Valerio Romitelli
email: valerio.romitelli@unibo.it

On Enthusiasm in Politics: 12 Hypotheses

Keywords: Enthusiasm, Democracy, Masses, Historic Sequence, Neoliberalism, Communism

The author suggests twelve hypotheses to revive enthusiasm as a political category and, eventually, to restate the possibility of political action. In this perspective, the enthusiasm of the masses was the essential political element of "The Glorious Thirty Years" that followed the Second War World, characterized by a strong tension to realize a greater social justice. It was also at the base of the "Long Sixty-Eight" and of the processes of decolonization. Enthusiasm is thus thought in opposition both to the monopolistic pretension of political innovation forwarded by communist parties in the past and to the exclusive competence on political-economical decisions claimed today by democratic élites. Re-thinking the enthusiasm that animated these historic sequences of the twentieth century should offer the possibility to respond to "the sad passions" that dominate the era of neoliberal administration of global democracy.

Introduction

The hypotheses suggested in this essay originate from the studies that I have undertaken on thought, passion and political bodies. My research work examines the possibilities of political materialism in a post-Marxist perspective, yet far from any anti-Marxist notion fashionable today. My main goal is to go beyond the conceptions which reduce politics to a superstructure without its own power of determination. The propositions presented here collide with many polemical fronts: first of all, they clash against the conception which sees the turning point from Fordism to Post-Fordism as the key to explaining the entire history of the last hundred years. In other words, I reject the concept that everything that happens in politics is due to the capitalistic objectivation of the subjective creativity of the working class. I also take a distance from any idea that considers history as a history of a struggle between opposite economic interests. I think it's necessary to acknowledge that there is an infinity of phenomena, events and sequences that can't be analyzed as an effect of the capitalistic transformations, more or less determined by the class struggles. For instance, it must be recognized that in the name of socialism, communism, fascism and even of the New Deal it was deemed possible to change destiny, in every instance in a singular way, and to understand this we require adequate categories of analysis. Using Lacan's terms, we can approach this kind of phenomena as subjective phenomena: "subjectifications" or "subjective processes". However, it is important not to abuse this language by removing it from its field of relevance. A mistake made, for example, by Ernesto Laclau in his book On the Populist Reason, where he attempts to interpret Lacan's thought with the idea of "inter-subjective relationships". This is an interpretation which fails. Wherever Lacan speaks about the subject, he speaks about the interior division of every subject in front of its own singular object of desire: therefore, Lacan's thought does not allow any meta-psychological generalization and this is not the way in which his teaching could help the search for new post-Marxist views. In this perspective it is far more interesting to think about how lacanian psychoanalysis finds objectivity, i.e. inside subjectivity rather than outside it.

1.

'The Glorious Thirty Years' as a singular age of social justice

The principal feature of this period can be described as follows: it was the high point of the last hundred years in terms of wages, social security and public education, resulting in the reduction of social differences and the creation of the middle class; the epicenter was in Europe, but at the same time there was also the spread of policies of 'de-colonization', which were immediately criticized as 'neo-colonialism'; at the geopolitical level, this era was also marked by the Cold War first, then by 'peaceful coexistence' and 'détente' between the two 'superpowers', and in the meantime by the emergence of the 'Movement of Non-Aligned countries'.

There are three sets of characteristics which, when combined, can lead to a conclusion of major importance: that the exceptional peak of social justice of this era was initially made possible by the rivalry between the spheres of influence of American capitalism and Soviet socialism, in a time when the latter was still credible as a regime devoted to the creation of greater social justice. The expansion of that kind of regime aroused both love and fear, which can therefore be identified as the driving forces behind this golden era of universal decrease in the gap between the rich and the poor.

2.

The "enthusiasm of the masses for Socialism" as a key category to understand 'the Glorious Thirty Years' Speaking about the "enthusiasm of the masses for Socialism", Mao alluded to the passion aroused by those political bodies that were inspired by the same Communist idea, though differently interpreted (i.e. the party-state of the USSR, China and that half of the world where their models were imitated). Even then there were good reasons to doubt that the socialist countries were really creating social justice, but the words of Mao and the actual universal trends of reducing social differences show that, in this age, such doubts were less important than both the "enthusiasm" and the fear for the spreading of communism. Happiness was a condition of politics, not only because prosperity was the obligatory promise of every regime, but also because the future was a source of hope in the socialist countries as well as in the capitalist ones. Saying that the "East wind" was blowing, the Chinese at that time were alluding to the fact that socialist States and communist parties were on the offensive on the world scene.

The 'Long Sixty-Eight' as an explosion of 'non-aligned particles'

The 'Long Sixty-Eight' is a widely accepted term in historical literature. I use it to indicate the political season that was possible worldwide between the Sixties and the Seventies, thanks to a multiplicity of organizations which were mostly inspired by Communist ideals, but not by Communist parties. At that time, in fact, these parties began to decline because of their subordination to the geo-political relationships between the Soviet and the American "superpowers". By using the term "particle" to characterize the political groups of the 'Long Sixty-Eight', I aim to stress the ambiguous relationships between these groups and every kind of party: on the one hand, they tried to be organized as a party, but chose not to participate in elections nor to lead governments; on the other hand, their main goal was to change radically the political systems in order to promote greater social justice. Furthermore, by applying to these groups the attribute of 'non-aligned', I aim to stress their convergence with what is known as the 'Non-Aligned Movement', i.e. Third World governments demanding greater justice, but at the geopolitical level. Symbols of these phenomena were, for instance, the committees organized by the youth of every capitalistic metropolis in solidarity with the resistance of Vietnam. It is not a simple coincidence that both the political groups of the 'Long Sixty-Eight' and the movement of Third World governments were virtually contemporaneous in terms of beginnings, developments and final dispersion.

The Chinese Cultural Revolution as a crucial sequence of the 'Long Sixty-Eight'

The Cultural Revolution in China saw its enemy in the culture of a communist party leading a socialist State, but the repercussions of this event were universal during the 'Long Sixty-Eight': this peculiar kind of revolution was cited everywhere by every political group born during this time. It is always relevant to ask ourselves why.

The assumption maintained by the culture of those communist parties which were born in 1921 was shared by other political cultures: it concerned the irreducible pre-eminence of experts. It is the famous assumption of Kautsky renewed by Lenin that theory does not arise from social movements, which are considered historically crucial, but politically blind. In the Leninist doctrine as it was established by Stalin and thereafter shared by all the communists faithful to the USSR, the perspective was even more restricted: so much so that the party's central committee remained the only source of legitimate knowledge. Cultural pluralism was always openly defended in the capitalist world, but at the same time there was an essentially unchallenged assumption that in politics, as well as in other fields of culture, only experts are authorized to educate the non-experts. The Cultural Revolution destroyed this certitude. During this sequence, not only Mao's thought was to be exalted, but also that of the masses. The investigations conducted by the experts among the masses seem today only one of many totalitarian injunctions which justified deportations and other ill-treatments of dissenters. However, this understanding fails to explain why the Cultural Revolution became a universal reference of the 'Long Sixty-Eight'. It is not a coincidence that the investigations among the masses, even if otherwise called and with completely other aims, were thereafter considered an absolutely necessary condition of any political or economic strategy even in capitalist countries.

The end of the 'Long Sixty-Eight' as the conclusion of the most relevant political experimentation in the last forty years

The word 'revolution' nowadays is often used in an equivocal way as a synonym to 'regime change', i.e. to support the destabilization of local regimes with the aim of promoting various foreign interferences by the most powerful countries. The issue therefore is to restore political dignity to this word and all that it meant from the end of the Seventeenth to the end of Twentieth centuries. My proposition is that of rethinking this long cycle as a season of political experimentations. This choice is justified if we think that the category of revolution was first used in those astrophysical experiments which made the transition "from the closed world to the infinite universe" possible, according to the title of the famous book by Alexandre Koyré. Robespierre, for instance, was absolutely clear about the fact that the French revolution should be a continuation in the political field of what the Copernicus discovery had begun in the field of physics.

Obviously, this experimental way to approach politics has to be distinguished from other kinds of approaches. In order to simplify this distinction, I propose to divide the latter into two types: the first one is normally called conservative politics and the other one is generally known as reactionary politics. Reducing the political issue to these three major categories (experimental, conservative and reactionary) has multiple theoretical consequences. But the basic assumption remains: the aim of conservative politics is to conserve the continuity of institutions, whereas the aim of reactionary politics is to react in front of an explosion of disorder, so as to restore an imaginary order. Both these types of politics are, however, different from the experimental one in a crucial feature: the latter is only possible if it is thought and experienced as a singular sequence, in discontinuity with all that precedes as well as with all that follows. It implies that we can draw a political lesson from each of these experiences only if we recognize that each one is deeply different from the others of the same type. But we also have to consider another point in this regard: just as in scientific or artistic experimentations, also in political ones, only the course of time decides on the success or failure. It is very significant that only a few years ago even the former President of France Nicolas Sarkozy declared that it was necessary to break with the aftermath of 'Sixty-Eight'. The recent decline of his public image suggests that these consequences were stronger than he thought. As a matter of fact, every time conservative and reactionary politics become powerless and the need for new political experimentations emerges again, thinking about the 'Long Sixty-Eight' is unavoidable.

6.

There

is a correlation between the conclusion of 'the Glorious Thirty Years of the welfare state' and the end of Socialist states' and Communist parties' credibility

In the second part of the Seventies, China definitely took the capitalist path that had always been opposed by the Cultural Revolution during the ten years of its different waves. In the same years, the USSR seemed to obtain some success with its traditional Cold War enemy in the diplomatic plan of the so called 'Détente' (for instance, in 1976 at the Conference of Helsinki which first recognized the division of Germany). At that time, the USA were deeply wounded by the end of the Vietnam war and by Nixon's impeachment. Nevertheless, in both the Chinese and the Soviet instances, Communist universalism is explicitly abandoned for the benefit of nationalist interests. Neither enthusiasm nor fear could be generated within these countries anymore. All the socialist and communist parties and class organizations started to decline or to break definitely with their traditions. It was precisely in those years that 'the Glorious Thirty' came to the end.

7.

Starting from the Eighties, the global leadership has belonged to the neoliberal democratic regimes which follow the pattern of the so called 'biggest democracy in the world' Towards the end of the Seventies, an unprecedented opportunity presented itself for the US. From the second post-war period until the Sixties, its influence was limited by the rise of Communism in half of the world and by the enthusiasm thus aroused. After the 'Long Sixty-Eight', the Vietnam defeat and Nixon's impeachment presented problems for the credibility of the 'biggest democracy in the world', which reached a low ebb. However, these problems were overcome as soon as a Hollywood actor entered the White House. While China was folding in on itself in order to reestablish a capitalist system, the USSR was trying to recover from 'stagnation' by embarking on the Soviet-Afghan war, which indeed contributed to its definitive collapse. Following this, Communism, class parties and class organizations lost all credibility, and therefore so did the ideals of social justice they conveyed. More than one century of experiences in the name of these ideals was reduced to nothing. This bleak void is one of the first conditions of the epochal turning point which became possible during the Eighties: on one hand, the particular pattern of American democracy can work as a universal model for democracy; on the other hand, this pattern can also impose a new version of economic liberalism. Nothing will be able to hinder the global hegemony of neoliberal democracy anymore. Thanks to its different interpretations, now more democratic, now more neoliberal, this kind of regime will be adopted by all the governments of the richest countries, which each suffer from every kind of criticism and opposition, but remain more unified around the same strategic mainstream than ever before.

The present democratic regimes are all constituted by neoliberal political ideals

In other words, it is vain both to fear the existence of a capitalism without democracy (as Žižek and Sloterdijk maintain) and to hope in a democracy beyond capitalism (as Hardt and Negri maintain). Neo-liberalism is often criticized nowadays, but always as a limitation of (or a threat to) democracy. On the contrary, it is important to admit that capitalism needs both democracy and neo-liberalism (as Badiou maintains). The point is that in our time, despite all the increasing riots and general disorders, there is 'no alternative'. In this absence of political and intellectual alternatives, the leading world democracies can reduce the traditional distinction between the rulers and the ruled, the capitalists and the workers, the rich and the poor, to a biological distinction between the worthier and the less worthy: that is, between the fit and the less fit to survive. Helping disadvantaged populations has become less of a problem for governments. The solution they propose is to increase philanthropic operations, whose aim is to extend financial power to the utmost limits. Worldwide, people such as the 'ninja' (no income, no job, no asset), the homeless or the poor have been seduced by capitalism and integrated into the system of debts and credits. This is the result of carefully calculated capitalist interests, but also of the unchallenged belief that the so called rules of the market are the most democratic way to select and reward people and communities.

9.

"Superclass" and "the bottom billiard": beyond the ruled world

As democratization implies a single political strategy, its main result is the global increase in social differences. During the Nineties, some poor countries began to become richer, but within every one of these countries the gap between the poor and the rich increased dramatically. In 2008 Rothkopf (author of the book Superclass) counted six thousand "Very Important Persons" who live far from public eye, but are able to condition the destinies of the six billion or more people who live on the Earth. Whereas Collier counted in his book A Bottom Billion of people condemned to fight hunger, thirst, illness and local wars only to survive.

The class struggle theory was the most important interpretation of the classical theory of the division between rulers and ruled, but I maintain it is no longer so. What is missing nowadays is the social battlefield itself: the most important political problem has become the global injustice suffered by all the growing populations explicitly abandoned by every government. It is not enough to analyze and to emphasize the contingent conflicts that those populations fight against exterior effects of the domination of the "superclass": the most important political question is how to unify the "bottom billion" around new strategic ideas.

10. Do we live in the "era of sad passions" (as described by Benasayag and Schmit)? Definitely, in one of political apathy

What is necessary for survival depends today not directly on nature, but on the decisions of the powers able to make natural resources available: and that is true everywhere. But these powers are limiting their range of action more and more. Increasing populations thus see their hopes reduced. Not only in the world peripheries, but also at the heart of capitalistic countries where both unemployment and wages too low for workers to survive are already considered normal. The time when the future was a source of hope is only an old memory. Nobody knows how or where the faith in progress can come back. In the psychoanalytical field, a growth of sad passions is noted. In the political one, we simply notice an absence of passions.

In this regard the principal problem is, in my view, that the conditions for experimenting with the ideals of social justice are completely new: beyond every conflict between rulers and the ruled, without any certainty of global progress. If that is the case, first of all intellectual work is necessary to renew a debate about the way to conceive these kinds of experiments.

Beyond the juxtaposition between democracy and dictatorship

I don't know if it is the case elsewhere, but in Europe we are often told that the rulers have to do their 'homework' as dictated by the requirements of markets. If we consider that the word "dictatorship" comes exactly from the verb "to dictate", it follows that the "dictatorship of markets" is something more than a fictional story created by incurable extremists. On the contrary, the assumption that the many countries allied with the US are ruled in a democratic way is not at all realistic. In effects, this assumption works only to hide the power of the "super-class" who is able to condition markets and politicians. The self-evident fact is that no political action (meeting or demonstration, negotiation or choice of candidates, election or management of election results, and so on) is possible, if its conditions are not preliminarily dictated by some entity which has the power to decide them.

Recognizing that doesn't imply the nihilistic conclusion that all politics are dictatorial, yet entails an admission ever more important today: that in any politics there is always a dictatorial "feature" (according to the terminology of Mao's text On the Correct Handling of Contradictions Among the People (1957)', which can be treated into two opposite ways; on the one hand, this feature could be declared, democratically discussed so that to limit its consequences; on the other hand, if this feature is hidden, its consequences are unlimited and democracy is reduced to simulacrum. But there is also another important implication coming from the admission that no democracy can exist without "dictating". This implication is that no people can have force (according to the etymological root of the word "democracy"), if there aren't intellectual authors able to dictate how and when this force could be conceived and organized. It is not a coincidence that among all the greatest political thinkers it is impossible to find one who believes in democracy as such. And the fact that this belief is virtually obligatory nowadays, shows just how low the passions for politics are.

12. Rethinking the "enthusiasm" of the 'Long Sixty-Eight'

That the engine of capitalism is greed is not news.

In the last thirty years the novelty has rather been the absence of ideas and organized bodies searching for greater social justice. It is for this reason that today the desire to become richer and richer has no limits, even if fewer and fewer people can be satisfied in this quest. Technical propositions to exit from the crisis are not lacking. The lack is instead of a politics able to experiment with these propositions. Numerous economists invoke their arrival, but the years pass and there is no sign of it.

Therefore, it is not an academic or nostalgic exercise to rethink 'the Glorious Thirty Years', the 'Long Sixty-Eight' and the 'enthusiasm' for a greater social justice aroused by the 'East wind'. At the beginning, all that was possible thanks to the opposition between the two parties present everywhere in the period after the Second World War: i.e. the communist and the anticommunist ones. Nevertheless, further experiments seeking to establish greater social justice were pushed to the most extreme consequences above all by small groups with big ideas and passions. Isn't this the best example of what there still is to think and do in politics? How to learn from the masses, through adequate inquiries? How to experiment with models for establishing greater social justice, starting from where governments are absent? How to organize a new politics, whose aim is not simply to gain power? These three and many other fundamental questions of the 'Long Sixty-Eight' are certainly not anachronistic.


Thomas Bulteau
email: t.bulteau@brgm.fr

Gonéri Le Cozannet

Uncertainties of shoreline change projections along the sandy coasts of Aquitaine (southwestern France)

Uncertainties of shoreline change projections along the sandy coasts of Aquitaine (southwestern France)

Thomas Bulteau, Gonéri Le Cozannet

Far from being stable, sandy shorelines are constantly evolving, threatening occasionally human assets such as buildings or transport infrastructures. In these environments, sea-level rise will exacerbate coastal erosion to an amount which remains poorly understood. To investigate this issue, we apply two suites of coastal evolution models applicable at decadal to centennial timescales on the dune-beach systems of the Aquitaine coast (southwestern France): one based on the Bruun rule, and the other based on a simplified version of a dune toe evolution model. Both modelling approaches incorporate the natural variability of shoreline change at seasonal, interannual and multidecadal time scales. They are fed with probabilistic sea-level projections and complementary information on coastal subsidence, in order to design locally applicable sea-level rise projections. For the model based on physical processes of erosion/accretion of the dune toe, a full temporal Monte Carlo simulation of multivariate storm conditions is performed so as to represent hundreds of years of realistic sea conditions.

The results show that even wide dune field has the potential to vanish completely after 150 to 200 years. However, the models predict important shifts towards erosion not earlier than the mid-21st century, suggesting that there remains several decades before current risk management practices become obsolete. Moreover, the results illustrate that mitigation of climate change has large benefits for these coastal systems, as it limits the risks of large sandy shoreline retreats according to the models outcome. As a limitation, probabilistic frameworks ignore the epistemic uncertainties affecting both sea-level projections and shoreline changes models. Here, extra-probabilistic uncertainties theories provide useful complementary information by exploring the fuzzy boundary between possible and impossible future shoreline changes.


Clément Jailin

Andreea Carpiuc

Kyrylo Kazymyrenko

Martin Poncelet

Hugo Leclerc

François Hild

Stéphane Roux

Virtual hybrid test control of sinuous crack

Keywords: A, Crack propagation and arrest, B, Concrete, Crack mechanics, Sensors and actuators, C, Finite element

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Mechanical and civil engineering simulations are based on increasingly complex constitutive models. They enable the behaviour of materials and real structures to be predicted with better accuracy at the expense of more demanding calibration steps. On the one hand, such models have the potential to reduce the need for very expensive large scale tests in service conditions. On the other hand, calibration of the constitutive parameters and assessment of their domain of validity becomes more challenging and requires either new tests (potentially at small scale), or new procedures to exploit at best a richer information. Enhanced sensitivity to model parameters thus becomes a crucial issue in designing optimal experimental tests.

For quasi brittle materials such as concrete, there are different ways of modelling the development of cracking.

Smeared crack models were first introduced [START_REF] Rashid | Ultimate strength analysis of prestressed concrete pressure vessels[END_REF][START_REF] De Borst | Nonorthogonal cracks in a smeared finite element model[END_REF]. Continuum damage mechanics is an alternative route with local [START_REF] Mazars | A description of microscale and macroscale damage of concrete structures[END_REF] and non-local formulations [START_REF] Pijaudier-Cabot | Nonlocal Damage Theory[END_REF][START_REF] Peerlings | Gradient-enhanced damage for quasi-brittle materials[END_REF][START_REF] Desmorat | Nonlocal anisotropic damage model and related computational aspects for quasi-brittle materials[END_REF][START_REF] Jirásek | Damage and Smeared Crack Models[END_REF][START_REF] Lorentz | Gradient damage models: Toward full-scale computations[END_REF][START_REF] Xenos | Calibration of nonlocal models for tensile fracture in quasi-brittle heterogeneous materials[END_REF]. Cohesive zone models were also proposed to handle initiation and propagation in such materials [START_REF] Hillerborg | Analysis of crack formation and crack growth in concrete by means of fracture mechanics and finite elements[END_REF][START_REF] Pandolfi | A recursive-faulting model of distributed damage in confined brittle materials[END_REF][START_REF] Carrara | A coupled damage-plasticity model for the cyclic behavior of shear-loaded interfaces[END_REF].

In crack band models, the finite width of the process zone is accounted for [START_REF] Bažant | Crack band theory for fracture of concrete[END_REF].

Variational formulations can also describe these two different stages of propagation [START_REF] Francfort | Rivisiting brittle fracture as an energy minimization problem[END_REF]. Discrete descriptions based on lattice calculations are considered to account for multiple cracking [START_REF] Delaplace | Progressive damage in discrete models and consequences on continuum modelling[END_REF].

All these models need to be calibrated with possibly discriminating experiments.

Stress/strain data are the typical information for calibrating smeared crack and local damage models [START_REF] Hamon | Modèle d'endommagement de Mazars[END_REF]. For non-local damage models, load/deflection curves of so-called size effect tests can be used [START_REF] Bellégo | Calibration of nonlocal damage model from size effect tests[END_REF]. The width of the process zone was shown to be very useful for tuning such models [START_REF] Iacono | Calibration of a higher-order continuum model using global and local data[END_REF].

Point measurements were used by these authors. The process zone width can be revealed via digital image correlation (DIC) [START_REF] Skarźyński | Measurements and calculations of the width of the fracture process zones on the surface of notched concrete beams[END_REF]. Similarly, the roughness of the fractured surface was also measured and considered for calibration purposes [START_REF] Xenos | Calibration of nonlocal models for tensile fracture in quasi-brittle heterogeneous materials[END_REF].

Conversely, many models only require ultimate strength properties to be known to get first order estimations of their parameters (e.g., see Refs. [START_REF] Lorentz | Gradient damage models: Toward full-scale computations[END_REF][START_REF] Carpiuc | Innovative tests for characterizing mixed-mode fracture of concrete : from pre-defined to interactive and hybrid tests[END_REF]). From 2 this analysis it is concluded that very few studies use full-field measurements for identification and validation purposes. DIC was used to monitor complex experiments [START_REF] Carpiuc | A complex mixed-mode crack propagation test performed with a 6-axis testing machine and full-field measurements[END_REF] and calibrate a cohesive zone model [START_REF] Ferreira | On the analysis of notched concrete beams: From measurement with digital image correlation to identification with boundary element method of a cohesive model[END_REF]. The aim of the present work is to propose such a procedure via virtual testing that explicitly considers full-field measurements and their uncertainties.

A recent series of crack propagation tests on concrete samples [START_REF] Carpiuc | A complex mixed-mode crack propagation test performed with a 6-axis testing machine and full-field measurements[END_REF] has been performed to achieve complex and discriminating crack patterns (and partly used in an on-going benchmark [START_REF] Carpiuc | CARPIUC Benchmark: Crack Advance, Reorientation, Propagation and Initiation Under Complex loadings[END_REF]) by modifying the loading path during the experiment. The mechanical loading was prescribed on the top and bottom part of the sample by a hexapod [START_REF] Flohic | A 3D displacement control by digital image correlation for the multiaxial testing of materials with a Stewart platform[END_REF][START_REF] Ismail | Novel hexapod-based unidirectional testing and FEM analysis of the RNC isolator[END_REF][START_REF] Guo | Development of a material testing machine with multi-dimensional loading capability[END_REF] (see Figure 1) and the crack tip position was obtained thanks to 2D DIC [START_REF] Sutton | Determination of displacements using an improved digital correlation method[END_REF][START_REF] Hild | Digital image correlation: from displacement measurement to identification of elastic properties-a review[END_REF]. First, the results of these tests show that it is possible to manually control the stability and the deflection of a crack in a quasi brittle material using plane loading, namely, a combination of tension, shear and in-plane rotation. Then it has been shown that these complex cracks with many changes of propagation direction and branching were sensitive to parameters of a state-of-the-art non-local damage model [START_REF] Carpiuc | Innovative tests for characterizing mixed-mode fracture of concrete : from pre-defined to interactive and hybrid tests[END_REF]. the loading so that the crack path successively propagates along predetermined directions [START_REF] Carpiuc | A complex mixed-mode crack propagation test performed with a 6-axis testing machine and full-field measurements[END_REF] Manual control of the loading path in a multiaxial testing machine to achieve such crack propagation experiment is thus possible, but extremely delicate and definitely non-optimal. Automated load control would be highly desirable. It would allow for a much more reactive feedback on loading directions and magnitudes, hence leading to an enhanced accuracy in following a predetermined crack path, and provide the needed flexibility to reveal parameters that are difficult to calibrate. This, undoubtedly, would pave the way to 'smart' and adaptive experiments, a kind of holy Grail in optimised mechanical tests.

The design of such control loops is challenging. The first generation of mechanical tests appeared in the late eighteenth century, which were based on a load defined before performing the experiment. The second generation of tests followed when they were controlled by a local and measurable quantity of interest (e.g., strain). This was also true for multiaxial experiments [START_REF] Hayhurst | Creep rupture under multi-axial states of stress[END_REF][START_REF] Hayhurst | A biaxial-tension creep-rupture testing machine[END_REF][START_REF] Demmerle | Optimal design of biaxial tensile cruciform specimens[END_REF]. Such experimental possibilities now flourish up to a full 3D control, which with 6 degrees of freedom becomes very demanding in terms of servo-control [START_REF] Flohic | A 3D displacement control by digital image correlation for the multiaxial testing of materials with a Stewart platform[END_REF][START_REF] Ismail | Novel hexapod-based unidirectional testing and FEM analysis of the RNC isolator[END_REF][START_REF] Guo | Development of a material testing machine with multi-dimensional loading capability[END_REF].

In spite of this remarkable progress, none of these types of experiments meets the above objective. It is the authors' belief that the control of experiments via numerical simulations (e.g., finite element computations) updated in realtime with full-field measurements constitutes a major breakthrough and open new perspectives. As such they can be seen as a sort of '3rd generation experiments.' It also enters the framework of dynamic data-driven application systems [START_REF] Darema | Dynamic Data Driven Applications Systems: A New Paradigm for Application Simulations and Measurements[END_REF] whose emergence is one critical aspect of simulation-based engineering sciences [START_REF] Oden | Simulation-based engineering sciences[END_REF]. Therefore, an interaction between the mechanical loading, the measurements and the model must be considered, thereby defining the basis of a hybrid testing procedure (Figure 2). In the literature, most hybrid experiments reported so far are the so-called 'sub-structured hybrid tests' [START_REF] Taucer | CASCADE PROJECT : Recent advences and futur need in experimental earthquake engineering[END_REF][START_REF] Saouma | Hybrid Simulation -Theory, implementation and application[END_REF][START_REF] Wagg | Substructuring of dynamical systems via the adaptive minimal control synthesis algorithm[END_REF][START_REF] Bursi | Modern testing techniques for structural systems -Dynamics and control[END_REF]. The latter ones address the dynamic response of generally large-scale structures, and are currently a very active and promising field of research. The spirit of the method is to split the structure into at least two parts. Among them, those that have a simple mechanical behaviour are computationally simulated and referred to as numerical sub-domains (or substructures). The other parts, which are critical and complex (e.g., involving non-linearities, irreversibility), the so-called physical sub-domains, are kept as substructures actually tested in the laboratory. Pseudo-dynamic testing, continuous pseudo-dynamic testing, fast hybrid testing, real-time substructure testing and real-time dynamic substructure testing [START_REF] Nakashima | Development of Real-time Pseudo dynamic Testing[END_REF][START_REF] Darby | Improved control algorithm for real-time substructure testing[END_REF][START_REF] Wagg | Substructuring of dynamical systems via the adaptive minimal control synthesis algorithm[END_REF][START_REF] Jung | Performance evaluation of a real-time pseudodynamic test system[END_REF] are methodologies developed within this hybrid framework and where part of the structure is substituted by computations. On the one hand, these hybrid tests can greatly reduce the overall cost since most of the structure is simulated numerically, while the much smaller key elements are studied experimentally. Moreover this enhanced cost-effectiveness involves no prejudice to the realism of the mechanically tested part. On the other hand, the models used for the numerical sub-domains are to be well-mastered and simple enough so that computations can be performed quickly to perform dynamic tests. In addition, the measurement devices used for feedback are generally poor and frequently reduced to few local strain gauges or LVDTs and often no more than one single actuator is used. These limitations are nothing but the manifestations of fairly recent and novel developments. The main objective is to focus the experiment on a substructure, and substitute the environment -expected to be simple and well-understood -by a model and controlled actuators.

In contrast, although the proposed hybrid approach shares a number of common features and tools, it does not rely on a partition of the structure into parts.

In the above-mentioned desired monitoring, the tested and simulated structure or material specimen are shared, and perfectly superimposed. Hence, an additional challenge to be faced is that the behaviour is not yet fully determined at the stage where control is to take action. To perform these so-called 3rd generation innovative tests, the hybrid loop is composed of three main parts:

• An experimental loading device (e.g., hexapod) applying multiaxial loading histories to samples enabling cracks to propagate in stable increments along various directions.

• A measurement technique (e.g., DIC, Integrated-DIC) to determine the position of the crack tip [START_REF] Roux | Stress intensity factor measurements from digital image correlation: post-processing and integrated approaches[END_REF][START_REF] Mathieu | Identification of a crack propagation law by digital image correlation[END_REF][START_REF] Réthoré | Automatic crack tip detection and stress intensity factors estimation of curved cracks from digital images[END_REF] and injecting it in a control algorithm. The virtual test is an important contribution to the development and the optimization of the control technique. Equally important is the assessment, from the virtual test, of the feasibility of various crack paths and their figures of merit for identification purposes.

In the present study, a sinusoidal crack path is chosen as the target. The motivations for this choice are numerous:

• changes in propagation directions bring a wealth of information that a straight path cannot reveal, such as mode mixity, damage ahead of the crack tip, or other physical microscale mechanisms affecting crack propagation may be masked by symmetry,

• the periodicity of the sinusoidal shape allows the repeatability of the experiment to be probed, at least approximatively,

• the mean (large scale) crack direction remains straight and hence the path can be kept distant from boundaries,

• the amplitude and wavelength of the path can be tuned to optimise the test.

The paper is organised as follows. First, the general methodology and composition of the 3rd generation test loop are presented with special attention paid to the control algorithm. Then, a virtual test-case with a sinusoidal crack path is shown. The feasibility of the method and the relevance of the choice of the crack path are the main focus of the present work. Last, an application to parameter identification is proposed.

Control Principle of the hybrid test

In this section the content of control part of the hybrid loop (see Figure 3), which can be used for real or virtual hybrid tests, is described. The possible experimental setup is not specified in this work since it could be rather generic, provided that it possesses two important features. First, boundary conditions have to be applied to the specimen using at least two independent degrees of freedom, which influence crack propagation in different ways (otherwise propagation and orientation will not be controlled independently).

Second, the measurement of the crack tip position should be possible during crack propagation (otherwise the error calculation will not be possible). For example, a well-instrumented multiaxial testing machine (e.g., hexapod) could be a suitable candidate to conduct such an experiment [START_REF] Carpiuc | A complex mixed-mode crack propagation test performed with a 6-axis testing machine and full-field measurements[END_REF]. The crack tip measurement may rely on Integrated Digital Image Correlation (I-DIC) using, for example, Williams' series [START_REF] Williams | On the Stress Distribution at the Base of a Stationary Crack[END_REF][START_REF] Roux | Digital image correlation and fracture: an advanced technique for estimating stress intensity factors of 2D and 3D cracks[END_REF][START_REF] Mathieu | Identification of a crack propagation law by digital image correlation[END_REF], or any other suitable technique.

For a virtual test, the experiment is replaced by a finite element (FE) simulation of crack propagation. In the present case, a non-local damage model is used. The crack tip position is obtained by a damage field analysis, but any other numerical procedure based on, say, FE results could also be considered.

The control part is used to find the boundary conditions to obtain the a priori chosen crack path, which is established before running the test. It consists of three steps: (1) a deflection angle has to be determined and linked to the stress intensity factors (SIFs), (2) these SIFs have to be converted into boundary conditions, (3) a criterion, which controls the advance of the crack per step, has to be satisfied.

This inverse problem and the evaluation of the SIFs are computed using the 'low-level' model, which is in the present case a Linear Elastic Fracture Mechanics (LEFM) analysis coupled with the geometry updating procedure provided by the eXtended Finite Element Method (X-FEM) [START_REF] Liu | XFEM for direct evaluation of mixed mode SIFs in homogeneous and bi-materials[END_REF][START_REF] Geniaut | A stable 3D contact formulation using X-FEM[END_REF][START_REF] Colombo | An implicit geometrical approach to level sets update for 3D non planar X-FEM crack propagation[END_REF].

Crack deflection angle

During mixed-mode propagation, the crack may move away from the chosen path. To correct the path by applying a new loading a deflection angle criterion has to be used. (It is worth noting that this method will not be applicable if the process zone is larger than the crack path curvature, because the crack will not be influenced by the short-term fluctuation of the loading). Let us consider a given crack path and the measured crack propagation, which is simulated by the 'high-level' (i.e., damage) model. In the same spirit as in the field of robotics and automation for a robot following a path [START_REF] Coulter | Implementation of the Pure Pursuit Path Tracking Algorithm[END_REF], a 'look-ahead distance' is chosen. A first idea (Figure 4 (a)) is to look for the intersection between the chosen (input) crack path and a circle of radius dr, which can be considered as a path discretization parameter. This intersection becomes the next target point and the deflection angle is that between the last propagation step and this new target.

For sufficiently large values of dr it is impossible to describe the full crack path due to its curvature. The shorter dr, the more precise the final crack surface. However, if during the experiment, the 'real' crack path happens to be out of dr circle the procedure may fail. Moreover the smaller dr, the more sensitive to measurement uncertainty, material heterogeneities and propagation fluctuations. Special attention should be paid while choosing dr, namely, smaller than the radius of curvature of the crack path but larger than the process zone.

To address the previous issue, another simpler method is chosen herein. Because the crack propagation component along the y-axis is small compared with the component along x (i.e., the crack propagates mainly in the x-direction), the circle can be replaced by a straight line parallel to the y-axis, with an offset of dx ≈ dr. The deflection angle is thus chosen as the angle to reach the intersection between the input crack and the vertical line (see Figure 4 (b)).

The advantage of this simplified procedure is that the algorithm does not fail, even though the determined crack orientation angle is restricted to cracks whose propagation never recedes along the x-axis.

(a) (b) With the chosen procedure and dr set, the predetermined crack path may not be reached if the propagation step is shorter than dr or if the deflection angle reaches its maximum threshold value. Consequently, the crack will follow in the best case a path that will be called the command crack path, namely, the discretised version of the initial input path.

Inverse problem

In the previous section, an algorithm has been proposed, which allows the deflection angle of the next step to be selected once the crack tip is detected.

The purpose is then to get the boundary conditions that generate this angle for further loading. LEFM is assumed to yield a reasonable approximation for small crack propagation increments. Within this theory, in-plane cracks are fully described by two SIFs K I and K II . Let us stress that such is not the case for what is here considered as the ground truth because of the non-locality of damage. However, in the limit when the internal length scale of the model tends to 0, it is expected that LEFM is recovered.

In the following test-case with a hexapod, three degrees of freedom are used to keep the experiment without significant out-of-plane motion. The other three degrees of freedom correspond to in-plane motions, namely, tension, shear and in-plane rotation applied to the upper part of the sample. However, the latter degrees of freedom are not directly those on which the control conditions are expressed. Rather, the target is to tune the boundary conditions to produce a given SIF pair, which is an inverse problem. However, the two SIFs that control crack propagation are linearly dependent on the sample boundary conditions, and can be bijectively related to two degrees of freedom. One last degree of freedom remains available, and it can be made orthogonal to the previous ones, so that it preserves the SIF pair objective, and the planar character of the test.

It is proposed to use this last degree of freedom in order to ensure stable crack propagation.

Many methods were developed in the past for crack stability control, starting from the simple SIF gradient or principal stresses analysis, going to more complex force-displacement snap-back identification to finally end up with highly non-linear Rayleigh quotient minimization problem. As a first approximation we limit ourselves to linear elasticity and hence introduce the stability criterion as some general linear constraint on three in-plane motion amplitudes. The advantage of such description is that the whole relationships remain linear and read as a 3×3 system ({k} = [S]{b})

         K I K II κ          =      S 1 I S 2 I S 3 I S 1 II S 2 II S 3 II S 1 κ S 2 κ S 3 κ               b 1 b 2 b 3          (1)
where K I , K II are the total SIFs that control crack propagation, S i I and S i II the sensitivity matrix elements, i.e., SIFs generated by elementary boundary loadings b i (e.g., kinematic (translation and rotation), static (force and torque) components), and S i κ a set of parameters that defines the stabilising motion constraint

κ = S 1 κ b 1 + S 2 κ b 2 + S 3 κ b 3 (2)
where S i κ parameters could be either set during the whole experiment or may vary depending on the crack tip position. The easy way to establish all sensitivity matrix elements S i I,II is to perform numerical simulations to assess the influence of each elementary loading on the SIFs. To summarise, in order to inverse and solve this system (Equation (1)), three main ingredients are needed, 260 namely, deflection, arrest and stability criteria:

• Deflection criterion. Within the LEFM framework it is commonly assumed that, first, the SIF amplitude sets the crack propagation threshold, and second, the SIF ratio K II /K I influences the crack deflection angle.

While many different deflection criteria exist (e.g., maximum normal stress criterion [START_REF] Erdogan | On the crack extension in plates under plane loading and transverse shear[END_REF], local symmetry criterion [START_REF] Amestoy | Déviation infinitésimale d'une fissure dans une direction arbitraire[END_REF][START_REF] Qian | Mixed mode fatigue crack growth: A literature survey[END_REF]), in order to simplify the analysis the deflection angle is given by an implicit function of SIF ratio

f (θ) = K II K I (3)
• Arrest criterion. With the ratio of load amplitude b i , i = 1-3, set by the deflection angle and K I gradient, a proportional loading is applied until the crack increment length reaches dr. The advantage of this iterative methodology is that it is directly controlled by the measured quantity of interest, i.e., the crack length. In another class of possible strategies (e.g., prediction method ) the missing equation needed to get the boundary conditions should be provided. For example, the boundary amplitude may be chosen to satisfy the LEFM propagation criterion K I = K Ic . One of these additional equations described above is an arrest criterion on the proportional boundary conditions that controls the incremental crack advance.

• Stability criterion.

The stability of the crack is an essential feature to be taken into account. Ideally (i.e., for a rigid testing machine), it is based on the change of the elastic energy with crack advance [START_REF] Glucklich | On crack stability in some fracture tests[END_REF]. Such an assumption will be made in the following. In a real test, the rigidity of the testing machine may have to be taken into account to avoid unstable crack propagation [START_REF] Nakayama | Crack Stability in the Work-of-Fracture Test: Refractory Applications[END_REF]. Based on experience from real experiments [START_REF] Carpiuc | A complex mixed-mode crack propagation test performed with a 6-axis testing machine and full-field measurements[END_REF],

in our application case, the stabilising condition is set to be the rotational loading. The idea is to obtain the crack to open by edge rotation about the crack tip centre, which creates a relative compression zone in front of the crack thereby penalising its subsequent sudden propagation (Figure 5). This empirical procedure helps avoiding the crack to become unstable and to instantaneously propagate through the ligament. The corresponding loading is a rotation about a moving centre, which should be close to the crack tip. Represented as a superposition of tension and fixed centre rotation it could be written within a single linear constraint (Equation ( 2)). an approach is that the computations are carried out only using the sound mesh.

In summary, the procedure consists of the following steps:

1. From the current and targeted crack path, compute the incremental crack direction, and hence the ratio K II /K I .

Compute [S] matrix using a low-level model. Exploiting linearity and

hence the superposition principle one computation of {k} is needed for each b i .

3. The desired direction of propagation and the stability heuristic rule provide a direction for vector {k}, from which a direction for the loading {b} is derived. (One may choose K I = K Ic to use a convention that can hold for the entire experiment, although it has no consequence.)

4. Until the crack has propagated by dx, apply the loading λ(t){k} where λ(t) is the magnitude of the loading along the direction {k}, which is slowly increased with time.

5. Repeat steps 1-4 until the crack has propagated over the desired length.

Proof of concept

Test-case

To show the feasibility of the new method and the identification procedure, the test-case of a concrete sample with a sinusoidal crack path is considered.

This test-case is based on the multiple crack deflection experiments presented in Ref. [START_REF] Carpiuc | A complex mixed-mode crack propagation test performed with a 6-axis testing machine and full-field measurements[END_REF]. In these tests, concrete samples have been loaded in a hexapod using tension, shear and in-plane rotation as control variables. The cracks that initiated at the notch propagated slowly while the boundary conditions were manually changed in order to reorient the crack towards the desired angle. The result of such test is a Z -shaped (i.e., zigzagged) cracked surface with multiple controlled turns. The following virtual experiment is proposed in the same spirit. Identical specimen geometry and type of loading will be used hereafter.

The boundary conditions are applied on the upper face while the lower face is motionless. The kinematic loadings correspond to in-plane rigid body motions, namely, global tension (i.e., translation along y), global shear (i.e., translation along x) and in-plane rotation.

The specimen is a 200 × 200 × 50 mm 3 concrete parallelepiped with a 5 × 25 mm 2 notch (see Figure 6).

The virtual test follows the same principle (see Figure 7) as the real one, namely, the numerical control is the same as used in real tests, the experimental 330 loading device is replaced by a numerical simulation performed using a damage model in order to obtain realistic responses. The assessment of the crack tip is also performed by numerical means. For each iteration i, the crack propagates because of the i-th loading and the final i-th position of the tip is the input in the control part for iteration i + 1. Finally the sample is unloaded and the 335 boundary conditions are computed for the next iteration. Because crack propagation is simulated with a damage model, the crack tip position is numerically found by analysing damage field contours. For a virtual test, the testing machine, the specimen and the measurement can be seen as a black box where boundary conditions are sent in and a crack path is obtained.

Deflection criterion

In the following, the simulated crack path and its position will be considered as a true representation of crack tip propagation and the numerical results as experimental reference. The chosen deflection criterion is the maximum normal stress. The crack tip is assumed to turn in the direction where the normal stress (σ θθ in a local crack frame) is maximum. The local SIF ratio that has to be applied to the specimen is thus given by (see Equation ( 3))

K II K I = sin(θ) 1 -3 cos(θ) (5)

Inverse problem

The linear system that uses the boundary conditions to deflect, propagate and stabilise the crack is presented. The three needed boundary conditions are on the upper face tension b 1 , shear b 2 and a plane rotation about the upper face centre b 3 . In this test-case, the stabilization of the crack path is controlled with a moving Instantaneous Centre of Rotation (ICR) at a determined x position.

The x position of the centroid, which depends on tension and rotation, enables tensile and compressive zones to be created in the sample. In order to stabilise crack propagation, the ICR has been located at a horizontal distance ∆x from 350 the crack tip. This distance has to be chosen with caution. It is bounded by

• the prediction of models (i.e., with LEFM, ∆x ∈ [0, ∞] to ensure K I > 0 at the current crack tip)

• the real experimental behaviour (e.g., intermittent propagation of the crack, heterogeneity)

355

• the capacity of the machine (i.e., the maximum level of tension, compression and torque the hexapod is able to perform)

• the material behaviour in other parts of the sample (i.e., damage on the compressive zone and on the edges) From previous experience [START_REF] Carpiuc | A complex mixed-mode crack propagation test performed with a 6-axis testing machine and full-field measurements[END_REF], it has been chosen to set this horizontal distance from the crack tip ∆x = 58 mm. The position (i) of the ICR is thus updated for each crack propagation by (i) = C(i) + ∆x where C(i) is the horizontal crack position for the corresponding time step, and (0) = -16.5 mm.

This condition is rewritten as

b 1 + (i)b 3 = 0 (6)
The linear system (7) that has to be inverted to find the value of the boundary conditions reads

         K I K II 0          =      S 1 I S 2 I S 3 I S 1 II S 2 II S 3 II 1 0 (i)               b 1 b 2 b 3          (7)
When inverting and solving the system with Equation ( 7), the sought ratio of boundary conditions is obtained

                                      K I K II 0            =       S 1 I S 2 I S 3 I S 1 II S 2 II S 3 II 1 0 (i)                  b 1 b 2 b 3           
Inverse problem ( 7)

K II K I = sin(θ) 1 -3 cos(θ)
Deflection criterion [START_REF] Peerlings | Gradient-enhanced damage for quasi-brittle materials[END_REF] Arrest criterion (e.g., iterative methodology, deflection criterion [START_REF] Erdogan | On the crack extension in plates under plane loading and transverse shear[END_REF])

In the following test-case, the advance of the crack has been set according to 360 the arrest criterion. This corresponds to an advance of the crack by dx = 6.6 mm

per step C(n) = C(0) + n × dx.
A graphical representation of the system is shown in Figure 8 in the (K I , K II ) plane when normalised by the toughness K Ic .

The influence of the two boundary conditions is different and complementary, and a linear combination has to be chosen to get the adequate SIF ratio.

Mechanical loading and measurement

For the considered virtual test, a damage model simulates crack propagation.

The computation is performed with a non-local damage model [START_REF] Lorentz | Gradient damage models: Toward full-scale computations[END_REF][START_REF] Lorentz | Convergence of a gradient damage model toward a cohesive zone model[END_REF] suitable for quasi brittle materials like concrete. The model considers the growth of and does not take into account crack closure in the current formulation.

Damage is described by a scalar field a(x) defined on the entire sample, a = 0 corresponding to sound material and a = 1 to a fully damaged state. With the increase in damage the stiffness of the material, A(a), decreases (i.e., A(0) = 1 and A(1) = 0). The stress/strain relationship takes into account stiffness loss

σ = A(a)E : ( 9
)
where E is the stiffness tensor. The following potential functional is used to describe the material state

Φ( , a) = A(a)Γ( ) + ka + c 2 ( a) 2 (10)
where is the infinitesimal strain tensor, ka the dissipated energy, and c the parameter that controls the size of the damage localization zone D. The 1D analytical solution [START_REF] Lorentz | Convergence of a gradient damage model toward a cohesive zone model[END_REF] opens the way for the parameters k and c to be identified, which can be linked to Griffith energy G f by

k = 3 4
G f D c = 3 8 DG f (11)
The Γ function is expressed as

Γ( ) = c T tr( ) + c H tr 2 ( ) + c S 3 2 dev : dev 2 ( 12
)
where c H , c T and c s are numerical parameters that can be identified from three physical measurements, namely, the triplet of tensile strength, compressive strength and shear strength.

The yield surface is obtained by deriving the functional Φ with respect to the damage field a(x)

f ( , a) = - ∂Φ ∂a (13)
The consistency (Kuhn-Tucker) conditions read f ≤ 0, ȧ ≥ 0, ȧf = 0 [START_REF] Francfort | Rivisiting brittle fracture as an energy minimization problem[END_REF] constant. The real yield surface [START_REF] Lee | Biaxial behavior of plain concrete of nuclear containment building[END_REF] of concrete is described in the tension-shear domain by adjusting the model parameters (Figure 9) Last, the stiffness loss function A(a) defines the post-peak behaviour of the material. It is written by using two parameters

c T = E 2 α 1 -2ν 1 2α - 1 2β c H = α 2 E (1 -2ν) 2 1 2α - 1 2β 2 - 1 3γ 2 (15) c S = Eα 2 6γ 2 (1 + ν) 2
A(a) = (1 -a) 2 (1 -a) 2 + ma (1 + pa) ( 16
)
where m defines the initial damage rate

m = 3 2 EG f Dα 2 (17)
The parameter p is more important as it influences the size of the process zone. The identification of the latter is very challenging for classical fracture experiments. That is why the identification presented in the next section will deal with this scalar parameter.

For the computations, the numerical parameter D has to be adapted to the size of the mesh to cover at least 3 elements inside the damaged zone of size 2 × D (i.e., finer meshes are not needed as the results converge quickly to a cohesive zone response [START_REF] Lorentz | Convergence of a gradient damage model toward a cohesive zone model[END_REF]). In the current study D is set to 1 mm and the length scale p to 1.5. The mesh is composed of 31,296 6-noded triangular (T6)

elements with an average size equal to 1 mm.

The damage model is implemented in Code Aster [START_REF] Code-Aster | Analyse des Structures et Thermomécanique pour des Etudes et des Recherches, version 13, EDF R & D[END_REF]. The measurement of the crack tip is obtained at each step from the damage field with the Code Aster module Post Endo Fiss [START_REF] Bottoni | Procédure POST ENDO FISS[END_REF]. With this method, the crack path is detected when the damage level is greater than a characteristic threshold. During each iteration, the new crack tip is detected and injected in the control part. Consequently the crack geometry has to be updated after each propagation increment. The threshold on the damage variable to detect the crack is set to a = 0.1.

In this study with a numerical simulation of the crack path, the arrest criterion is an iterative method based on a certain amount of damage increase per increment. This is possible to obtain since a numerical simulation is used and coupled with the damage control continuation method implemented in Code Aster [START_REF] Kazymyrenko | Réalisation de calculs d'endommagement en quasistatique (rupture fragile)[END_REF]. This functionality is very convenient since the position of the crack tip cannot be known on the fly during loading with the FE code.

The crack tip position and the damage level are strongly linked with the crack propagation length per iteration, as will be shown in the next section.

Results

Once the 33 reorientation iteration steps have converged, the last thin ligament tends to break in an unstable manner and can no longer be controlled.

The result obtained after all 33 steps is shown for the horizontal and vertical displacement fields in Figure 10 and the crack propagation path in Figure 11. In Figure 11, the target crack path and that obtained by the simulated procedure (red circle) are compared. Even if there is a visible deviation between both crack paths, it is observed that the simulated path is close to the target.

The mean relative error is 6.3 %, which is deemed acceptable. The maximum displacement of the top left and right corners of the sample applied for each increment before unloading is reported in Figure 12. Even if a global sinusoidal shape can be seen on the shear boundary conditions, the precise loading history is not trivial and hence not easily guessed a priori. The crack length for each iteration is plotted in Figure 13. The increase of the crack length and the variation of boundary conditions mean that the crack continuously turns during the experiment. The length of the crack vs. the number of increments is close to a linear curve.

The mean crack advance is found to be ≈ 5 mm per iteration. This length does 420 not exactly correspond to the advance of the ICR (≈ 6.6 mm). Even if the tensile gradient slightly decreases during the test, it is still important to stabilise the crack. During propagation, the mean absolute value of the deflection angle is |θ| = 6 • , i.e., the crack does not rotate much for each iteration because of small propagation increments and the rather smooth prescribed crack path.

With the help of the iterative procedure the inverse damage problem is solved recursively, i.e., the boundary conditions (Figure 12) could be directly applied to the sample from the start until the end (i.e., 33rd step) to obtain the desired sinusoidal path if the behaviour of the machine and the specimen were exactly those of the numerical simulation.

The question of the relationship between the 'low-level' and 'high-level' models is worth mentioning. In the present case, it is natural to observe that if the internal length scale of the model is much smaller than all characteristic sizes of the problem (here the smallest crack path radius of curvature), the highlevel model becomes identical to the low-level one, that is LEFM. Hence a very good control is expected, unless precisely when the internal length scale plays a significant role, and hence the present approach naturally provides a very discriminating tool to reveal such model enrichment as explored in the following subsection.

However, one may wonder whether a cruder 'low-level' model may still provide a viable route to experimentally control such a sinuous crack path. To this end, a very poor version of the 'low level' model has been used to test the control principle. Instead of an LEFM prediction, a simple proportional-integral (PI) controller with incremental change of the shear stress and a constantly growing tension/rotation permits us to drive the crack as desired during the first 12 steps. However the inaccuracies of this low level model produce spurious damage patterns from which a secondary crack may branch whenever the main crack path reaches some unfavourable direction. More details are shown in Appendix

A. The feasibility of the iterative control of a crack propagation, even though limited, is proved.

Application to identification

The aim of the testing procedure is to be sensitive enough to the model parameters to enable for their identification. A rough estimate of the sensitivity to the length scale can be directly obtained from the analysis of the crack path trajectory with respect to the studied parameter variation. As shown 455 in Figure 14 the crack path is sensitive to the length scale parameter p, but it is difficult to assess its best fit value, as its variation is really important (i.e., a factor of 3). For a more quantitative analysis, an identification procedure must be performed. The previous test-case will be used for identification purposes with 460 artificial noise added in order to prove its robustness.

X [mm]

Global DIC

In practice, a global DIC approach [START_REF] Hild | Comparison of local and global approaches to digital image correlation[END_REF] will be used to measure the displacement field for identification purposes [START_REF]Full-Field Measurements and Identification in Solid Mechanics[END_REF]. It consists of the registration of an image f (x) in the reference configuration and a series of pictures g(x, t) in the deformed configuration indexed by time t. The correlation code minimises the sum of squared differences between the deformed image corrected by the measured displacement u(x, t) and the reference image

χ 2 f (t) = 1 2γ 2 f |Ω| Ω ((g(x + u(x, t), t) -f (x)) 2 (18)
with respect to the nodal displacement vector {u m } of a finite element mesh defined by the shape functions Φ i and unknown degree of freedom u i such that u(x, t) = i Φ i (x)u i (t), where x ∈ Ω is any considered pixel, Ω the Region Of Interest (ROI) and γ f the standard deviation (expressed in grey levels) of the Gaussian white noise assumed to affect each image independently (including the reference one, which is responsible for the factor 2).

The minimization of this functional is solved by successive linearisations and corrections using modified Gauss-Newton scheme [START_REF] Mathieu | Estimation of elastoplastic parameters via weighted FEMU and integrated-DIC[END_REF] [M ]{δu m } = {b}

where [M ] is the DIC matrix, {δu m } the vector gathering all corrections to the measured degrees of freedom, and {b} the residual column vector, which cancels out when the grey level conservation is satisfied everywhere. Since the measured displacement field is corrupted by noise, the DIC matrix is related to the covariance matrix [C u ] of the measured degrees of freedom by [START_REF] Hild | Comparison of local and global approaches to digital image correlation[END_REF] [

C u ] = 2γ 2 f [M ] -1 (20)
Once a discretization has been chosen for displacement field measurements, and the level of acquisition noise is known, this derivation gives access to the sensitivity of the measured degrees of freedom to acquisition noise [START_REF] Mathieu | Estimation of elastoplastic parameters via weighted FEMU and integrated-DIC[END_REF].

Weighted FEMU

Several identification methods have been adapted or developed to quantitatively use full field measurement results to calibrate material parameter [START_REF] Avril | Overview of identification methods of mechanical parameters based on full-field measurements[END_REF][START_REF]Full-Field Measurements and Identification in Solid Mechanics[END_REF].

The most used method is Finite Element Model Updating (FEMU), which is based on the minimization of the differences between the measured quantities (i.e., forces F m and displacement fields u m ) and the corresponding computed quantities from a numerical simulation, namely, F c and u c . This simulation, which utilises a constitutive model, is performed with Dirichlet boundary conditions extracted from, say, DIC analyses.

Weighted displacement-based FEMU (i.e., FEMU-U) consists in computing the set of (dimensionless) constitutive parameters {p} that minimises the functional χ 2 u [62]

χ 2 U = 1 N u {u m -u c }[C u ] -1 {u m -u c } (21)
where χ 2 U is the weighted squared difference between the measured and calculated displacements, and N u the total number of kinematic degrees of freedom.

The weighted chi-squared error therefore refers to the [M ]-norm associated with the DIC matrix (up to a constant prefactor 1/(2γ 2 f )). The reaction forces may also be measured and computed. Thus, the same approach is followed with the reaction forces for which χ 2 F is minimised (FEMU-F) [START_REF] Mathieu | Estimation of elastoplastic parameters via weighted FEMU and integrated-DIC[END_REF]

χ 2 F = 1 N F {F m -F c }[C F ] -1 {F m -F c } (22)
where N F is the number of load measurements and [C F ] the covariance matrix of the measured load (in the present case it is assumed that the load measurements are uncorrelated so that [C F ] = γ 2 F [I], with γ F the standard load uncertainty and [I] the identity matrix).

The identification based on the combined displacement field and reaction force (FEMU-UF) [START_REF] Pottier | Contribution of heterogeneous strain field measurements and boundary conditions modelling in inverse identification of material parameters[END_REF] is achieved by minimising the global functional χ 2 U F in a Bayesian framework [START_REF] Mathieu | Estimation of elastoplastic parameters via weighted FEMU and integrated-DIC[END_REF]

χ 2 U F = N u N u + N F χ 2 u + N F N u + N F χ 2 F ( 23
)

Comparison of two different tests

The so-called Nooru-Mohammed tests [START_REF] Nooru-Mohamed | Mixed-mode fracture of concrete: an experimental approach[END_REF] have become a reference for mixed mode propagation of cracks in quasi brittle samples. One of these crack paths will be chosen to be compared with the sinusoidal path obtained with the present method. The input data for identification purposes are the measured boundary conditions of the sample (i.e., force and displacement field). These boundary conditions applied to the top and bottom surface are known for a Nooru- The predicted behaviour is simulated with the non-local damage law presented previously. In the following the most difficult parameter to identify, namely, the length scale p that controls the size of the process zone, is studied. The other parameters (i.e., E, ν, α, β, γ and G f ), which can be more easily identified with standard tests are assumed to be known. The values of the parameters are reported in Table 1. Even though the experiment has not

been performed yet, it is possible to analyse its sensitivity to various quantities (e.g., material parameters, sample geometry, loading path). The sensitivity analysis can be performed on any of the three identification residuals (i.e., χ 2 u , χ 2 F , χ 2 uF , see Refs. [START_REF] Bertin | Optimization of a cruciform specimen geometry for the identification of constitutive parameters based upon full-field measurements[END_REF]67]). In the present case, the dimensionless Hessian is computed when only kinematic data are considered

[H u ] = [S u ] t [C u ] -1 [S u ] (24)
with

{S u } i = {u c (p i + δp)} -{u c (p i )} (25)
where {S u } i denotes the column vector of displacement sensitivities with respect to the i-th material parameter when δp is equal to 10% of p i . The eigenvalues of [H u ] are very helpful to probe the sensitivity of a given experimental configuration to the chosen parameters {p} [START_REF] Bertin | Optimization of a cruciform specimen geometry for the identification of constitutive parameters based upon full-field measurements[END_REF]. If only the length of the process zone parameter (i.e., p) is studied, the diagonal component p of the dimensionless Hessian [H u ] is to be compared. In the present case, its level is equal to 1.5 × 10 -3 for the Nooru-Mohammed test, and 9.0 × 10 -2 for the proposed experiment. The norm of the sensitivity for the length scale parameter is 59 times higher for the sinusoidal test, which proves the interest of such test.

More globally, the spectrum of eigenvalues associated with the dimensionless

Hessian [H u ] can be compared to probe the global sensitivity of the two experiments to the whole set of material parameters for the chosen damage model (i.e., the higher any eigenvalue, the more sensitive the experiment). Figure 15 shows the spectrum for the two configurations. On average there is a two order of magnitude difference in favour of the new configuration. These results

show that the new configuration is significantly more sensitive to the material parameters. The coupling between the parameters is analysed with the covariance nor-malised by the diagonal terms. The corresponding correlation matrix is formed

(with no index summation) [ H -1 u ] ij = [H -1 u ] ij [H -1 u ] ii [H -1 u ] jj (26)
For the sinusoidal crack path the correlation matrix shows an anti-correlation between the Poisson's ratio and the scale parameter p. The correlation matrix for the Nooru-Mohammed test (obtained with the same mesh) appears to be strongly coupled whereas the sinusoidal experiment is less coupled (Figure 16).

535

For the force uncertainty, γ F is assumed to be proportional to the magnitude of the load level with a coefficient ρ 2 1 [START_REF] Bertin | Integrated digital image correlation applied to elastoplastic identification in a biaxial experiment[END_REF]. A minimum uncertainty for the load cells is included whose variance ρ 2 0 is independent of the load level so that the load uncertainty reads

γ 2 F = ρ 2 1 |F | 2 + ρ 2 0 ( 27
)
where the parameters, which are representative of nowadays cameras and load cells, are given in Table 2. is more sensitive to the length scale p and less to noise. Hence the identification procedure will converge more precisely and faster [START_REF] Dufour | Displacement and Mechanical Properties from Isogeometric Multiview Stereocorrelation[END_REF].

Length scale variation in percent

Conclusion

This paper presented an experimental procedure to perform tests with new quantities of interest (e.g., SIF, crack tip position) in order to validate models and calibrate their parameters. A virtual test-case has shown the feasibility of the described method by following a sinusoidal crack path on a concrete sample.

The loading was in tension, shear and in-plane rotation and crack propagation was simulated with a non-local damage model. The extracted boundary conditions enable the sensitivity of this crack path to be assessed relative to a more standard crack path for mixed mode crack propagation in quasi brittle materials. It has been shown that the crack path obtained with the proposed method was about two orders of magnitude more sensitive especially for the length scale identification.

The sinusoidal shape has been arbitrarily selected and was not optimised for identification purposes. The use of the proposed virtual method allows different crack paths to be analysed very quickly. An optimization step of the crack path for the identification of a specific parameter can be performed based on the presented framework [START_REF] Bertin | Optimization of a cruciform specimen geometry for the identification of constitutive parameters based upon full-field measurements[END_REF]. Last, achieving real 3rd generation tests is the next step of the present study to calibrate internal length scales of damage models, which are very difficult to estimate with current tests and procedures. The results are that the deflection of the crack is less smooth and creates bifurcation zones. However, for small increments, the control of the 'high level model' was performed by a simple proportional integral (PI) corrector (as a 575 qualitative illustration, see Figure 19). Such a control strategy provides an acceptable response for the main crack up to an inflection point of the trajectory. This result illustrates the benefit of having a low level model that is close enough to the expected ground truth.

Last, let us note that for the inversion of a non-linear control model, the implementation of a Gauss-Newton routine, for example, may solve the problem

Figure 1 :

 1 Figure 1: (a) Multiaxial testing machine (hexapod) with six degrees of freedom and (b) vertical displacement field on a concrete sample measured via DIC after having manually monitored

Figure 2 :

 2 Figure 2: Principle of 3rd generation tests

Figure 3 :

 3 Figure 3: Block diagram of 3rd generation tests

Figure 4 :

 4 Figure 4: Target point with the two proposed methods, (a) at a distance dr from the previous measured point, and (b) at a projected distance dx on the x-axis

Figure 5 :

 5 Figure 5: Example of a rotational loading stabilising crack propagation. The distance ∆x between the crack tip and the projected instantaneous centre of rotation (ICR) is introduced in the studied case

Figure 6 :

 6 Figure 6: Test-case geometry and possible loading for the sinusoidal crack path

Figure 7 :

 7 Figure 7: Block diagram of virtual 3rd generation tests

  340

Figure 8 :

 8 Figure 8: Fracture threshold (maximum normal stress criterion [50]) and influence of the boundary conditions

Figure 9 :

 9 Figure 9: Elastic domain (plane stress) shape in the bitensile quadrant of the experimental data for concrete[START_REF] Lee | Biaxial behavior of plain concrete of nuclear containment building[END_REF]

Figure 10 :

 10 Figure 10: Displacement field at the end of the virtual sine test. (a) Horizontal (x-axis), and (b) vertical (y-axis) components

Figure 11 :

 11 Figure 11: Comparison between (a) the measured crack path, (b) the target crack path, and (c) the input crack (never reached because of the step discretization)

Figure 12 :

 12 Figure 12: Maximum applied displacement at the top left and right corners of the sample vs. step number. (a) Tension on the top left corner (positive values) and on the top right corner, and (b) shear (uniform)

Figure 13 :

 13 Figure 13: Total crack length vs. step number

Figure 14 :

 14 Figure 14: Crack path sensitivity to the length scale parameter p

495

  Mohammed test. The material and behaviour are the same as in the sinusoidal test-case. A double notched sample is analysed. The most used crack path of all Nooru-Mohammed tests is the number 5. As in the real Nooru-Mohammed test, the load starts with shear up to 10 kN, then the sample is loaded with a tensile displacement while maintaining constant the shear force. This load leads 500 to a curved crack path in mixed mode I and II.

Figure 15 :

 15 Figure 15: Comparison of the 7 eigenvalues of [Hu] with a sinusoidal test and Nooru-Mohammed test

Figure 16 :

 16 Figure 16: Correlation matrix for (a) the Nooru-Mohammed test (b) the sinusoidal experiment

Figure 17 .

 17 Figure 17. The shape of the functional for the sinusoidal path is much steeper 540

χ

  UF Sinusoidal path χ UF Nooru-Mohammed path

Figure 17 :

 17 Figure 17: Weighted norm of the error χ U F as a function of the length of the process zone for the two test configurations

Figure 18 :

 18 Figure 18: Control loop of the crack path with a low level (PI) control model. Us is the shear displacement boundary condition, which is mainly responsible for the deflection of the crack, and ∆Us an elementary variation of the shear component

Figure 19 :

 19 Figure 19: Damage field (red is for the damage variable greater than 0.95) with multiple branching zones for a crack controlled with the 'low level' (PI control) model

Table 1 :

 1 Parameter values

	Parameter	p	ν	β	G f	E	γ	α
	Units	-	-	(MPa) (J/m 2 ) (GPa) (MPa) (MPa)
	Value	1.05-3 0.2	8.5	100	20	3.25	3.9
	Number	1	2	3	4	5	6	7

Table 2 :

 2 Parameters for the analysis of noise sensitivity

Acknowledgement

The support of CNRS (French National Centre for Scientific Research) for the CONCRET project is gratefully acknowledged.

Appendix A

To go along the same lines as Section 3 but exploring yet a cruder example of low level prediction, a poor version of the control model has been tested in the same spirit as the manual tests performed by Carpiuc et al. [START_REF] Carpiuc | A complex mixed-mode crack propagation test performed with a 6-axis testing machine and full-field measurements[END_REF]. Instead of working with an LEFM prediction, the control was performed by an incremental with successive updated sensitivities on the boundary conditions.