Eva Toulouze

Vasili Lytkin and the latinisation of Komi

The so-called episode of the "latinisation" in Russia in the 1930ies is a curious one and deserves to be examined in detail. It played a certain role in the beginning and the development of a written culture in different languages of the URSS and the sequence of the events connected with it illustrates most eloquently the tragic unwinding of the Soviet tale in the two decades between the Revolution and the so-called Great Patriotic War. Vasily Lytkin was one of the main actors of this story in the Komi territory and the aim of this article is to unfold and comment it. Still, in order to understand what was at stake, I shall start by two introductory chapters: one about the general context of latinisation 1 and the other about the peculiar situation in Komi. Only when these two contextual questions have been reviewed, we may be able to analyse Lytkin's role in it and his motivations.

Latinisation in Russia

The adoption of Latin alphabet for a large amount of the languages spoken in the Soviet Union is a process that covers almost two decades. In the eventual triumph of the Russian alphabet that happened in 1936-1940 and has been lately crowned by the decision of the State Duma in 2002 of denying the languages of Russia the right to be written in any other alphabet than the Cyrillic2 , one can symbolically recognise the victory of the hegemonic principles ruling Russian politics.

When the Bolshevik revolution could assert its rights on the territory of the former Russian empire, the languages spoken in this huge territory were written with different alphabets -Cyrillic for the Slavic languages, Latin for example in Carelia (but most of the languages written with the Latin alphabet were spoken in territories lost by Soviet Russia), Arabic for the Turkic languages as well as specific alphabets for Georgian and Armenian and others, as the Mongol and the Chinese circulating in Asia. This diversity in itself was not seen by the first Bolshevik leaders as a problem, but in their attempt to obtain control of the territory with its diverse population, they were mainly concerned by the use of the Arabic alphabet.

The Arabic alphabet presented different peculiarities that could displease the Soviet authorities both from the symbolic and from the practical side. From the symbolic point of view, the Arabic alphabet was of religious origin and thus it connected its users with a world the Bolshevik strived to eliminate. From the political point of view, the Arabic alphabet had its centre far from Russia and from the revolutionary system -it was mainly connected with the Arab world and with Mecca. This could represent a danger in a Soviet Russia harassed by solitude in an environment that did not follow its revolutionary example. From the practical point of view, the use of the Arabic alphabet for Turkic languages was doubly exposed to criticism: first of all, from a non-political, merely linguistic point of view, its peculiarity of not noting vowels could not be fit to languages in which the vowel system is so developed and important for distinguishing meanings. Therefore, and this was a political issue in Soviet Russia, the use of the Arabic alphabet made literacy a privilege of the elite and hindered mass schooling (Zak, Isaev 1966: 5-6), while education and literacy were vital for the Bolshevik in order to achieve their goals: they were among the main instruments used by the new rulers to strengthen their grip on the country3 . Therefore, quite early (in 1922), Lenin proclaimed that "latinisation was Revolution for the East" and the fate of the Arabic alphabet in Russia was decided: it was to be replaced by the Latin alphabet. This orientation was adopted in the 1920s, i. e. much earlier than in Turkey4 .

Undoubtedly, Atatürk was inspired in his reforms by the example of Turkey's great Russian neighbour. But how can we explain the choice of the Latin alphabet, which was to be so thoroughly criticised later? The legacy, the time and the revolutionary leadership explain it convincingly. The legacy: for the revolutionary leaders, whose policy was supposed to be radically different from the tsarist regime's, it wasn't acceptable to promote the use of the Russian alphabet as an alternative, as the memory of the missionaries' endeavours was still fresh. Nikolay Il'minsky, since the 1860s, had forced a Cyrillic transcription of Tatar, based on the language of the countryside people, and more fit than the Arabic transcription to the language features (Ilminsky 1870:13). Cyrillic was the symbol of evangelisation, and evangelisation meant Russification. The time: Lenin pronounced his famous sentence in a period in which there was urgency and in which the first priority was to convince the nationalities to support the Bolshevik. The beginning of the 1920s is not a time for russification: "Russian chauvinism" is criticised and often Moscow supports claims by ethnic groups against Russian endeavours5 . The political nature of the choice of the Latin alphabet is clearly expressed in the resolution of a commission dedicated to the question: as both alphabets were fit upon linguistic criteria, the choice was political and fell on the Latin (Ubriatova 1977: 113). Moreover, the Bolshevik leaders in the first years of the Revolution are still very much the small group of intellectual that had been active before 1917: many had lived abroad in exile. For them, unlike for their followers, the Latin alphabet was quite familiar.

The movement started with the initiative of the Azeri, who were the first to abandon the Arabic alphabet and to latinise their written language in 1921. The Yakuts had also started a parallel movement as early as 1920 (Zak, Isaev 1966: 6). They were followed by several other people, among which the Bachkirs in 1926(Iskhakov 1934). The last to accept this trend were the Tatars. The Tatars had a national elite, which had been shaped by Muslim culture and the intellectual sphere was thoroughly arabised6 . They resisted longer than others and the intellectual debate about the way of writing the language was fierce. It was on old debate in the Tatar intellectual sphere. But eventually, in 1929, the Tatars were compelled, as the others, to write their language in the Latin alphabet (see [START_REF] Seid-Zade | [END_REF].

Still this process was not completely spontaneous. The first period was characterised by separate attempts (Grande 1934: 21-23). But there was a need for coordination and there were decisions to take about the transcription of different phonemes in the new alphabet.

The base was Latin, but it had to be adapted. After the Conference of Turkic languages speaking peoples called in 1926, this bacame the responsibility of the New Turkic Alphabet Committee (KHTA, Комитет Нового Tюркского Aлфавита). The Turkic languages being quite close to one another, the Committee's task was not impossible to achieve: the phoneme systems allowed a great regularity in transcription based on the principle "one phoneme, one grapheme" (Grande 1934: 14-15, 19-21). It was to be a test, on a simple sample, of what would be generalised later.

The next stage was the generalisation of the experience acquired by the Turkic languages.

The question arose in 1929, when the decision was officialised to provide several nonwritten languages with a written norm. The implementation of this decision required a huge amount of preliminary work: languages that had not been thoroughly studied previously had to be analysed from all points of view, a dialectal base had to be chosen and, last but not least, an alphabet had to be selected. The late experience of the Turkic languages provided a solution that was most coherent with the Bolshevik way of thinking. In 1930 the KTA was transformed into a Committee of the New Alphabet (KНA, Комитет Нового Алфавита), and started elaborating an alphabet that would be used for writing all the newly planned languages, the languages of small peoples of Siberia (Zak, Isaev 1966: 9). The project grew: it appealed very much to the Soviet aspiration to logical standardisation and to rational human interference. The idea was to create a kind of bank of phonemes/graphemes that would be fit for all the languages existing in the socialist space. For the same phoneme, every language would use the same grapheme. The graphemes corresponding to non existing phonemes in a given language would only be skipped in the alphabet of that language. The systematic approach was emphasised by the main linguist involved: Professor N.F. Yakovlev proposed in 1928 a mathematic formula for building an alphabet [START_REF] Yakovlev | [END_REF]. This project required enormous efforts aimed at identifying the phonological system of each language in order to apply the needed graphemes allowing the creation of a coherent alphabet. More visibly than this requiring scientific work, the Committee of the New Alphabet developed and ideology and a global project. The ideology was that the use of the Latin alphabet in the Western countries was chaotic; it served the interests of the local bourgeoisies and left the simple people in ignorance, because of the unsteady valour of each letter. In the socialist framework, this alphabet, the one in which Marx and Engels had written their works, would serve the interests of the working class and of universal literacy.

The project of having all the languages of the USSR passing to the Latin alphabet implied that Russian, and other languages in which the Cyrillic alphabet was well rooted, were to change.

The peak was reached in 1933, when most of the languages were written in Latin letters, as it was the official orientation of the authorities. Still, even in that period, the unification was not totally achieved. There were several graphemes that represented different phonemes in different languages (Grande 1934:128-129). In 1934, we already see a tendency to seek "errors" in the field of the "latinisers" (BSE 1934), and two years later the course is entirely changed and all the previously latinised languages were undergoing a new transformationgoing over to Cyrillic alphabet, under its Russian form.

Komi written culture in the 1920s

The Komi language is among those languages in which the Cyrillic tradition was quite well rooted. Notwithstanding an older original tradition -St Stephen of Perm's invented alphabet called abur -komi had been written in a scarce amount of documents since the 17 th century with the help of the Cyrillic alphabet (Lytkin 1952: 50-59)7 . In the early 19 th century, it was in a Cyrillic notation that the first literary texts were composed by a generation of literate young men, among whom the most famous is the poet Ivan Kuratov -who wrote for himself poems that have become classics. It is also in this alphabet that Georgi Lytkin wrote both his translations into Komi and his original work -Зырнский край при епископaх пермских и зырнский язык. Пособие при изучении зырянами русскаго языка -written in 1889. So the tradition existed. But the existing transcriptions were not entirely satisfactory and in 1920 a linguist, Vladimir Molodcov (1886Molodcov ( -1940)), proposed a strictly phonemic alphabet, conceived for Komi, which was adopted. This is a unique case in the Soviet Union: the process launched by the Revolution concerning the emancipation of nationalities led only in Komi to the creation of an alphabet peculiar to a particular language. The Molodcov alphabet was a very rational alphabet for the Komi language; it had been very carefully conceived. It was composed of letters from the Cyrillic alphabet8 , some with additions on Molodcov's own. It was implemented in 1920, and all printed materials issued in Komi during the decade and later in the second half of the 1930s used this alphabet.

The question of going over to the latin alphabet had been discussed in 1923-1924, even before it became a political trend all over the Soviet Union. Its followers considered the Molodcov's alphabet was narrowly nationalist, relying on its uniqueness (Oboturov 1934: 191). It had been rejected and the Molodcov alphabet had been confirmed. Some years later, however, the generalisation of the Latin alphabet for all new written languages in the country and the processes mentioned above led to a new debate in the Komi oblast' about the better way of writing the language. This discussion between the followers of the Molodcov alphabet and those who supported the Latin alphabet was concluded by the victory of the Latin alphabet, which was then officially supported by the Soviet authorities, both in Syktyvkar and in Moscow. A linguistic conference took place in 1929 and not even Molodcov expressed further opposition to the adoption of the Latin alphabet (Popov 1989: 59); it was followed by a resolution by the Party and the State organs confirming the decision (Baraksanov 1963: 180). Actually there was no difference in the conception of both alphabets: the Latin version was as phonemic as the previous one, only the form of the letters was changed. It was implemented in 1930, and for two years, it coexisted with the Molodcov alphabet (Baraksanov 1993: 179).

The experience lasted 6 years, but it seems that the first critical reports started earlier than in other regions: already at the end of 1934 reports in State organs draw the conclusion that it would be reasonable to go back to the Molodcov's alphabet (Popov 1989: 57). The assessment was more and more critical, until the Latin alphabet was condemned, as elsewhere in the Soviet Union: it was accused of driving apart the Komi language from Russian (Baraksanov 1963: 181). In 1936, the Komi went back to the Molodcov alphabet and resisted for two years the pressure to adopt the Cyrillic one, which came into being in 1938 and is still in use. It is interesting to observe that the accusation of nationalism, which had been already produced in connection with the Molodcov's alphabet, was also issued against the adepts of the Latin alphabet, as Lytkin. Any choice of anything else than the Russian system appeared, in the times of repression, as nationalistic…

Vasily Lytkin and latinisation

Vasily Lytkin (1885-1981) is a key figure in Komi intellectual picture covering the 20 th century. He has been a witness and an actor of the main changes in that period, from the Revolution until the Brezhnev era, being himself a victim in the Stalinist. He had enjoyed an enviable privilege: after having achieved his studies in Moscow, he was the only scholar among Russia's Finno-Ugrians, who was allowed to pursue Finno-Ugric studies outside the Soviet Union, in Finland and Hungary (Jelsova 1994: 15). And besides being a scholar, he was also a well-known poet under the name of Illya Vas'. But this aspect of his activity is beyond the scope of this article.

Vasily Lytkin had thus a wider experience of the world than most of his colleagues. He had already widened his perspectives during the Civil War: he has been a soldier first in the tsarist army, and later as a volunteer in the Red Army in the North, in the Urals, on the Polish front; he was even made prisoner in Prussia (Oni ljubili 1993: 139;Baraksanov, Zherebcov 1994: 63, 65;Turkin 1995a: 208).

The years he spent in Moscow studying linguistics were mightily useful for his subsequent work as a Finno-Ugrist. Although he could not study Finno-Ugristics, for this discipline did not exist as such, he studied Russian linguistics and dialectology and his professors took advantage of his interest and his knowledge of Komi for orienting him towards comparative linguistics. They also supported him in his desire to pursue his studies and in 1925 he was admitted as a doctoral student at the Institute of the Oriental peoples9 , whose director was the later famous academic Marr and the vice-rector a linguist (not indifferent to our purpose), N.F. Yakovlev (Turkin 1995a: 209). He was supported and got the scholarship that allowed him to spend two years, one in Finland and one in Hungary, and to visit bot Germany and Estonia. He returned home in 1929 (Turkin 1995a: 209, Turkin 1995b: 291). He took good advantage of this exceptional possibility. In Finland, in 1926, he attended lectures about basic Finno-Ugristics and about Finno-Ugric cultures (Turkin 1997: 22-25); He took advantage of it to visit Estonia during 10 days (Turkin 1995: 210, Turkin 1997: 23-24). In Hungary, he was awarded the title of Doctor for his work about the Permian suffixes (1927), he learned Hungarian and discovered the Hungarian school of Finno-Ugristics (Oni ljubili 1994: 145, Turkin 1995c: 211n, Turkin 1995a: 212, Turkin 1997: 25).

The idea of latinising the writing of Komi is an old one for Lytkin. Even before his leaving for Moscow University, in 1923, he had expressed it, alongside recognition for Molodcov's work: the perspective was fixed for a later time but existed (Baraksanov 1989: 48). At the same time, it did not seem to be an issue for him at that time: the important was "to have a practical and scientific alphabet" (Oni ljubili... 1993: 150). Then, for some time, the idea was overpowered by other concerns -as the creation of two Komi languages, Komi-Zyrien and Komi-Permiak, against which Lytkin positioned himself clearly. But actually it has been noticed that those who were favourable to the Latin alphabet were also favourable to a unique Komi language… (Oni ljubili… 1993: 149).

Vasily Lytkin became a promoter of the Latin alphabet for Komi. He was the person in charge of introducing, in coordination with the overall process, the Latin alphabet in Komi. He was called at the board of the Committee of the New Alphabet and was expected to propose a project both for Komi and for Udmurt. The result of his work was accepted by the Party in November 1931 and the Latin alphabet was introduced at school in 1932 (Isayev 1979:203).

Lytkin's more resolute positions in favour of the Latin alphabet appeared after he returned from his studies abroad. Certainly, the years he spent in Finland and in Hungary increased his familiarity with the Latin alphabet. Not only in these countries the languages were written in the Latin alphabet, but moreover, the linguistic study of the Permian languages was also based on a Latin transcription and not on the existing official literary language. The wish to keep closer to the kin languages and of the languages of Western Europe played certainly a role in Lytkin's choice of the Latin alphabet.

More personal concerns might also have been determinant: on the one hand, his conflict with Molodcov. Molodcov seemed to be permanently at the core of conflicts, he has, according to scholars, quite a difficult personality (Oni ljubili 1993: 210). At the beginning, Lytkin defended him, for example against A.A. Tsember, who criticised his alphabet and called him, in 1921, "an evil enemy of the Komi people" (Oni ljubili 1993: 211). Quite early, Molodcov had expressed his objections against the adoption of a generalised Latin alphabet: according to him, many languages spoken in the USSR have very different phonetic systems from the ones transcribed with the Latin alphabet; the internationalisation of the alphabet is a waste for weak written cultures, the more so if it is not coordinated (Molodcov 1924: 155-156).

But perhaps even more important might have been Lytkin's closeness to linguists directly implicated in the latinisation process. The main ideologist of the latinisation movement was the same N.F. Yakovlev, who had recommended Lytkin as a doctoral student and who was the vice-rector of the institution that received him. He called Lytkin to be part of a commission he chaired, and whose aim was to propose a Latin alphabet system for Russian.

Besides the natural sympathy Lytkin could have for the Latin alphabet, these personal contacts may have strengthened his involvement in the struggle for it.

His involvement in the defence of the Latin alphabet for Komi was certainly a proof of his wide approach: he didn't want his language to be encompassed in narrow frames, but wished it to dialogue, to be situated in an ample space. From this point of view, Lytkin was certainly not an obtuse nationalist. But he was a "nationalist" from another point of view: he was dedicated to his culture and devoted himself to it. If the debate could have been led further and could have been free, we may only suppose all the benefit the Komi culture would have derived from it.

The latinisation episode was closed and presented by the Party as a mistake. Perhaps it was a mistake from the point of view of the development of literacy in the vernacular. The permanent change of alphabet -four times in two decades -brought certainly instability.

And this instability was probably not favourable to increase the popularity of the written form of Komi. Still the attempt became suspicious not only because it complicated the task of the teachers, of the students and of the literate people, but also because, as earlier the Arabic alphabet, le Latin one was connected with a world -although without a recognised centre -that was more and more felt as threatening and dangerous. Some years later, Lytkin could not have studied abroad as he was allowed to -Kuzebay Gerd, the Udmurt writer and ethnographer, who dreamt to imitate him, did not obtain the necessary scholarship and permission.

The fact he had been involved as the main actor in a direction that was later disowned by the Party was certainly suspicious. But Lytkin was eliminated from public life before he could be reproached his role in the latinisation of Komi: he was arrested as soon as 1933 -when the promotion of the Latin alphabet was still not exposed to political accusations. There were other reasons for which Lytkin was considered an intellectual to eliminate: his poems were more and more criticised and, above all, he had committed the crime of spending a long time abroad and to have met during his travels people who were known for their hostility towards the Soviet regime.

Still Lytkin may be considered as lucky: not only he came back from prison and exile, but he came back to his job as a linguist and was able to contribute further to Permian studies.

His participation in the Latinisation episode, although central, has never been emphasised by later researchers before and after 1991, although other contributions to the development of the Komi language after the Revolution have been remarked, as his work on textbooks while he was a student in Moscow. His conflict with Molodcov is only rapidly mentioned [START_REF] Baraksanov | В.И. Лыткин (к 80-летию со дня рождения)[END_REF]: 5, Oni ljubili 1993: 140-143). Still, this episode has its place in the history of the Komi written culture and I feel honoured to recall it.

Bibliography

Baraksanov 1964 = Бараксанов, Г.Г., Коми литературный язык, история его формирования и диалектная основа, Автореферат кандидатской диссертации, Москва 1964

On this question in general, see Toulouze 1997.

This decision was taken by the Duma, and later (in 2004) confirmed by the Constitutional Court in connexion with the attempt of Tatarstan to go back to the Latin alphabet adopted at the end of the 1920s (see below).

About the criticism of the Arabic alphabet, seeUbriatova 1977: 113, Zak Isaev 1966: 5-6, Musaev 1965:10-11; about the political advantages of the Latin alphabet see Nurmakov 1934:5-6.

The Latin alphabet was adopted for Turkish only in 1928.

A good example is Moscow's position in several disputes between the Party's organisation of Izhevsk, whihc was strongly Russian and hostile to the Udmurts, and the Udmurt leader Trofim Borisov.Moscow, in 1923 and 1925, protected Borisov and promoted him, while in his region, the Russian leader attempted to eliminate him(Kuznecov 1994: 27-28).

Missionary Nikolay Il'minsky, who had been trained in his knowledge of Tatar by his teacher Kazem-bek, had written his first translation of religious texts in this language. He had later discovered, in living longer in the countryside, that the Tatar population spoke a much less arabised kind of Tatar than the city intellectuals Ilminsky 1870).

Those early monuments are mainly tranlations of religious literature(Martynov 1985:71).

Such as it was before the orthographic reform for Russian in 1918, for example with the letter « i », which exists no longer in Russian, but is present in other forms of Cyrillic and in the Latin alphabet.

In Russian : Научно-исследовательский институт этническох и национальных культур народов Востока СССР.


Stéphane Le Corre
email: le.corre.stephane@hotmail.fr

DARK MATTER, A DIRECT DETECTION

Keywords: gravitation, gravitic field, dark matter

In a previous paper, we demonstrated that the linearized general relativity could explain dark matter (the rotation speed of galaxies, the rotation speed of dwarf satellite galaxies, the movement in a plane of dwarf satellite galaxies, the decreasing quantity of dark matter with the distance to the center of galaxies' cluster, the expected quantity of dark matter inside galaxies and the expected experimental values of parameters Ω dm of dark matter measured in CMB). It leads, compared with Newtonian gravitation, to add a new component (gravitic field) to gravitation without changing the gravity field (also known as gravitomagnetism). In this explanation, dark matter would be a uniform gravitic field that embeds some very large areas of the universe. In this article we are going to see that this specific gravitic field, despite its weakness, could be soon detectable, allowing testing this explanation of dark matter. It should generate a slight discrepancy in the expected measure of the Lense-Thirring effect of the Earth. In this theoretical frame, the Lense-Thirring effect of the "dark matter" would be a value between around 0.3 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚/𝑦𝑦𝑚𝑚𝑚𝑚𝑚𝑚 and 0.6 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚/𝑦𝑦𝑚𝑚𝑚𝑚𝑚𝑚. In the LAGEOS or Gravity Probe B experiments, there was not enough precision (around 10% for the expected 6606 𝑚𝑚𝑚𝑚𝑚𝑚. 𝑦𝑦 -1 geodetic and 39 𝑚𝑚𝑚𝑚𝑚𝑚. 𝑦𝑦 -1 frame-dragging precessions). In the GINGER experiment, there could be enough, the expected accuracy would be around 1%. If this discrepancy were verified, it would be the first direct measure of the dark matter.

Overview

General relativity implies the existence of two gravitational components. In addition of the gravity field, there is a gravitic field (together giving what is called the gravitomagnetism) just like the magnetic field in electromagnetism. The new gravitic field can be measured by its precession effect, known as Lense-Thirring effect. Several experiments have validated this effect for the Earth gravitic field, NASA's LAGEOS satellites or Gravity Probe B [START_REF] Adler R | The three-fold theoretical basis of the Gravity Probe B gyro precession calculation[END_REF] with an accuracy of around 10%. Some new experiments will try to obtain a higher accuracy, for example GINGER [START_REF] Ruggiero | Sagnac Effect, Ring Lasers and Terrestrial Tests of Gravity[END_REF] with an expected accuracy of around 1%.

In [START_REF] Corre | Dark matter, a new proof of the predictive power of General Relativity[END_REF], a solution is proposed to explain the dark matter. This explanation leads to the assumption that we are embedded in a relatively uniform gravitic field generated by larger structures than galaxies (likely the clusters). Just like the Earth gravitic field can be measured, this hypothetical embedded gravitic field could be measured by its precession effect. Such a measure will be a direct measure of the "dark matter". We are going to see that the magnitude of this effect is at the limit of our detection. And even, in the most advantageous case, the accuracy of 1% (as expected in GINGER experiment) could be enough to detect it.

First, I recall the theoretical idealization used in this article and in [START_REF] Corre | Dark matter, a new proof of the predictive power of General Relativity[END_REF].

Gravitation in linearized general relativity

From general relativity, one deduces the linearized general relativity in the approximation of a quasi-flat Minkowski space (𝑔𝑔 𝜇𝜇𝜇𝜇 = 𝜂𝜂 𝜇𝜇𝜇𝜇 + ℎ 𝜇𝜇𝜇𝜇 ; |ℎ 𝜇𝜇𝜇𝜇 | ≪ 1). With following Lorentz gauge, it gives the following field equations [START_REF] Hobson | Relativité générale[END_REF] (with

= 1 𝑐𝑐 2 𝜕𝜕 2 𝜕𝜕𝜕𝜕 2 -∆): 𝜕𝜕 𝜇𝜇 ℎ � 𝜇𝜇𝜇𝜇 = 0 ; ℎ � 𝜇𝜇𝜇𝜇 = -2 8𝜋𝜋𝜋𝜋 𝑚𝑚 4 𝑇𝑇 𝜇𝜇𝜇𝜇 (𝐼𝐼) With: ℎ � 𝜇𝜇𝜇𝜇 = ℎ 𝜇𝜇𝜇𝜇 - 1 2 𝜂𝜂 𝜇𝜇𝜇𝜇 ℎ ; ℎ ≡ ℎ 𝜎𝜎 𝜎𝜎 ; ℎ 𝜇𝜇 𝜇𝜇 = 𝜂𝜂 𝜇𝜇𝜎𝜎 ℎ 𝜎𝜎𝜇𝜇 ; ℎ � = -ℎ (𝐼𝐼𝐼𝐼)
The general solution of these equations is:

ℎ � 𝜇𝜇𝜇𝜇 (𝑚𝑚𝑐𝑐, 𝑥𝑥 ⃗) = - 4𝜋𝜋 𝑚𝑚 4 � 𝑇𝑇 𝜇𝜇𝜇𝜇 (𝑚𝑚𝑐𝑐 -|𝑥𝑥 ⃗ -𝑦𝑦 ⃗|, 𝑦𝑦 ⃗) |𝑥𝑥 ⃗ -𝑦𝑦 ⃗| 𝑚𝑚 3 𝑦𝑦 ⃗
In the approximation of a source with low speed, one has: 𝑇𝑇 00 = 𝜌𝜌𝑚𝑚 2 ; 𝑇𝑇 0𝑖𝑖 = 𝑚𝑚𝜌𝜌𝑢𝑢 𝑖𝑖 ; 𝑇𝑇 𝑖𝑖𝑖𝑖 = 𝜌𝜌𝑢𝑢 𝑖𝑖 𝑢𝑢 𝑖𝑖 And for a stationary solution, one has:

ℎ � 𝜇𝜇𝜇𝜇 (𝑥𝑥 ⃗) = - 4𝜋𝜋 𝑚𝑚 4 � 𝑇𝑇 𝜇𝜇𝜇𝜇 (𝑦𝑦 ⃗) |𝑥𝑥 ⃗ -𝑦𝑦 ⃗| 𝑚𝑚 3 𝑦𝑦 ⃗
At this step, by proximity with electromagnetism, one traditionally defines a scalar potential 𝜑𝜑 and a vector potential 𝐻𝐻 𝑖𝑖 . There are in the literature several definitions [START_REF] Mashhoon | Gravitoelectromagnetism: A brief review[END_REF] for the vector potential 𝐻𝐻 𝑖𝑖 . In our study, we are going to define:

ℎ � 00 = 4𝜑𝜑 𝑚𝑚 2 ; ℎ � 0𝑖𝑖 = 4𝐻𝐻 𝑖𝑖 𝑚𝑚 ; ℎ � 𝑖𝑖𝑖𝑖 = 0
With gravitational scalar potential 𝜑𝜑 and gravitational vector potential 𝐻𝐻 𝑖𝑖 :

𝜑𝜑(𝑥𝑥 ⃗) ≡ -𝜋𝜋 � 𝜌𝜌(𝑦𝑦 ⃗) |𝑥𝑥 ⃗ -𝑦𝑦 ⃗| 𝑚𝑚 3 𝑦𝑦 ⃗ 𝐻𝐻 𝑖𝑖 (𝑥𝑥 ⃗) ≡ - 𝜋𝜋 𝑚𝑚 2 � 𝜌𝜌(𝑦𝑦 ⃗)𝑢𝑢 𝑖𝑖 (𝑦𝑦 ⃗) |𝑥𝑥 ⃗ -𝑦𝑦 ⃗| 𝑚𝑚 3 𝑦𝑦 ⃗ = -𝐾𝐾 -1 � 𝜌𝜌(𝑦𝑦 ⃗)𝑢𝑢 𝑖𝑖 (𝑦𝑦 ⃗) |𝑥𝑥 ⃗ -𝑦𝑦 ⃗| 𝑚𝑚 3 𝑦𝑦 ⃗
With 𝐾𝐾 a new constant defined by: 𝜋𝜋𝐾𝐾 = 𝑚𝑚 2 This definition gives 𝐾𝐾 -1 ~7.4 × 10 -28 very small compare to 𝜋𝜋.

The field equations (𝐼𝐼) can be then written (Poisson equations):

∆𝜑𝜑 = 4𝜋𝜋𝜋𝜋𝜌𝜌 ; ∆𝐻𝐻 𝑖𝑖 = 4𝜋𝜋𝜋𝜋 𝑚𝑚 2 𝜌𝜌𝑢𝑢 𝑖𝑖 = 4𝜋𝜋𝐾𝐾 -1 𝜌𝜌𝑢𝑢 𝑖𝑖 (𝐼𝐼𝐼𝐼𝐼𝐼)
With the following definitions of 𝑔𝑔 ⃗ (gravity field) and 𝑘𝑘 �⃗ (gravitic field), those relations can be obtained from following equations:

𝑔𝑔 ⃗ = -𝑔𝑔𝑚𝑚𝑚𝑚𝑚𝑚 ����������⃗ 𝜑𝜑 ; 𝑘𝑘 �⃗ = 𝑚𝑚𝑚𝑚𝑐𝑐 ������⃗ 𝐻𝐻 � �⃗ 𝑚𝑚𝑚𝑚𝑐𝑐 ������⃗ 𝑔𝑔 ⃗ = 0 ; 𝑚𝑚𝑚𝑚𝑑𝑑 𝑘𝑘 �⃗ = 0 ; 𝑚𝑚𝑚𝑚𝑑𝑑 𝑔𝑔 ⃗ = -4𝜋𝜋𝜋𝜋𝜌𝜌 ; 𝑚𝑚𝑚𝑚𝑐𝑐 ������⃗ 𝑘𝑘 �⃗ = -4𝜋𝜋𝐾𝐾 -1 ȷ p ��⃗
With relations (𝐼𝐼𝐼𝐼), one has:

ℎ 00 = ℎ 11 = ℎ 22 = ℎ 33 = 2𝜑𝜑 𝑚𝑚 2 ; ℎ 0𝑖𝑖 = 4𝐻𝐻 𝑖𝑖 𝑚𝑚 ; ℎ 𝑖𝑖𝑖𝑖 = 0 (𝐼𝐼𝐼𝐼)
The equations of geodesics in the linear approximation give:

𝑚𝑚 2 𝑥𝑥 𝑖𝑖 𝑚𝑚𝑐𝑐 2 ~-1 2 𝑚𝑚 2 𝛿𝛿 𝑖𝑖𝑖𝑖 𝜕𝜕 𝑖𝑖 ℎ 00 -𝑚𝑚𝛿𝛿 𝑖𝑖𝑖𝑖 �𝜕𝜕 𝑖𝑖 ℎ 0𝑖𝑖 -𝜕𝜕 𝑖𝑖 ℎ 0𝑖𝑖 �𝑑𝑑 𝑖𝑖
It then leads to the movement equations:

𝑚𝑚 2 𝑥𝑥 ⃗ 𝑚𝑚𝑐𝑐 2 ~-𝑔𝑔𝑚𝑚𝑚𝑚𝑚𝑚 ����������⃗ 𝜑𝜑 + 4𝑑𝑑 ⃗ ∧ �𝑚𝑚𝑚𝑚𝑐𝑐 ������⃗ 𝐻𝐻 � �⃗ � = 𝑔𝑔 ⃗ + 4𝑑𝑑 ⃗ ∧ 𝑘𝑘 �⃗
From relation (𝐼𝐼𝐼𝐼), one deduces the metric in a quasi flat space:

𝑚𝑚𝑚𝑚 2 = �1 + 2𝜑𝜑 𝑚𝑚 2 � 𝑚𝑚 2 𝑚𝑚𝑐𝑐 2 + 8𝐻𝐻 𝑖𝑖 𝑚𝑚 𝑚𝑚𝑚𝑚𝑐𝑐𝑚𝑚𝑥𝑥 𝑖𝑖 -�1 - 2𝜑𝜑 𝑚𝑚 2 � ��𝑚𝑚𝑥𝑥 𝑖𝑖 � 2
In a quasi-Minkowski space, one has:

𝐻𝐻 𝑖𝑖 𝑚𝑚𝑥𝑥 𝑖𝑖 = -𝛿𝛿 𝑖𝑖𝑖𝑖 𝐻𝐻 𝑖𝑖 𝑚𝑚𝑥𝑥 𝑖𝑖 = -𝐻𝐻 � �⃗ . 𝑚𝑚𝑥𝑥 ����⃗
We retrieve the known expression [START_REF] Hobson | Relativité générale[END_REF] with our definition of 𝐻𝐻 𝑖𝑖 :

𝑚𝑚𝑚𝑚 2 = �1 + 2𝜑𝜑 𝑚𝑚 2 � 𝑚𝑚 2 𝑚𝑚𝑐𝑐 2 - 8𝐻𝐻 � �⃗ . 𝑚𝑚𝑥𝑥 ����⃗ 𝑚𝑚 𝑚𝑚𝑚𝑚𝑐𝑐 -�1 - 2𝜑𝜑 𝑚𝑚 2 � ��𝑚𝑚𝑥𝑥 𝑖𝑖 � 2
Remark: Of course, one retrieves all these relations starting with the parameterized post-Newtonian formalism. From (CLIFFORD M. WILL, 2014) one has:

𝑔𝑔 0𝑖𝑖 = - 1 2 (4𝛾𝛾 + 4 + 𝛼𝛼 1 )𝐼𝐼 𝑖𝑖 ; 𝐼𝐼 𝑖𝑖 (𝑥𝑥 ⃗) = 𝜋𝜋 𝑚𝑚 2 � 𝜌𝜌(𝑦𝑦 ⃗)𝑢𝑢 𝑖𝑖 (𝑦𝑦 ⃗) |𝑥𝑥 ⃗ -𝑦𝑦 ⃗| 𝑚𝑚 3 𝑦𝑦 ⃗
The gravitomagnetic field and its acceleration contribution are:

𝐵𝐵 𝑔𝑔 ����⃗ = ∇ � �⃗ ∧ �𝑔𝑔 0𝑖𝑖 𝑚𝑚 𝚤𝚤 ���⃗ � ; 𝑚𝑚 𝑔𝑔 ����⃗ = 𝑑𝑑 ⃗ ∧ 𝐵𝐵 𝑔𝑔 ����⃗
And in the case of general relativity (that is our case): 𝛾𝛾 = 1 ; 𝛼𝛼 1 = 0 It then gives:

𝑔𝑔 0𝑖𝑖 = -4𝐼𝐼 𝑖𝑖 ; 𝐵𝐵 𝑔𝑔 ����⃗ = ∇ � �⃗ ∧ �-4𝐼𝐼 𝑖𝑖 𝑚𝑚 𝚤𝚤 ���⃗ �
And with our definition:

𝐻𝐻 𝑖𝑖 = -𝛿𝛿 𝑖𝑖𝑖𝑖 𝐻𝐻 𝑖𝑖 = 𝜋𝜋 𝑚𝑚 2 � 𝜌𝜌(𝑦𝑦 ⃗)𝛿𝛿 𝑖𝑖𝑖𝑖 𝑢𝑢 𝑖𝑖 (𝑦𝑦 ⃗) |𝑥𝑥 ⃗ -𝑦𝑦 ⃗| 𝑚𝑚 3 𝑦𝑦 ⃗ = 𝐼𝐼 𝑖𝑖 (𝑥𝑥 ⃗)
One then has:

𝑔𝑔 0𝑖𝑖 = -4𝐻𝐻 𝑖𝑖 ; 𝐵𝐵 𝑔𝑔 ����⃗ = ∇ � �⃗ ∧ �-4𝐻𝐻 𝑖𝑖 𝑚𝑚 𝚤𝚤 ���⃗ � = ∇ � �⃗ ∧ �4𝛿𝛿 𝑖𝑖𝑖𝑖 𝐻𝐻 𝑖𝑖 𝑚𝑚 𝚤𝚤 ���⃗ � = 4∇ � �⃗ ∧ 𝐻𝐻 � �⃗ 𝐵𝐵 𝑔𝑔 ����⃗ = 4𝑚𝑚𝑚𝑚𝑐𝑐 ������⃗ 𝐻𝐻 � �⃗
With the following definition of gravitic field:

𝑘𝑘 �⃗ = 𝐵𝐵 𝑔𝑔 ����⃗ 4 One then retrieves our previous relations:

𝑘𝑘 �⃗ = 𝑚𝑚𝑚𝑚𝑐𝑐 ������⃗ 𝐻𝐻 � �⃗ ; 𝑚𝑚 𝑔𝑔 ����⃗ = 𝑑𝑑 ⃗ ∧ 𝐵𝐵 𝑔𝑔 ����⃗ = 4𝑑𝑑 ⃗ ∧ 𝑘𝑘 �⃗
A last remark: The interest of our notation is that the field equations are strictly equivalent to Maxwell idealization (in particular the speed of the gravitational wave obtained from these equations is the light celerity). Only the movement equations are different with the factor "4". But of course, all the results of our study could be obtained in the traditional notation of gravitomagnetism with the relation

𝑘𝑘 �⃗ = 𝐵𝐵 𝑔𝑔 �����⃗ 4 .

Gravitic field: a way to measure it

Just like for the electromagnetism, this gravitic field implies a phenomenon of precession. It is known as the Lense-Thirring effect. Instead of taking into account only the own gravitic field of the earth, we are also going to take into account the hypothetical external gravitic field that explains the dark matter. We are first going to recall what the equations in the general relativity are for the Lense-Thirring effect. And secondly, we will use it to test our solution by calculating the contribution to the precession effect generated by our gravitic field that explain the dark matter.

Gravitic field and precession effect

The equations of the motion for the spin four-vector 𝑆𝑆 𝜇𝜇 has been studied in several papers. In general relativity, it leads to a precession of 𝑆𝑆 𝜇𝜇 . For example, with (ADLER, 2015), one can write the following equations:

𝑆𝑆 ⃗ ̇= ��𝛾𝛾 + 𝛼𝛼 2 � 1 𝑚𝑚 2 �𝑔𝑔𝑚𝑚𝑚𝑚𝑚𝑚 ����������⃗ 𝜑𝜑 ∧ 𝑑𝑑 ⃗� + 1 4 (𝛾𝛾 + 𝛼𝛼)𝑚𝑚𝑚𝑚𝑐𝑐 ������⃗ ℎ �⃗ � ∧ 𝑆𝑆 ⃗
Which lead to define a geodetic vector field Ω 𝐺𝐺 �����⃗ and a "gravitomagnetic" vector field Ω 𝐿𝐿𝐿𝐿 �������⃗ :

Ω 𝐺𝐺 �����⃗ = �𝛾𝛾 + 𝛼𝛼 2 � 1 𝑚𝑚 2 �𝑔𝑔𝑚𝑚𝑚𝑚𝑚𝑚 ����������⃗ 𝜑𝜑 ∧ 𝑑𝑑 ⃗� ; Ω 𝐿𝐿𝐿𝐿 �������⃗ = 1 4 (𝛾𝛾 + 𝛼𝛼)𝑚𝑚𝑚𝑚𝑐𝑐 ������⃗ ℎ �⃗
These expressions use the PPN formalism. As seen before, for general relativity, one has: 𝛾𝛾 = 1 ; 𝛼𝛼 = 1 It leads to:

Ω 𝐺𝐺 �����⃗ = 3 2𝑚𝑚 2 𝑔𝑔𝑚𝑚𝑚𝑚𝑚𝑚 ����������⃗ 𝜑𝜑 ∧ 𝑑𝑑 ⃗ ; Ω 𝐿𝐿𝐿𝐿 �������⃗ = 1 2 𝑚𝑚𝑚𝑚𝑐𝑐 ������⃗ ℎ �⃗
In our notation:

𝐻𝐻 � �⃗ = ℎ �⃗ 4 ; 𝑘𝑘 �⃗ = 𝑚𝑚𝑚𝑚𝑐𝑐 ������⃗ 𝐻𝐻 � �⃗
One then has

Ω 𝐿𝐿𝐿𝐿 �������⃗ = 2𝑘𝑘 �⃗

Measure of the dark matter

In our solution, around the Earth, 𝑘𝑘 �⃗ represents the addition of two terms, the own gravitic field of the earth 𝑘𝑘 𝐸𝐸 ����⃗ and the external uniform gravitic field 𝑘𝑘 0 ����⃗ :

𝑘𝑘 �⃗ = 𝑘𝑘 𝐸𝐸 ����⃗ + 𝑘𝑘 0 ����⃗
In the same way, the Lense-Thirring effect Ω 𝐿𝐿𝐿𝐿 �������⃗ is then composed of the own Earth gravitic field term Ω 𝐿𝐿𝐿𝐿_𝐸𝐸 �����������⃗ and of a new supplementary term of "dark matter" Ω 𝐿𝐿𝐿𝐿_𝐷𝐷𝐷𝐷 ��������������⃗

Ω 𝐿𝐿𝐿𝐿 �������⃗ = 2𝑘𝑘 𝐸𝐸 ����⃗ + 2𝑘𝑘 0 ����⃗ = Ω 𝐿𝐿𝐿𝐿_𝐸𝐸 �����������⃗ + Ω 𝐿𝐿𝐿𝐿_𝐷𝐷𝐷𝐷 ��������������⃗
The term Ω 𝐿𝐿𝐿𝐿_𝐸𝐸 �����������⃗ is the traditional frame-dragging precession:

𝐻𝐻 𝐸𝐸 �����⃗ = ℎ 𝐸𝐸 ����⃗ 4 = � 𝜋𝜋 2𝑚𝑚 2 𝑚𝑚 3 � �𝑚𝑚 ⃗ ∧ 𝐽𝐽 ⃗ � ; Ω 𝐿𝐿𝐿𝐿_𝐸𝐸 �����������⃗ = 𝜋𝜋 𝑚𝑚 2 � 𝐽𝐽 ⃗ 𝑚𝑚 3 - 3𝑚𝑚 ⃗ 𝑚𝑚 5 �𝑚𝑚 ⃗ • 𝐽𝐽 ⃗ ��
In the Gravity Probe B experiment, the expected values were:

�Ω 𝐺𝐺 �����⃗ � = 6606 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚/𝑦𝑦𝑚𝑚𝑚𝑚𝑚𝑚 �Ω 𝐿𝐿𝐿𝐿_𝐺𝐺𝐺𝐺𝐵𝐵 ���������������⃗ � = 39 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚/𝑦𝑦𝑚𝑚𝑚𝑚𝑚𝑚 Let's evaluate the order of magnitude of the external gravitic field (our dark matter) around the Earth. From [START_REF] Corre | Dark matter, a new proof of the predictive power of General Relativity[END_REF], an average value is:

�Ω 𝐿𝐿𝐿𝐿_𝐷𝐷𝐷𝐷 ��������������⃗ � = 2�𝑘𝑘 0 ����⃗ �~2 × 10 -16.5 𝑚𝑚 -1
It then gives

�Ω 𝐿𝐿𝐿𝐿_𝐷𝐷𝐷𝐷 ��������������⃗ � = 0.4 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚/𝑦𝑦𝑚𝑚𝑚𝑚𝑚𝑚
In fact, from the sample of galaxies studied in [START_REF] Corre | Dark matter, a new proof of the predictive power of General Relativity[END_REF], one obtains the following possible interval for 𝑘𝑘 0 :

10 -16.62 < �𝑘𝑘 0 ����⃗ � < 10 -16.3
If we assume that these galaxies can be representative of our own galaxy, the expected discrepancy should be in the following interval (in 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚/𝑦𝑦𝑚𝑚𝑚𝑚𝑚𝑚): 0.3 < �Ω 𝐿𝐿𝐿𝐿_𝐷𝐷𝐷𝐷 ��������������⃗ � < 0.6 2015) we have seen that at this distance the gravitic field of the galaxy could be of the same magnitude. Therefore the expected value should be around 2 times greater than �Ω 𝐿𝐿𝐿𝐿_𝐷𝐷𝐷𝐷 ��������������⃗ �. One also have to take into account the unknown direction of 𝑘𝑘 0 ����⃗ , implying that the magnitude of the effect could be reduced. Furthermore the effect of precession �Ω 𝐿𝐿𝐿𝐿_𝐷𝐷𝐷𝐷 ��������������⃗ � could be spread on the two components �Ω 𝐺𝐺 �����⃗ � and �Ω 𝐿𝐿𝐿𝐿 �������⃗ � and then decrease the discrepancy.

Discussion

Conclusion

In the better case, a precision of 1% could reveal a discrepancy in the measure of the expected precession effects. The next generation of experiments (as GINGER) will have such an accuracy. In our solution the expected discrepancy should be in the following interval (in 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚/𝑦𝑦𝑚𝑚𝑚𝑚𝑚𝑚) 0.3 < �Ω 𝐿𝐿𝐿𝐿_𝐷𝐷𝐷𝐷 ��������������⃗ � < 0.6. But without any detection, a higher accuracy will be required to definitively declare that this solution is irrelevant. In particular if the direction of 𝑘𝑘 0 ����⃗ is very disadvantageous.

  �Ω 𝐿𝐿𝐿𝐿_𝐺𝐺𝐺𝐺𝐵𝐵 ���������������⃗ � is only known with a precision of 10%. We need to have a better accuracy on this kind of experiments to hope to detect this discrepancy. GINGER experiment should have a precision of 1%. It could be enough to detect a discrepancy. But there are several aspects of the experiment that can play a role in decreasing or increasing this discrepancy. The Sun is at around 8kpc from the galactic center. In (LE CORRE,

	�Ω 𝐿𝐿𝐿𝐿_𝐷𝐷𝐷𝐷 ��������������⃗ � represents around 1% of �Ω 𝐿𝐿𝐿𝐿_𝐺𝐺𝐺𝐺𝐵𝐵 ���������������⃗ �. But until
	now,


Ivan Giorgio
email: ivan.giorgio@uniroma1.it

Daria Scerrato

Multi-scale concrete model with rate-dependent internal friction

Keywords: Internal Friction, Nonlinear systems, LuGre model, dissipation in concrete, micromorphic continuum

In this paper, a micromorphic, non-linear 3D model aiming to describe internal friction phenomena in concrete is considered. A reduced two-degrees-of-freedom model is employed for the sake of easy handling to explain dissipative loops which have been observed in some concrete specimens tested under cyclic uniaxial compression loading with different frequencies and having various amplitudes but never inducing large strains. As (linear or non-linear) viscoelastic models do not seem suitable to describe neither qualitatively nor quantitatively the measured dissipation loops, we propose to introduce a multi-scale micro-mechanism of Coulomb-type internal dissipation associated to the relative motion of the faces of the micro-cracks present in the material and to the asperities inside the micro-cracks. We finally present numerical simulations showing that the proposed model is suitable to describe some of the available experimental evidence.

Introduction

Concrete is the most widely employed structural material in the world, thus, its environmental impact is enormous considering the huge amounts of water and heat energy required to produce it, or the problem of its recycling when it is no longer usable. In this context, all efforts provided to improve concrete performance or its durability are really effective both from a social and an economic point of view. The paper is particularly focused on the design and characterization of earthquake-proof concrete, aiming to obtain a material with high durability and good performances for earthquake engineering. Indeed, some experimental campaigns [START_REF] Madeo | Effect of micro-particle additions on frictional energy dissipation and strength of concrete[END_REF] have shown that the addition of some ad-hoc fillers can significantly increase the internal dissipation under dynamic cyclic loading, even inside the conventional yield surface characterizing -with the commonly accepted approximation-the 1 elastic domain. This first experimental evidence has justified the pursuit of the experiments for a larger range of frequencies [START_REF] Scerrato | Effect of Micro-Particle Addition on Frictional Energy Dissipation and Strength of Concrete: Experiments and Modelling[END_REF] and the development of a novel constitutive model. Thus we have proposed such a model [START_REF] Scerrato | A simple non-linear model for internal friction in modified concrete[END_REF][START_REF] Scerrato | A micro-structural model for dissipation phenomena in the concrete[END_REF] to take into account the internal friction phenomena occurring in the enriched concrete under cyclic loading and to provide a mathematical tool aiding in the design of advanced concretes with high dynamic performances for earthquake engineering.

Concrete is a highly heterogeneous material and has a complex microstructure. The microscopic complexity of such a material thus could be modeled, at a macro-level scale by higher gradient or micromorphic continua in the more general context of generalized materials (see for more details e.g. [START_REF] Jouan | Using a second gradient model to simulate the behaviour of concrete structural elements[END_REF][START_REF] Carcaterra | Macroscopic description of microscopically strongly inhomogenous systems: A mathematical basis for the synthesis of higher gradients metamaterials[END_REF][START_REF] Dell'isola | The postulations á la D'Alembert and á la Cauchy for higher gradient continuum theories are equivalent: a review of existing results[END_REF][START_REF] Dell'isola | Cauchy tetrahedron argument applied to higher contact interactions[END_REF]); indeed, the standard Cauchy model is too poor to describe this material in our opinion. As a matter of fact concrete consists of a heterogeneous distribution of different kinds and amounts of solid phases, pores, and micro-cracks. By making suitable changes in the microstructure of this material, its properties can be modified (see e.g. [START_REF] Misra | Stabilization characteristics of clays using class C fly ash[END_REF]). Adding admixtures to the cement paste, it is possible to modify mechanical characteristics of the medium therefore pinpointing all the features of admixtures that are capable of improving the concrete dynamic behavior. Due to its multi-scale structure, this material can be susceptible to cracking when subjected to the influence of tensile stresses induced by differential movements between aggregates and cement paste. Such differential movements commonly arise, for example, either on drying or on cooling of concrete. A concrete may thus have micro-cracks even before it is loaded. Of course, impact loads, drying shrinkage, freeze and thaw cycles and sustained loads at high stress levels will have the effect of increasing the size and the number of micro-cracks. Micro-crack size can vary significantly, it may range from 0.1 to 500 µm [START_REF] Mehta | Concrete: microstructure, properties, and materials[END_REF]. On the other hand, different concrete models are based on the idea of the existence of an interfacial transitional zone around aggregate grains, constituting a third phase, different from the aggregate and bulk cement paste phases (see e.g. [START_REF] Dell'isola | A phenomenological approach to phase transition in classical field theory[END_REF][START_REF] Yeremeyev | The stability of the equilibrium of two-phase elastic solids[END_REF][START_REF] Dell'isola | Dynamics of solids with microperiodic nonconnected fluid inclusions[END_REF] to model some relevant aspects of the interaction between different phases).

Many formulations have been proposed to account for the dissipation in solid media (see e.g. [START_REF] Carcaterra | Fractional dissipation generated by hidden wave-fields[END_REF][START_REF] Carcaterra | Dissipation in a finite-size bath[END_REF][START_REF] Altenbach | Analysis of the viscoelastic behavior of plates made of functionally graded materials[END_REF][START_REF] Lamarque | Study of a rheological model with a friction term and a cubic term: deterministic and stochastic cases[END_REF][START_REF] Biswas | A reduced-order model from high-dimensional frictional hysteresis[END_REF]), however, herein, with aiming at analyzing the internal mechanisms of dissipation in concrete when no relevant damage phenomena occur at the macroscopic level, the internal dissipation is assumed to be related to the relative motion of micro-crack faces following the same spirit of several papers [START_REF] Kachanov | A microcrack model of rock inelasticity part I: Frictional sliding on microcracks[END_REF][START_REF] Hernandez-Olivares | Static and dynamic behaviour of recycled tyre rubber-filled concrete[END_REF][START_REF] Pensée | Micromechanical analysis of anisotropic damage in brittle materials[END_REF][START_REF] Shamy | Microscale energy dissipation mechanisms in cyclically-loaded granular soils[END_REF][START_REF] Wrzesniak | A two-scale time-dependent model of damage: Influence of micro-cracks friction[END_REF][START_REF] Zhu | Micromechanical modelling of anisotropic damage in brittle rocks and application[END_REF][START_REF] Tonazzi | Experimental and numerical analysis of frictional contact scenarios: from macro stick-slip to continuous sliding[END_REF]. The basic idea was to introduce a new (scalar) variable characterizing, in an average way, these micro-sliding along the micro-crack faces with a proper constitutive relation describing the evolution of this variable and its coupling with the Coulomb friction force. The validation of this model was fully convincing for a certain range of frequencies [START_REF] Scerrato | A micro-structural model for dissipation phenomena in the concrete[END_REF][START_REF] Scerrato | Effect of Micro-Particle Addition on Frictional Energy Dissipation and Strength of Concrete: Experiments and Modelling[END_REF] -above 2 Hz-, but it appears to not be applicable for lower frequencies. The purpose of this paper is now to introduce a second scalar variable to characterize not only the relative sliding of the micro-crack faces, but also the micro-structure of the asperities inside the micro-cracks -as a second step. These asperities are described by taking into account their heights through a probability distribution. During cyclic loading, all these asperity heights will essentially play a different role in a sliding direction and in its reverse, allowing to generalize the first model.

The paper is basically divided in three parts. In the first one, a review of dynamic friction models is provided by concluding the one which will be considered to build the present model. In the second part, the model is presented and the differences with the first model are emphasized and discussed.

Motivation

Since frictional forces arise as a results of a relative velocity of porous surfaces in a concrete matrix, we performed experimental tests in order to account for this issue. Specifically, we considered cyclic tests with varying frequency of load ranging from 0.1 Hz to 0.8 Hz, i.e. relative micro-velocity, and with a load included between 19.62 and 392.4 kN. Following the procedure described in [START_REF] Madeo | Effect of micro-particle additions on frictional energy dissipation and strength of concrete[END_REF][START_REF] Scerrato | Effect of Micro-Particle Addition on Frictional Energy Dissipation and Strength of Concrete: Experiments and Modelling[END_REF], the dissipated energy for different kinds of specimen is estimated. Table 1 summarizes the dissipated energy for the different frequencies tested. In particular, Tab. 1 shows a relative increase compared to the frequency 0.8 Hz of 75%, 133% and 206% for the frequencies 0.5, 0.3 and 0.1 Hz, respectively. This analysis thus shows that the dissipated energy increases with decreasing frequency (see Fig. 1). This means that the friction in tested concrete presents a sort of 'Stribeck effect', i.e. a friction phenomenon that gives rise at low velocity decreasing friction with increasing velocity [START_REF] Stribeck | Die wesentlichen Eigenschaften der Gleit-und Rollenlager: Untersuchung einer Tandem-Verbundmaschine von 1000 PS[END_REF][START_REF] Jacobson | The Stribeck memorial lecture[END_REF]. To take this phenomenon into account, it is possible to address this issue with models that include this effect: examples of such a formulation are the LuGre model proposed by Canudas de Wit [START_REF] De Wit | A new model for control of systems with friction[END_REF] or the Leuven model proposed by Swevers et al. [START_REF] Swevers | An integrated friction model structure with improved presliding behavior for accurate friction compensation[END_REF].

Friction models

Friction phenomena occurring in solids should be models which belong to two main groups: i) static friction models derived from constitutive relationships, chosen to fit a specific behavior, between the friction force and a proper macroscopic variable, ii) dynamic friction models in which the friction force depends also on internal hidden variables that evolve according to differential equations conceived to match experimental measures [START_REF] Olsson | Friction models and friction compensation[END_REF][START_REF] Wojewoda | Hysteretic effects of dry friction: modelling and experimental studies[END_REF]. For instance, Coulomb model, which assumes the friction force to be independent of sliding velocity, falls into the first group. Many other examples can be taken into account; among these it is worth noting the static model proposed in [START_REF] Makkar | A new continuously differentiable friction model for control systems design[END_REF]. The constitutive relation for the friction force F in term of the sliding velocity ẋ is

F ( ẋ) = γ 1 [tanh(γ 2 ẋ) -tanh(γ 3 ẋ)] + γ 4 tanh(γ 5 ẋ) + γ 6 ẋ (1)
where the term [tanh(γ 2 ẋ) -tanh(γ 3 ẋ)] is related to the so-called Stribeck effect [START_REF] Stribeck | Die wesentlichen Eigenschaften der Gleit-und Rollenlager: Untersuchung einer Tandem-Verbundmaschine von 1000 PS[END_REF][START_REF] Jacobson | The Stribeck memorial lecture[END_REF], i.e. a phenomenon in which the increase of velocity implies a decrease of friction force; the second term is a regularization of the function signum characterizing the Coulomb behavior and the last term stands for a viscous effect that becomes significant at high velocities.

Several dynamic friction models have been proposed in literature. Among these, Dahl [START_REF] Dahl | A solid friction model[END_REF][START_REF] Dahl | Solid friction damping of mechanical vibrations[END_REF] proposed a mathematical model of friction that was based on the assumption that the friction force F is only a function of sliding displacement x, and monotonically approaches the Coulomb friction force F C as long as ẋ is positive. When the direction of motion is reversed, the friction force approaches -F C . The friction function slope dF (x)/dx, however, remains always positive, although ẋ changes sign. This hysteretic behavior could be mimicked with a friction slope function of the form

dF (x) dx = σ 1 - F F C sgn( ẋ) β sgn 1 - F F C sgn( ẋ) (2)
where the quantity σ is the slope of the force-deflection curve at F = 0 or the rest stiffness. The value of the parameter β defines the basic shape of the friction function and thus it is chosen to characterize the friction law being used. A more useful way to employ Dahl's assumption consists in considering the time evolution of the friction force by means of the simple equation Ḟ = F ẋ. The Dahl friction force could be interpreted as a nonlinear elastoplastic model with a nearly linear elastic behavior for small deflections, which approaches a plastic behavior for large deflections (see Fig. 2).

Figure 2 shows typical Dahl's cycles obtained by varying the rest stiffness σ in dimensionless form.

In [START_REF] Dankowicz | On the modeling of dynamic friction phenomena[END_REF] Dankowicz suggests that the origin of frictional force between two rough solid surfaces coming into contact lies in the interaction of their asperities. The asperities are assumed to have different heights -relative to a reference plane-according to a probability distribution, while all the other properties remains unchanged. The imposition of a tangential displacement entails that the shorter asperities, which are under less compression, begin to slide before the longer ones and, only after a sufficiently large tangential displacement, the resultant tangential load is within a given deviation from maximum tangential load. To take account of this quasi-static behavior, an average shearing deformation, z, is introduced which describes the collective asperity deformation and the friction force is assumed as

F = F Max z δ = µN e -∆d/s z δ ( 3
)
where µ is the friction coefficient, N is the normal load applied, ∆d is the change in distance between the reference planes of the interacting surfaces, s is the standard deviation of the probability distribution of asperity heights and δ is the average deformation obtained under maximum tangential load. To consider the lag between the change in direction of motion and the tangential load, the deformation z is described by an ordinary differential equation according to the evolution rule

dz dt = ẋ 1 -sgn( ẋ) z δ (4)
x being the macroscopic tangential displacement.

The so-called Lund-Grenoble (LuGre) model proposed in [START_REF] De Wit | A new model for control of systems with friction[END_REF] is also based on the average behavior of the bristles whose average deflection is denoted by z. Similarly to the previous model, the evolution equation is

dz dt = v - |v| g(v) z (5)
in which g, i.e. the maximum value of z, depends explicitly on the sliding velocity. In the Dankowicz model a dependence on the sliding velocity is introduced in the maximum shear force through the surface separation ∆d considered as a dynamic variable evolving according to its own differential equation. The expression assumed for g to describe the Stribeck effect is

σ 0 g(v) = F C + (F S -F C ) exp - v v s 2 (6)
where σ 0 is the rest stiffness, F C is the Coulomb friction force, F S is the stiction force which plays the role of a static friction force, and v s , is a reference velocity also called the Stribeck velocity. The constitutive relationship for the friction force due to the bending of the bristles is given by

F = σ 0 z + σ 1 ż + σ 2 v (7)
The Dahl model is based on a Coulomb friction with a lag in the change of the friction force when the direction of motion reverses. The LuGre model generalizes the model of Dahl enabling it to describe the Stribeck effect, in which at low-velocity range the friction force decreases as velocity increases. Figure 3 shows, in dimensionless form, the dependence of LuGre's model on the sliding velocity. We can observe that when the maximum velocity of the cycle approaches zero the friction force tends to the static friction force F S , while for high velocities the friction force is about equal to the Coulomb friction F C . Anyway, if F C = F S there is not Stribeck effect and the LuGre model, which is rate dependent, reduces to the Dahl model, which is rate independent.

Finally it is worth mentioning the Leuven model, an extension of the LuGre model, proposed by Swevers et al. in [START_REF] Swevers | An integrated friction model structure with improved presliding behavior for accurate friction compensation[END_REF]. The key idea is to fit transition curves of arbitrary forms. The friction force is modeled as function of an auxiliary variable z, namely the average deformation of the asperities of the contact surfaces. The constitutive equation for the friction force is

F = F h (z) + σ 1 ż + σ 2 v (8)
where

F h (z) = F b + F d (z)
is the hysteresis friction force. F d (z) is the transition curve which is active at a certain time, F b is the value of the friction force at the beginning of a transition curve. The evolution equation of the state variable z is

dz dt = v 1 -sgn F d (z) S(v) -F b F d (z) S(v) -F b β (9)
in which the function S(v) characterizes the sliding regime near zero velocity, and is given by

S(v) = F C + (F S -F C ) exp - v v s δ (10)
In order to achieve a better match between model predictions and the experimental tests performed and recalled in Sec. 1.1, we propose to exploit a LuGre-type model adapted to the continuous case to take account of a 'Stribeck effect', as will be explained in detail in the rest of the article. Multi-scale models are a powerful tool to deal with complex materials which are increasingly used in engineering applications. Specifically, motivated by the studies referred to above, a model is herein developed for an engineering material which is a standard or enriched concrete. This building material has to be investigated from a dynamical point of view with three different scales, a bulk scale, a meso-scale of pores inside the solid matrix and finally a micro-scale of asperities characterizing the porous surfaces that come into contact with each other under a compression (see Fig. 4). The results obtained employing the proposed model may depend on the shape of interfaces which can be treated as rough surfaces. For such type of interfaces the averaging procedure proposed in [START_REF] Eremeyev | On effective properties of materials at the nano-and microscales considering surface effects[END_REF] using various models of surface elasticity can be taken into account. The model for the macro-scale is based on a Saint-Venant theory (see for more details [START_REF] Scerrato | A simple non-linear model for internal friction in modified concrete[END_REF] and [START_REF] Dell'isola | Saint-Venant's problem for porous linear elastic materials[END_REF]). The constitutive equations at the meso-scale contains elastic nonlinearities, which can be observed through experimental testings. The model for the micro-scale is derived from the LuGre model that accounts for Coulomb friction and includes the Stribeck effect. The proposed model is encompassed in the framework of the micromorphic continuum theory (many examples of such an approach can be found in the literature, see e.g. [START_REF] Yang | Micromechanics based second gradient continuum theory for shear band modeling in cohesive granular materials following damage elasticity[END_REF][START_REF] Berezovski | Interfaces in micromorphic materials: Wave transmission and reflection with numerical simulations[END_REF][START_REF] Madeo | A continuum model for the bio-mechanical interactions between living tissue and bio-resorbable graft after bone reconstructive surgery[END_REF][START_REF] Maurini | Extension of the Euler-Bernoulli model of piezoelectric laminates to include 3D effects via a mixed approach[END_REF][START_REF] Porfiri | Modeling and design of passive electric networks interconnecting piezoelectric transducers for distributed vibration control[END_REF][START_REF] Placidi | On the role of grain growth, recrystallization and polygonization in a continuum theory for anisotropic ice sheets[END_REF][START_REF] Dell'isola | What are the dominant thermomechanical processes in the basal sediment layer of large ice sheets[END_REF][START_REF] Placidi | Thermodynamics of polycrystalline materials treated by the theory of mixtures with continuous diversity[END_REF]). Thus, to obtain a proper kinematical description aiming to describe internal friction phenomena, we choose the common variable u as a displacement field defined on B (a simple connected open domain), the meso-structural variable ϕ as a scalar field, which may be interpreted as average slip inside the concrete matrix, i.e. an average relative displacement between opposite faces of microcracks, and the micro-structural variable z as a scalar field which describes the collective asperity deformation. In this framework, the 'internal forces' acting on a given subdomain D of B have a virtual work which may be written as

δW (i) = - D δΨ(u, ϕ) dV (11)
Ψ being the elastic potential (see [START_REF] Scerrato | A simple non-linear model for internal friction in modified concrete[END_REF][START_REF] Scerrato | A micro-structural model for dissipation phenomena in the concrete[END_REF])

Ψ = 1 2 λ [tr(E)] 2 + µtr(E 2 ) + 1 2 k 1 ϕ 2 + 1 3 k 2 ϕ 3 + 1 4 k 3 ϕ 4 + αϕ II devE ( 12
)
where E is the linearized strain tensor, II devE 1 , related to the coupling term, is the second invariant of the deviatoric component of E, and represents a measure of shear deformations that is responsible for a relative displacement ϕ. This representation of the coupling term is very close to the common expression used in porous materials (see e.g. [START_REF] Biot | Generalized theory of acoustic propagation in porous dissipative media[END_REF][START_REF] Cowin | Linear elastic materials with voids[END_REF] and more recently [START_REF] Sciarra | A solid-fluid mixture model allowing for solid dilatation under external pressure[END_REF][START_REF] Dell'isola | A micro-structured continuum modelling compacting fluid-saturated grounds: The effects of pore-size scale parameter[END_REF][START_REF] Placidi | Variational formulation of pre-stressed solidfluid mixture theory, with an application to wave phenomena[END_REF]), even if in this

1 II devE = 1 2 tr[(devE) 2 ]
, where the deviatoric strain tensor devE = E -1 3 tr(E)1.

case, we use II devE instead of the trace of the small strain tensor; this choice is dictated by our physical interpretation of ϕ; the exchange of energy between the bulk and the microstructure is related to a deformation at macro level that induces a sliding on micro-cracks. In Eq. ( 12) the microstructure elastic force is assumed to be nonlinear. It is well-known that for a low level of external static compression the behavior of the concrete can be assumed in a satisfactory way linear. Nevertheless, tests performed with cyclic loads show that even in a small strain range the stressstrain relationship appears to be nonlinear. Therefore, the presence of an internal phenomenon that triggers the nonlinear behavior in presence of varying loads can be assumed. It is reasonable to attribute this effect to the microstructure involved in the micro-sliding. In other words, the presence of micro-cracks, very reasonably, introduces non-linear effects that are activated by time-varying loads (see for similar effects e.g. [START_REF] Placidi | A variational approach for a nonlinear one-dimensional damage-elasto-plastic secondgradient continuum model[END_REF][START_REF] Placidi | A variational approach for a nonlinear 1-dimensional second gradient continuum damage model[END_REF][START_REF] Oliveto | Incremental analysis of plane frames with geometric and material nonlinearities[END_REF]). We notice that the most convenient pair of elastic constants for the expression of the stored energy density of an isotropic material is that comprised of the bulk modulus κ and the shear modulus µ [START_REF] Federico | The linear elasticity tensor of incompressible materials[END_REF][START_REF] Federico | An energetic approach to the analysis of anisotropic hyperelastic materials[END_REF][START_REF] Federico | Considerations on incompressibility in linear elasticity[END_REF], even more so than the pair of Lamé's constants.

The 'external forces' acting on D are: i) volume forces, exerted on D by systems located outside B, giving the virtual work

δW (v) = D f i δu i dV (13)
ii) contact forces provided on D by the parts of B located outside D, giving the virtual work

δW (s) = ∂D τ i δu i dS (14)
where τ i is the usual traction acting on the boundary ∂D (see e.g. [START_REF] Piccardo | A refined model for calculating 3-D equivalent static wind forces on structures[END_REF][START_REF] Cazzani | Numerical aspects of coupling strongly frequency-dependent soilfoundation models with structural finite elements in the time-domain[END_REF][START_REF] Luongo | A continuous approach to the aeroelastic stability of suspended cables in 1:2 internal resonance[END_REF]). The virtual work due to inertial forces can be expressed by

δW (in) = - D ρü i δu i dV - D ρ ϕ φδϕ dV (15)
where ρ and ρ ϕ are the (apparent) mass density of bulk material and the effective mass density linked to the micro-structural variable ϕ respectively. The virtual work of friction forces can be specified as

δW (d) = - D F δϕ dV (16)
where, according to LuGre model, the tangential load can be assumed

F = F Max z δ(σ n , φ) + γ 1 ż + γ 2 φ ( 17
)
being γ 1 and γ 2 damping coefficients related to viscous effects at high frequencies, δ the collective maximum deformation under the maximum tangential load, i.e.

F Max = µ( φ) h(σ n ) (18)
which results in a Coulomb-like expression, where µ is the friction coefficient, is the specific surface of micro-cracks, i.e. the surface area of micro-cracks per unit volume of concrete and h is a suitable function of σ n = κ tr(E) i.e. the compressive stress that allows the contact of the opposite faces of the micro-cracks. The function h(σ n ) takes into account the 'real' condition of contact (Signorini-like conditions [START_REF] Signorini | Questioni di elasticità non linearizzata e semilinearizzata[END_REF]), indeed it can be expressed as

h(σ n ) = 0 σ n |σ n | < σ T |σ n | σ T ( 19
)
where σ T is a threshold stress denoting the minimum pressure required to close the microcracks and assure the contact between the opposite faces. Clearly, when |σ n | is less than σ T there is not contact at all and then friction force can not be exerted. In future developments, we can improve this contact model by involving also the presence of a micro-impact between the opposite faces of microcracks (see on this subject e.g. [START_REF] Andreaus | Experimental and numerical investigations of the responses of a cantilever beam possibly contacting a deformable and dissipative obstacle under harmonic excitation[END_REF][START_REF] Andreaus | Soft-impact dynamics of deformable bodies[END_REF]). In particular, the friction force can be rearranged F Max = μ( φ)ζ ĥ[tr(E)] 2 introducing only one material parameter ζ defined as µ s κ and the dimensionless coefficient μ chosen to yield a steady-state dependence of friction on the slide velocity φ which agrees with a Stribeck curve, as follows

3 μ = µ( φ) µ s = µ k µ s + 1 - µ k µ s exp - φ v s ς (20)
Parameters µ s , µ k and v s are respectively the static friction, the kinetic friction and the Stribeck velocity which must be matched to experimental observations. The exponent ς is a parameter which can be chosen in the range from 0.5 to 2 (see [START_REF] Bo | The friction-speed relation and its influence on the critical velocity of stick-slip motion[END_REF]). The collective maximum deformation δ is assumed to be

δ(σ n , φ) = F Max γ 0 (21)
By combining the expressions above we assume the following expression for the total tangential component of the surface interactions, or the friction force

F = γ 0 z + γ 1 ż + γ 2 φ ( 22
)
where γ 0 is the stiffness. As a result, according to the point of view of Dahl and LuGre models, an evolution equation for the variable z can be assumed as 2 Where the function ĥ is defined from the Eq. ( 19), by considering the relation: σn = κ tr(E). 3 In a similar way a Lorentzian function can be used:

μ = µ( φ) µs = µ k µs + 1 - µ k µs 1 1 + φ vs ς . ż = φ 1 - z δ(tr(E), φ) sgn( φ) β sgn 1 - z δ(tr(E), φ) sgn ( φ) (23)
The generalized principle of virtual work then becomes

δW (i) + δW (v) + δW (s) + δW (in) + δW (d) = 0 (24)
The proposed model can be fruitfully applied to the general case of triaxial compression loads. Indeed, concrete is commonly subjected to triaxial stress states, and it is well known that the stress level influences greatly its mechanical behaviour. The Eq. ( 24) can be solved with the Finite Element Method; however, given the complexity of the material under study, some computation issues such as locking or hourglass modes can arise. To tackle these problems an isogeometric analysis can be employed as a powerful numerical tool (on this formulation some relevant works are e.g. [START_REF] Cuomo | A variational model based on isogeometric interpolation for the analysis of cracked bodies[END_REF][START_REF] Greco | B-Spline interpolation of Kirchhoff-Love space rods[END_REF][START_REF] Turco | A three-dimensional B-spline boundary element[END_REF][START_REF] Cazzani | Constitutive models for strongly curved beams in the frame of isogeometric analysis[END_REF][START_REF] Cazzani | Isogeometric analysis: a powerful numerical tool for the elastic analysis of historical masonry arches[END_REF][START_REF] Della Corte | Referential description of the evolution of a 2D swarm of robots interacting with the closer neighbors: perspectives of continuum modeling via higher gradient continua[END_REF]). On the other hand, based-cement materials can exhibit an internal loss of stability due to the presence of high heterogeneous phases inside their solid matrix. To investigate this disagreeable behavior, we refer to some relevant analyzes [START_REF] Rizzi | On the postbuckling analysis of thin-walled frames[END_REF][START_REF] Ruta | A beam model for the flexural-torsional buckling of thin-walled members with some applications[END_REF][START_REF] Rizzi | Initial postbuckling behavior of thin-walled frames under mode interaction[END_REF].

To analyze the case of a simple compression (with Poisson effect) in cylindrical concrete specimens the Eq. ( 24) can be reduced to a system of OD equations much easier to handle. In the framework of the Saint-Venant theory, we can assume as kinematical descriptor for the bulk deformation, the strain along the longitudinal axis of the considered cylinder, ε. In this case this strain field is a scalar variable that does not depend on spatial coordinates, but only, possibly on time. The same assumption of a spatial uniform field is made even for the variable ϕ because of the assumed uniformly distributed random orientation of the micro-cracks and for the compressive load considered that is also uniform. With these assumption, the principle of virtual work [START_REF] Vescovo | Theoretical and experimental dynamic analysis aimed at the improvement of an acoustic method for fresco detachment diagnosis[END_REF] for any arbitrary virtual strain δε and virtual micro-sliding δϕ provides

         M ε ε + K ε ε + αϕ = f ext (t) m ϕ φ + k ϕ (ϕ) + αε + F (z, ż, φ) = 0 ż = φ 1 - z δ(ε, φ) sgn( φ) β sgn 1 - z δ(ε, φ) sgn ( φ) (25)
where the non-linear microstructure elastic force is expressed by the power-law below

k ϕ (ϕ) = k1 ϕ + k2 ϕ 2 + k3 ϕ 3 (26)
β is a parameter that defines the force-deflection curve, and the material parameters can be deter-Table 2: Values of the material parameters used in numerical simulations. mined by

K ε (Nm) α (N) k1 (N/m) k2 (N/m 2 ) k3 (N/m 3 ) 1.
M ε = B ρ ν 2 x 2 1 + x 2 2 + x 2 3 dV, m ϕ = B ρ ϕ dV, K ε = B µ (3 λ + 2 µ) λ + µ dV = B Y dV, α = B √ 3/3 (1 + ν) α dV, k1 = B k 1 dV, k2 = B k 2 dV, k3 = B k 3 dV,
being Y and ν respectively the Young modulus and the Poisson ratio. For the determination of all of these constitutive parameters, it is possible to resort to methods of identification as outlined in [START_REF] Placidi | Gedanken experiments for the determination of two-dimensional linear second gradient elasticity coefficients[END_REF][START_REF] Dietrich | Problems of identification of mechanical characteristics of viscoelastic composites[END_REF][START_REF] Vescovo | Theoretical and experimental dynamic analysis aimed at the improvement of an acoustic method for fresco detachment diagnosis[END_REF]. Note that since z is governed by an ordinary differential equation, it does not respond instantaneously to changes of slide velocity. This accounts for the lag between the change in direction of motion and the tangential load. It is worth noting that the LuGre model also satisfies a Lipschitz condition in z if β 1 and therefore has a unique solution; otherwise, if 0 β < 1, the Lipschitz condition is not satisfied but only the continuity [START_REF] Padthe | On the LuGre model and friction-induced hysteresis[END_REF].

Friction dissipates energy, however, since this model is dynamic, there may be time intervals where frictional visco-plasticity stores energy and others where it gives energy back. Barabanov and Ortega provided an answer to the question of passivity of the generalized LuGre friction model. The concept of passivity in this context is related to a dissipative system in which any motion starting at the time t 0 satisfies for all time T the inequality

℘ = T t 0 F φdt 0 (27)
Namely, they have shown that the model defines a passive operator ℘ if and only if the coefficients satisfy a proper inequality [START_REF] Barahanov | Necessary and sufficient conditions for passivity of the LuGre friction model[END_REF].

Results and discussion

To illustrate the features of the proposed model, we consider a basic example using some concrete specimens undergoing cyclic uniaxial compression loading with different frequencies. In this case Eqs. ( 25) can be utilized once all the material parameters are estimated on the base of a cylindrical sample whose diameter is D = 11.28 cm and having height h = 22 cm. Some criteria have been taken into account for parameter identification. Indeed, values of the parameter ϕ have to be compatible with the order of magnitude of known dimensions of typical cracks in concrete. The micro-force k ϕ (ϕ) is assumed to be elastic, and so monotonously increasing. The deformation energy needs to be definite positive and consequently suitable restrictions on the stiffness parameters as well as on the coupling between micro-and macro-motion must be considered. Table 2 summarizes the material parameters used in the simulations. We remark that non-linear effects are not negligible as it is clear from the non-linear stiffnesses. The equivalent mass M ε can be easily estimated by assuming that the volume mass density of material is homogeneous -in an averaged sense-and known , here it has been set ρ = 2300 kg/m 3 . The order of magnitude of the micro-parameter m ϕ , not directly measurable, was selected in order to better fit with experimental data, assuming reasonably that the micro-mass density, ρ ϕ is a suitable fraction of the bulk mass density ρ, here we set ρ ϕ = ρ/10. The other parameters used in the simulations and related to the dissipation are

µ k /µ s = 0.2, ς = 2, β = 1, γ 0 = 5 × 10 11 N/m 4 , v s = 1.9 × 10 -4 m/s, γ 1 = 1 N s/m 4 , γ 2 = 2 × 10 8 N s/m 4 .
The cyclic external load applied is f ext (t) = -F 0 -F 1 sin(2πf t) with a pre-load F 0 to ensure that we simulate a regime in which the micro-cracks are closed, and with frequency f = {0.1, 0.3, 0.5, 0.8} Hz. The values of F 0 and F 1 are respectively 206.01 kN and 186.39 kN, in order to have the same setup of the experimental tests.

Figure 3 shows the dissipative loops in a stress-strain plane obtained varying frequency. In this range of frequencies it is easy to observe that the dissipation energy, i.e. the area of the loops, decreases with increasing of frequency. That particular behavior -indeed, it is opposite to the behavior expected for a viscous dissipation-matches with measured cycles shown in Fig. 1. In Tab. 3 are reported the dissipated energy for the frequencies tested as obtained by the numerical simulations. Comparing these values with those measured and listed in Tab. 1, we can observe a good matching between previsions of the proposed model and measurements.

Figure 6 displays the friction force for two cycles of frequency 0.1 Hz. It is possible to note the sharp shape of the force in the neighborhood of reversal velocities points at 0, 5, 10 and 15 s. In addition, between these reversal points, at the center of the loading and unloading curves, it is possible to recognize the effect due to the pressure σ n ; indeed, the amplitude of the friction force is varied in accordance to the amplitude of the modulating pressure σ n , which change the level of the interaction between the faces of the micro-cracks, implying a greater or lesser friction force.

Figure 7 exhibits the power ℘ of the friction force F . It is worth noting that even if some instantaneous values of the power are positive, the passivity of the system considered, which is recalled in Eq. ( 27), is respected. This behavior can be justified attributing, as predicted by the LuGre model, an elastic behavior to the micro-asperities, which release elastic energy when the sliding velocity becomes zero. If µ s = µ k , i.e. the friction coefficient is constant, the LuGre model adopted turns in the Dahl model that is rate independent as mentioned above. Figure 8 exhibits the dissipative loops for a concrete mixture with Dahl model and frequencies 0.1 Hz and 1 Hz, and it is easily observable that the area of a dissipative loop does not depend on the frequency.

Conclusions

In this paper, an already tested mathematical model is applied for describing some dissipation phenomena which are observed in some experimental tests carried out on cylindrical concrete specimens subjected to uniaxial compressive loading. In order to achieve a better match between the model proposed and the experimental tests performed with different frequencies, we propose to exploit a LuGre-type model adapted to a continuum medium to take into account of the observed 'Stribeck effect'. Specifically, for the sake of simplicity, we use a reduced model constituted of three coupled ODEs which respectively describe macroscopic, mesoscopic and microscopic equilibrium of the material modeled.

The numerical simulations that have been carried out show a good agreement with the mea- surements carried out at different frequencies, allowing us to validate the model proposed in the frequency range explored. Finally, we have shown that if we consider the Dahl model to describe the friction force instead of the LuGre one, the dissipative loops do not depend on the frequency of the applied load. Thus, with a little change in the model, we can capture different behaviors that can arise from different mixtures or even from other materials that present a micro-structure similar to that considered here.

The model seems to be extendable to interpret friction phenomena in concrete also in conjunction with steel rebars. In this sense, an application to simplified models of concrete beams like the one described in [START_REF] Contrafatto | An enriched finite element for crack opening and rebar slip in reinforced concrete members[END_REF] appears to be very interesting and immediately deducible from the model. Thus, further developments are expected considering composite fiber-reinforced models and the interaction between the steel bars and the concrete based on suitable homogenizations (see e.g. [START_REF] Alibert | Second-gradient continua as homogenized limit of pantographic microstructured plates: a rigorous proof[END_REF][START_REF] Madeo | Wave propagation in pantographic 2D lattices with internal discontinuities[END_REF][START_REF] Dell'isola | A two-dimensional gradient-elasticity theory for woven fabrics[END_REF][START_REF] Af Cheviakov | Fully non-linear wave models in fiber-reinforced anisotropic incompressible hyperelastic solids[END_REF][START_REF] Dell'isola | Plane bias extension test for a continuum with two inextensible families of fibers: A variational treatment with Lagrange multipliers and a perturbation solution[END_REF][START_REF] Guo | Mechanical modeling of incompressible particle-reinforced neo-Hookean composites based on numerical homogenization[END_REF]).

Figure 1 :

 1 Figure 1: Measured dissipative loops for a concrete mixture and frequencies 0.8 Hz, 0.5 Hz, 0.3 Hz and 0.1 Hz.

Figure 3 :

 3 Figure 2: Tipical Dahl's cycles

Figure 4 :

 4 Figure 4: Three scale representation.

Figure 5 :Figure 6 :Figure 7 :

 567 Figure 5: Simulated dissipative loops for a concrete mixture and frequencies 0.8 Hz, 0.5 Hz, 0.3 Hz and 0.1 Hz.

Figure 8 :

 8 Figure 8: Dissipative loops for a concrete mixture with Dahl model and frequencies 0.1 Hz and 1 Hz.

Table 1 :

 1 Measured dissipated energy per cycle.Eventually, in final last part, the comparison with experimental results allows the investigation of the enlarged domain of validity of this new model particularly with respect to the frequency range.

	Frequency (Hz)	0.8	0.5	0.3	0.1
	Dissipated Energy (J/c) 0.53 ± 0.06 0.92 ± 0.03 1.22 ± 0.08 1.60 ± 0.02

Table 3 :

 3 Simulated dissipated energy per cycle.

	Frequency (Hz)	0.8	0.5	0.3	0.1
	Dissipated Energy (J/c)	0.59	0.93	1.31	1.61

Acknowledgments The authors would like to thank Félix Darve for helpful comments and advice through the study.