between 20 and 55°, as it is the case for most of the uncemented soils.

The optimisation computation involves a change in the value of the friction angle, and, consequently, a change in the values of the domain bounds p pl and p pa . Then, we have to finally control that the expression of the pressuremeter curve used in the optimisation process corresponds to the pressure level into the probe.

-APPLICATIONS

In order to show the effect of the strain softening and the effect of the definition of the compatibility rules on the value of the friction angle determined by optimisation, we present two pressuremeter tests, named SP1-8 and SP2-8. They were carried out in a clean fine sand formation at 8 meters depth (σ v ' = 128 kPa, K 0 = 1).

The value of the shear modulus G, after being corrected (Eq. 21), is 33 MPa. The dilation angle is correlated to the friction angle according to (Eq. 22). χ is set to 1 in the strain softening domain and to 0 in the critical state. The friction angle at critical state is 33°, then β is close to 0,8.

According to the procedure previously described, computations with the large strain hypothesis model by [START_REF] Yu | Finite cavity expansion in dilatant soils : loading analysis[END_REF]) (Eq. 17) lead to a value of the friction angle equal to 39,7° for the test SP1-8 and 51,0° for the test SP2-8. These values enclose the value of the friction angle ϕ' triax (ϕ' triax = 43°) obtained by means of laboratory triaxial tests if we also take into account the difference due to the stress path. The field data and the theoretical pressuremeter curves are compared in Figure (3a).

Considering the elastic perfectly plastic model with the small strain assumption (Eq. 14), the optimised value of the friction angle is respectively 34,1° for the test SP1-8 and 41,0° for the test SP2-8.

Finally, with our strain-softening model, the pair of optimised values (ϕ', α) is respectively (ϕ' = 43°, α = 30) for the test SP1-8 and (ϕ' = 43°, α = 1) for the test SP2-8. The value α = 30 means that the decrease of the strength is relatively fast after the peak whereas the value α = 1 corresponds to a slow decrease from the peak to the critical state.

The difference between the two tests could be attributed to different site conditions (local cementation between grains, different shear moduli, …). In particular, the determination of the earth pressure at rest is an usual problem in geomechanics. If we assume, for example, that K 0 = 0,5 instead of K 0 = 1, optimisation computations lead to a value of the friction angle equal to 48° for the test SP1-8 and the elastic perfectly plastic model using a large strain hypothesis (Eq. 17). We note that the result of the computations is very sensitive to the value of the earth pressure at rest that also depends on the quality of the borehole.

In all cases, the theoretical pressuremeter curves accurately fit the field data, as shown in Figure (3b) for the test SP2-8.

-CONCLUSIONS

We present a semi-analytical expression of the pressuremeter curve assuming a linear elastic plastic model with a post-peak strain softening. We confront this model with the expressions of the pressuremeter curve obtained considering a linear elastic perfectly plastic model, with a small strain hypothesis or a large strain one. These expressions are then implemented in a commercial software that also provides an optimisation tool.

Using the procedure developed to identify the friction angle of uncemented sands from field tests, we show the great effect of the strain softening and of the definition of the compatibility rules (small or large strain hypothesis) on the values of the friction angle determined by optimisation. The strain softening model can also provide information about the state of the soil in which the pressuremeter tests are carried out. But, the analytical resolution of the mechanical problem do not allow the simultaneous identification of many parameters. Further developments will be necessary to achieve this goal.

Figure 1 .

 1 Figure 1. The pressuremeter device and the resulting pressuremeter curve.

  Fig. 2. The constitutive model and the corresponding zone into the soil.

  Fig.3a

Figure 3 .

 3 Figure 3. Comparison of field data and computations.

Table 1 .

 1 Recapitulative table.