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International audience; We present and analyze two simple N-particle systems for the spread of an infection, respectively with binary and with multi-body interactions. We establish a convergence result, as N → ∞, to a set of kinetic equations, providing a mathematical justification of related numerical schemes. We analyze rigorously the time asymptotics of these equations, and compare the models numerically.

International audience; The equilibrium forms of pantographic blocks in a three-point bending test are investigated via both experiments and numerical simulations. In the computational part, the corresponding minimization problem is solved with a deformation energy derived by homogenization within a class of admissible solutions. To evaluate the numerical simulations, series of measurements have been carried out with a suitable experimental setup guided by the acquired theoretical knowledge. The observed experimental issues have been resolved to give a robust comparison between the numerical and experimental results. Promising agreement between theoretical predictions and experimental results is demonstrated for the planar deformation of pantographic blocks.

International audience

In this article we use a social representational perspective on a large sample of European students to consider the interplay between pacifist attitudes and representations of World War I (WWI). WWI gave rise to pacifist movements across the globe. Across 10 European countries (N = 1,347 undergraduate students), we invited participants to report the first 5 words that came to their mind when thinking about this event and measured their level of pacifism. Analyses of the reported words revealed the presence of seven lexical classes corresponding to 2 main perspectives on WWI. The first main perspective was characterized by "analytical" representations of the war, with a focus on the places and actors of WWI. The second perspective highlighted negative emotions and appraisals associated with war. Overall, we found that the pacifist attitudes we analyzed were associated with a view of the conflict in terms of negatively valanced words, both at the emotional level and in terms of concrete consequences. Conversely lower pacifist attitudes were linked with an emphasis upon weapons. The present results help fill a gap in the literature on attitudes toward peace and war by evidencing their interplay with the way individuals view war. These findings allow us to establish a connection between mainstream, individualcentered approach to attitudes relying on individual premises and social representation theory. Moreover, in line with historical scholarship, these results suggest that a common interpretational framework underlies a view of the conflict associated with the pacifist wave that emerged 100 years ago and current pacifist attitudes. Finally, the present study is the first large scale psychology study of the social representations of the Great War.

International audience; Environmental forensics assessments are often based on few analytical measurements. Biased data can, therefore, led the involved actors to be improperly charged or acquitted. Data reliability, considered in terms of uncertainty and variability, is determined by the analytical methods and the performed sampling procedures. The influence of this latter increases with highly heterogeneous materials characterized by the presence of trace, “rare” elements (both considered resources or contaminants), as occurring in the field of waste management. Nevertheless, the theoretical principles behind representative sampling are still not well understood in the waste community, which often underestimates its influence on the reliability of the resulting data and consequent decisions. Applying the binomial probability distribution, the theory of representative sampling is reviewed to facilitate its use for granular solid wastes. The concept of the number of particles n that shall compose a representative sample is introduced. Within n, the same fraction p shall be included of “rare” particles present in the lot to be characterized. Also, n depends on the minimum achievable variability of test results, i.e., coefficient of variation CV computable on a set of equivalent samples. Data are presented quantifying occurring intra and inter-laboratory variability of the most used chemical-physical and ecotoxicological analyses of homogeneous material. Accordingly, a specific formula to calculate n is proposed. Three requirements are then laid down to transform n into mass of sample. Representative samples must include a fraction p of particles considered “rare” in terms of i) presence/concentration of substances and ii) particles size distribution, since large waste particles are seldom characterized by the same composition of fine particles. Further, iii) the size of the sampling equipment shall prevent segregation of larger particles. The mass of a representative sample is calculated by reconciling these three conditions. According to the dedicated technical standards, the layout of a replication experiment is developed to measure the uncertainty arising from a sampling plan designed for waste classified with European Waste Code 191212 “other wastes including mixtures of materials from mechanical treatment of wastes”.Practical and design challenges that can affect the correct performance of the related experimental activities and the reliability of the outcomes are discussed.

International audience; We call nonlinear dilatational strain gradient elasticity the theory in which the specific class of dilatational second gradient continua is considered: those whose deformation energy depends, in an objective way, on the gradient of placement and on the gradient of the determinant of the gradient of placement. It is an interesting particular case of complete Toupin-Mindlin nonlinear strain gradient elasticity: indeed, in it, the only second gradient effects are due to the inhomogeneous dilatation state of the considered deformable body. The dilatational second gradient continua are strictly related to other generalized models with scalar (onedimensional) microstructure as those considered in poroelasticity. They could be also regarded to be the result of a kind of "solidification" of the strain gradient fluids known as Korteweg or Cahn-Hilliard fluids. Using the variational approach we derive, for dilatational second gradient continua the Euler-Lagrange equilibrium conditions in both Lagrangian and Eulerian descriptions. In particular, we show that the considered continua can support contact forces concentrated on edges but also on surface curves in the faces of piecewise orientable contact surfaces. The conditions characterizing the possible externally applicable double forces and curve forces are found and examined in detail. As a result of linearization the case of small deformations is also presented. The peculiarities of the model is illustrated through axial deformations of a thick-walled elastic tube and the propagation of dilatational waves. Keywords Strain gradient elasticity • Dilatational elasticity • Variational principle • Poroelasticity Communicated by Marcus Aßmus and Andreas Öchsner. V.A.E. and F.d'I. acknowledge the support by the Russian Science Foundation under Grant 20-41-04404 issued to the Institute of Applied Mechanics of Russian Academy of Sciences.

International audience; The derivation by variational asymptotic homogenization of a 2D-continuum model describing large elastic planar deformations of a discrete bi-pantographic structure is presented. A rectangular bi-pantographic specimen was additively manufactured and subjected to a bias extension test for macroscopic strains up to ca. 40%. The deformations of the bi-pantographic sample were measured via FE-based digital image correlation. Measured boundary conditions then drove the numerical model of the experiment. The gray level residuals were utilized to independently probe the kinematic hypotheses of DIC and FE simulations against the full video of the experiment for validation purposes.

International audience; We provide the proof of an existence and uniqueness theorem for weak solutions of the equilibrium problem in linear dilatational strain gradient elasticity for bodies occupying, in the reference configuration, Lipschitz domains with edges. The considered elastic model belongs to the class of so-called incomplete strain gradient continua whose potential energy density depends quadratically on linear strains and on the gradient of dilatation only. Such a model has many applications, e.g., to describe phenomena of interest in poroelasticity or in some situations where media with scalar microstructure are necessary. We present an extension of the previous results by Eremeyev et al. (2020 Z angew Math Phys 71(6): 1-16) to the case of domains with edges and when external line forces are applied. Let us note that the interest paid to Lipschitz polyhedra-type domains is at least twofold. First, it is known that geometrical singularity of the boundary may essentially influence singularity of solutions. On the other hand, the analysis of weak solutions in polyhedral domains is of great significance for design of optimal computations using a finite-element method and for the analysis of convergence of numerical solutions.

International audience; Metamaterials are often studied for their peculiar mechanical properties. However, few 4D studies were conducted on 3D printed pantographs. This study aims at analyzing an in situ torsion test in a lab tomograph. The acquired scans were used to measure displacement fields via digital volume correlation. The final goal was to analyze the deformation mechanisms of an Inconel pantograph and to rationalize potential Poynting effects.

International audience; Cosserat continuum model has attracted increasing interest for describing the mechanical behavior of microstructured solids. Existing formulation of Cosserat continuum model often overlooks chiral effects that arise from coupling between stretching deformations and the micro-rotation. Here, we introduce an extended Cosserat model that accounts for such a coupling. We discover the links of the extended model to the mechanics of granular materials and identify the continuum and grain-scale parameters that are the source of postulated chirality. The micro-rotation is shown to be related to the coupling of shear and normal responses of grain-pair interactions. The consequences of this chirality are then explicated with the aid of numerical examples. Through parametric studies we also demonstrate the possibility of measuring the effects of this type of chirality in experiments.

International audience; Among the most studied models in mathematical physics, Timoshenko beam is outstanding for its importance in technological applications. Therefore it has been extensively studied and many discretizations have been proposed to allow its use in the most disparate contexts. However, it seems to us that available discretization schemes present some drawbacks when considering large deformation regimes. We believe these drawbacks to be mainly related to the fact that they are formulated without keeping in mind the mechanical phenomena for describing which Timoshenko continuum model has been proposed. Therefore, aiming to analyze the deformation of complex plane frames and arches in elastic large displacements and deformation regimes, a novel intrinsically discrete Lagrangian model is here introduced whose phenomenological application range is similar to that for which Timoshenko beam has been conceived. While being largely inspired by the ideas outlined by Hencky in his renowned doctoral dissertation, the presented approach overcomes some specific limitations concerning the stretch and shear deformation effects. The proposed model is applied to get the solutions for some relevant benchmark tests, both in the case of arch and frame structures. It is proved that, also when shear deformation effects are of relevance, the enriched, yet simple, model and numerical computation scheme herein proposed can be profitably used for efficient structural analyses of non-linear mechanical systems in rather nonstandard situations.

The estimation of flood losses implies the use of vulnerability/exposure models for flood damage and risk assessment (Meyer et al. 2013; Zischg et al. 2018; Wagenaar et al. 2018; Molinari et al. 2019). According to Gerl et al. (2016), in central Europe there are 28 models with 652 functions to assess flood losses, whereas almost half of the functions refer to residential buildings. The main differences among damage models are: (i) spatial scale, (ii) metric (i.e. absolute or relative loss), (iii) exposure assessment (i.e. whole building or affected floors, market values or reconstruction-replacement costs), (iv) number of input variables, (v) calibration/validation with observed losses. These differences pose the issue of model transferability to other urban or environmental contexts. With a joint effort of eight research groups, the objective of this study is to test and compare damage models for residential buildings used or developed by each group, by applying them in a “blind exercise”, i.e. a common flood case study characterised by high availability of hazard and building data, with unknown information on observed losses in the implementation stage of the models. As the research groups use approaches representing many different types and characteristics of models (e.g. univariable – multivariable; absolute – relative; graduated – regression – machine learning), being calibrated based on empirical data stemming from different countries (Austria, France, Germany, Italy, Japan, Netherlands), with different landscapes, the blind exercise provides an extensive comparison of models and their transferability. The analysis of the differences in the ensemble of model outcomes aims at pointing out common patterns or divergent behaviours between model outcomes and with respect to observed losses.

International audience; The nonlinear dynamics of a horizontal, initially taut, elastic string, subjected to a traveling point-mass driven by an assigned time-dependent force, is studied. A kinematically exact nonlinear model for the free boundary problem, previously derived by the authors through a variational procedure, is resumed. The model is developed in the framework of a mixed formulations, ruling the time-evolution of the configuration variables and the reactive contact forces. The model is then consistently simplified, via a static condensation of the horizontal displacements of the string. Differently from most of papers in literature, here the dynamic tension is accounted in the equations of motion, as a nonlinear effect. The simplified equations are discretized by the Galerkin method, which leads to a set of ordinary differential equations in the amplitudes of the trial functions and in the unknown abscissa of the point-mass. As a further simplified model, the mass of the string is neglected, and a set of two ordinary differential equations in the coordinates of the point-mass is derived. Numerical results are obtained, aimed at exploring the influence of the mass of the string and of the dynamic tension. The focus is on analyzing peculiar ‘back and forth’ motions of the point-mass, when this is driven by small or even zero forces.

International audience; In the last decade the exotic properties of pantographic metamaterials have been investigated and different mathematical models (both discreteor continuous) have been introduced. In a previous publication a large part of the already existing literature about pantographic metamaterials has been presented. In this paper we give some details about the next generation of research in this field. We present an organic scheme of the whole process of design, fabrication, experiments, models and image analyses.

International audience; Tullio Levi-Civita (29 March 1873 to 29 December 1941) has been an Italian mathematician and mathematical physicist, known above all for his work on the absolute differential calculus. Levi-Civita came from a rigorous and creative school of mathematical physicists and was a pupil of Gregorio Ricci-Curbastro. LeviCivita’s work included outstanding results in pure and applied mathematics and in celestial and analytic mechanics but also celebrated textbooks. These last, even those written in Italian, have influenced mathematical physicists all over the world.Levi-Civita has perfected some conceptual tools of great importance in modern science, particularly in general relativity, number theory, and continuum mechanics.

International audience; There is a class of planar 1D-continua which can be described exclusively by their placement functions which in turn are curves in a two-dimensional space. In contrast to the Elastica for which the deformation energy depends on the projection of the second gradient to the normal vector of the placement function, i.e. the material curvature, the proposed continuum does also depend on the projection onto the tangent vector, introduced as the stretch gradient. Thus, the deformation energy takes into account the complete second gradient of the placement function. In such a model, non-standard boundary conditions and more generalized forces such as double forces do appear. The deformation energy of the continuum is obtained by applying a heuristic homogenization procedure to a family of slender discrete pantographic structures constituted by extensional and rotational springs. Within the homogenization process, the overall length of the system is kept fixed, the number of the periodically appearing sub-systems, called cells, is increased, and the stiffnesses are appropriately scaled. For two examples, we numerically compare the family of discrete systems with the continuum. The analysis shows that the continuum represents the behaviour of the discrete system already for a relatively moderate number of cells. In particular, the behaviour of the deformation energy error between the discrete and the continuum models when the number of cells tends to infinity is determined by the homogenization process.

International audience; In this paper, we account for the research efforts that have been started, for some among us, already since 2003, and aimed to the design of a class of exotic architectured, optimized (meta) materials. At the first stage of these efforts, as it often happens, the research was based on the results of mathematical investigations. The problem to be solved was stated as follows: determine the material (micro)structure governed by those equations that specify a desired behavior. Addressing this problem has led to the synthesis of second gradient materials. In the second stage, it has been necessary to develop numerical integration schemes and the corresponding codes for solving, in physically relevant cases, the chosen equations. Finally, it has been necessary to physically construct the theoretically synthesized microstructures. This has been possible by means of the recent developments in rapid prototyping technologies, which allow for the fabrication of some complex (micro)structures considered, up to now, to be simply some mathematical dreams. We show here a panorama of the results of our efforts (1) in designing pantographic metamaterials, (2) in exploiting the modern technology of rapid prototyping, and (3) in the mechanical testing of many real prototypes. Among the key findings that have been obtained, there are the following ones: pantographic metamaterials (1) undergo very large deformations while remaining in the elastic regime, (2) are very tough in resisting to damage phenomena, (3) exhibit robust macroscopic mechanical behavior with respect to minor changes in their microstructure and micromechanical properties, (4) have superior strength to weight ratio, (5) have predictable damage behavior, and (6) possess physical properties that are critically dictated by their geometry at the microlevel.

International audience; Recent advanced manufacturing techniques such as 3D printing have prompted the need for designing new multiscale architectured materials for various industrial applications. These multiscale architectures are designed to obtain the desired macroscale behavior by activating interactions between different length scales and coupling different physical mechanisms. Although promising results have been recently obtained, the design of such systems still represents a challenge in terms of mathematical modeling, experimentation, and manufacturing. In this paper, some research perspectives are discussed aiming to determine the most efficient methodology needed to design novel metamaterials. A multidisciplinary approach based on Digital Image Correlation (DIC) techniques may be very effective. The main feature of the described DIC-based approach consists of the integration of different methodologies to create a synergistic relationship among the different steps from design to fabrication and validation. Experimental techniques and modeling approaches are envisioned to be combined in feedback loops whose objective is to determine the required multiscale architectures of newly designed metamaterials. Moreover, it is necessary to develop appropriate mathematical models to estimate the behavior of such metamaterials. Within this new design approach, the manufacturing process can be effectively guided by a precise theoretical and experimental framework. In order to show the applicability of the proposed approach, some preliminary results are provided for a particular type of mechanical metamaterial, namely, pantographic metamaterials. Lastly, the most relevant challenges are highlighted among those that must be addressed for future applications.

The present study examines current social representations associated with the origins of the Great War, a major event that has profoundly affected Europe. A survey conducted in 20 European countries (N = 1906 students in social sciences) shows a high consensus: The outbreak of the war is attributed to the warring nations’ leaders while the responsibility of the populations is minimized. Building on the concept of social representation of history (Liu and Hilton, 2005), we suggest that the social representations of the Great War fulfill social psychological functions in contemporary Europe. We suggest that WWI may function as a charter for European integration. Their content also suggests a desire to distinguish a positively valued ingroup ("the people") from powerful elites, construed as an outgroup., SCOPUS: ar.j, info:eu-repo/semantics/published

International audience; Tullio Levi-Civita (29 March 1873 to 29 December 1941) has been an Italian mathematician and mathematical physicist, known above all for his work on the absolute differential calculus. Levi-Civita came from a rigorous and creative school of mathematical physicists and was a pupil of Gregorio Ricci-Curbastro. LeviCivita’s work included outstanding results in pure and applied mathematics and in celestial and analytic mechanics but also celebrated textbooks. These last, even those written in Italian, have influenced mathematical physicists all over the world.Levi-Civita has perfected some conceptual tools of great importance in modern science, particularly in general relativity, number theory, and continuum mechanics.

Part of the Advanced Structured Materials book series (STRUCTMAT, volume 87); International audience; In the standard asymptotic micro-macro identification theory, starting from a De Saint-Venant cylinder, it is possible to prove that, in the asymptotic limit, only flexible, inextensible, beams can be obtained at the macro-level. In the present paper we address the following problem: is it possible to find a microstructure producing in the limit, after an asymptotic micro-macro identification procedure, a continuum macro-model of a beam which can be both extensible and flexible? We prove that under certain hypotheses, exploiting the peculiar features of a pantographic microstructure, this is possible. Among the most remarkable features of the resulting model we find that the deformation energy is not of second gradient type only because it depends, like in the Euler beam model, upon the Lagrangian curvature, i.e. the projection of the second gradient of the placement function upon the normal vector to the deformed line, but also because it depends upon the projection of the second gradient of the placement on the tangent vector to the deformed line, which is the elongation gradient. Thus, a richer set of boundary conditions can be prescribed for the pantographic beam model. Phase transition and elastic softening are exhibited as well. Using the resulting planar 1D continuum limit homogenized macro-model, by means of FEM analyses, we show some equilibrium shapes exhibiting highly non-standard features. Finally, we conceive that pantographic beams may be used as basic elements in double scale metamaterials to be designed in future.

International audience; We present a geological map at 1:25,000 scale of the Plio-Quaternary L'Aquila-Scoppito intermontane basin (central Italy), which corresponds to the epicentral area of the 6th April 2009 L'Aquila earthquake (Mw: 6.29). The map is derived from geological field surveys at 1:5000 scale and takes into account previously published maps. It is supported by a fine-scale LiDAR digital elevation model, paleontological analyses, 14 C dating, well log analyses of deep boreholes, and geophysical data interpretation. By taking into account that the remarkable historical and present-day seismicity of the area is the result of its Plio-Quaternary geological evolution, the aim of this study is to produce a fine-scale geological map of the study area through a comprehensive analysis of the occurring Plio-Quaternary synthems and the tectonic processes that are taking place within it.

International audience; The combined effects of conservative and non-conservative loads on the mechanical behavior of an unshearable and inextensional visco-elastic beam, close to bifurcation, are investigated. The equations of motion and boundary conditions are derived via a constrained variational principle, and the Lagrange multiplier successively condensed, to get integro-differential equations. These latter, with the mechanical boundary conditions appended, are put in an operator-form, amenable to perturbation analysis. A linear stability analysis is carried out in the space of the two loading parameter, displaying the existence of codimension-1 and codimension-2 bifurcations. The influence of both internal and external damping on this scenario is thoroughly investigated. A post-critical analysis is carried out around a double-zero bifurcation, by using an adapted version of the multiple scale method, based on fractional series expansions in the perturbation parameter. The integro-differential problem is directly attacked, so that any a priori discretization is avoided. Emphasis is given to the interaction between the two damping coefficients. This reveals the existence, also in the non-linear range, of a phenomenon of destabilization, so far known only in the linear range.

International audience; Study of the tectonically active L'Aquila Basin offers new insights into both the creation of the extensional intermontane basins of the central Apennines of Italy and their tectono-sedimentary evolution through time. The combination of large mammal remains, ostracods, molluscs, Mousterian tools, and 14 C dating allows better definition of the onset and stratigraphic evolution of the L'Aquila Basin. Interpretation of a seismic reflection profile and well-log data allow evaluation of the subsurface setting of this sedimentary basin and its tectono-sedimentary evolution. The occurrence of a wedge-shaped seismic unit at the base of the basin sedimentary succession defines the first phase of basin fill during a late Piacenzian–Gelasian synrift stage. Activity along the main fault of the extensional fault system responsible for the onset and subsequent development of the western sector of the L'Aquila Basin (L'Aquila– Scoppito subbasin) migrated from southwest to northeast, reaching the presently active Mount Pettino normal fault only in the late Pleistocene–Holocene. The onset of sedimentation in the L'Aquila Basin was synchronous with the onset in the Tiberino Basin, and so the idea that these extensional GSA Bulletin; Month/Month 2017; v. 129; no. intermontane basins become progressively younger from the Tyrrhenian toward the Adriatic side of the central Apennines is rejected. In the northern and central Apen-nines, only two major syndepositional ex-tensional domains can be recognized: a late Miocene rifting area, which includes all the late Miocene extensional basins in Tuscany, and a late Pliocene to earliest Pleistocene rifting area, which possibly includes all the intermontane basins from the Tiberino Basin to the Sulmona Basin. The different time gaps between compressional and exten-sional deformation at any given locality in the central Apennines could indicate a partial decoupling of processes responsible for the migration of shortening and extension toward the foreland. Diachroneity between the eastward migration of shortening in the foreland and extension in the inner part of the orogen supports the notion that the central Apennines were created as a result of a partially decoupled collision zone. Study of the onset of the central Apennine extensional intermontane basins, together with their seismic activity, indicates that the central Apennine postorogenic extensional domain represents an archive of ~3 m.y. of continued crustal extension. These findings help to refine models of the long-term extensional rate of the central Apennines, and they provide a basis for more reliable seismotectonic models for one of the most seismically active sectors of the central Mediterranean area.

The present study addresses antecedents and consequences of collective victimhood in the context of World War I (WWI) across 15 European nations (N = 2423 social science students). Using multilevel analysis, we find evidence that collective victimhood is still present a hundred years after the onset of the war and can be predicted by WWI-related objective indicators of victimization at national and family levels. This suggests that collective victimhood is partly grounded in the actual experience of WWI. In addition, we show that sense of collective victimhood positively predicts acknowledgment of the suffering inflicted by one's nation on other countries during WWI. This is consistent with a social representation of WWI as involving a vast massacre in which nations were both victim and perpetrator. Finally, we find that objective indicators of victimization predict pacifism in divergent ways, with an indicator at the national level associated with more pacifist attitudes and an indicator at the family level being associated with less pacifist attitudes. This finding suggests that war-torn societies may have developed social representations favouring peaceful coexistence whereas, at the family level, victimization may still foster retaliatory tendencies., SCOPUS: ar.j, info:eu-repo/semantics/published

should write all the equations in scalar form and (inelegantly) drawa trivial bifurcation diagram x � 0 in one parameter. As a second strategy, also followed by some users, one could 1) expand the (parameter dependent) characteristic polynomial of the Jacobian matrix,leadingtotwoscalarequationsintherealandimaginaryparts oftheeigenvalue,inwhichtherealpartmustbevanished;2)consider these equations as the right-hand members of a fictitious dynamical system, in which two parameters are taken as dummy state variables and the imaginary part of the eigenvalue is taken as a dummy parameter; and 3) look for equilibrium paths for the dummy system, actually representing the bifurcation loci for the real system. If this procedure is employed, some of the bifurcation points found (e.g., Hopf points) could have no meaning for the true system. Although both of these procedures work, they are highly unsatisfactory from a formal point of view, since they are essentially based on tricks; moreover, theyare also unsatisfactory from a practical point of view, since they require handling a large number of data (scalar equations

International audience; This paper proposes a continuous perturbation treatment of the nonlinear equations of a cable, which is characterized by possible aeroelastic instability and internal resonance conditions. The objective is to evaluate the influence of stable modes (called passive) usually ignored when discretizing the model. The first step concerns the description of the structural equilibrium path under the action of the mean wind forces. Then, a multiple scale perturbation analysis of the integro-differential equations of motion is performed. Analyzing the stability of the reduced system, the existence of some limit cycles and of successive bifurcations is investigated. The comparison with previous papers, developed in the discrete field, allows clarification of the actual influence of the static equilibrium path and the contribution of passive modes.

International audience; A linearized coupled flexural two degree-of-freedom model, describing a lumped parameter system subjected to galloping, is analyzed. Through a perturbation approach, an approximated analytical solution for the eigenvalue problem is determined. Differently from the expressions existing in literature, the eigensolutions found here are valid both in quasi-resonant and non-resonant conditions. Discussing them allows depiction of the scenario of all the possible bifurcation mechanisms in the plane of the invariants of the aerodynamic damping matrix. In resonance conditions, both simple and double Hopf bifurcations are found, otherwise only simple Hopf bifurcations (eventually sequential) occur. In any case, both mono-modal and bimodal galloping can take place. A closed form expression for the critical velocity is derived; it coincides with the exact solution in the resonant case and presents very good agreement with the numerical solutions in quasi-resonant conditions. The critical velocities are compared with the Den Hartog velocity and the influence of the horizontal motion is thus evaluated. Numerical examples concerning technical cases highlight the accuracy of the proposed method.

International audience; Chains of nonlinear shear indeformable beams with distributed mass, resting on movable supports, are considered. To determine the dynamic response of the system, the transfermatrix approach is merged with the harmonic balance method and a perturbation method, thereby transforming the original space-temporal continuous problem into a discrete one-dimensional map Xk+1=F(Xk) expressed in terms of the state variables xk at the interface between adjacent beams. Such transformation does not imply any discretization, because it is obtained by integrating the single-element field equations. The state variables consist of both first-order variables, namely, transversal displacement and couples, and second-order variables, which are longitudinal displacement and axial forces. Therefore, while the linear problem is monocoupled, the nonlinear one becomes multicoupled. The procedure is applied to determine frequency-response relationship under free and forced vibrations.

International audience; A modified version of the traditional wave vector computational scheme for the dynamic analysis of long undamped periodic structures is presented. First, the consistency of the complex wave vector mathematical formulation is discussed, placing particular emphasis on the real or complex nature of the resulting characteristic equation from which the natural frequencies are derived. It is shown that the rearrangement in terms of complex waves entering the domain, devised to overcome ill-conditioning arising in the transfer matrix formulation, entails as a side effect an ill-posed problem, as it leads to a complex characteristic equation in a real unknown. Next, the proposed approach is described. It provides for transformation of frequency-dependent real transfer matrices for state vectors to real transfer matrices for wave vectors, thus avoiding an ill-conditioned and ill-posed problem. Finally, applications to mono-coupled and bi-coupled structures are illustrated, aiming at comparing the proposed method with the transfer matrix and complex wave vector approaches

International audience; Classes of motion of general multiresonant systems are derived through a geometrical algorithm based on a set representation. First, the elementary classes existing under simple resonance conditions are evaluated; rules governing the interaction between elementary classes belonging to different resonance conditions are then drawn up as applications of a unique theorem. Illustrative examples are given. The method also permits a hierarchical ordering of the amplitudes of the resonant modes, according to their participation in the classes; it also makes it possible to ascertain in advance the existence of a standard form for the amplitude modulation equations. The stability analysis of incomplete steady solutions is then addressed. Three classes of perturbation are distinguished, namely: in-class perturbations, out-of-class resonant perturbations, and out-of-class nonresonant perturbations. The structure of the Jacobian variational matrix is studied. The Jacobian matrix is shown to comprise three diagonal blocks associated with the three perturbation classes, so that the stability equations are uncoupled. Further possible uncouplings of one of the blocks are analyzed in relation to some of the geometrical properties of the classes.

International audience; A general multiresonant system is considered, in which the linear frequency and, possibly, a forcing frequency are involved in a set of linear conditions. The nature of the resonances is first discussed, by distinguishing independent and dependent equations, and both the analysis and design problems of the system are addressed. Rules are then given to construct the qualitative form of the AMEs to any desired order. Two families of terms are identified: improper resonant terms (not associated with any resonance conditions) and proper resonant terms (depending on the specific conditions), sub-divided into primary (of lower order) and secondary (of higher order). Theorems are proved to show that both improper and secondary resonant terms have no qualitative but only quantitative effects on classes of motion; reference is therefore made to reduced equations. An algebraic algorithm is illustrated to determine classes of motion, using only the integer resonance coefficients. The concept of degree of constraint of a given resonance condition is introduced, entailing a hierarchic order among the resonance conditions, the implications of which are discussed. Finally, some numerical simulations are shown to test the robustness of classes of motion to higher-order terms not accounted for in the asymptotic analysis.

International audience; The effects of a class of nonlinear Tuned Mass Dampers on the aeroelastic behavior of SDOF systems are investigated. Unlike classical linear TMDs, nonlinear constitutive laws of the internal damping acting between the primary oscillator and the TMD are considered, while the elastic properties are keept linear. The perturbative Multiple Scale Method is applied to derive a set of bifurcation equations in the amplitude and phase and a parametric analysis is performed to describe the postcritical scenario of the system. Both cubic- and van der Pol-type dampings are considered and the dependence of the limitcycle amplitudes on the system parameters is studied. These new results, compared with the previously obtained bifurcation scenario of a SDOF aeroelastic oscillator equipped with a linear TMD, show a detrimental effect on the maximum limit-cycle amplitude reduction of the nonlinear TMD. However, the analyses evidence that in the parameter region away from the perfect tuning condition the nonlinear connection can be used to tune the system with an enhancement of the limit-cycle amplitude reduction.

International audience; This paper proposes an approximate explicit probability density function for the analysis of external and parametric oscillation of a simply supported beam driven by random pulses. The impulsive loading model adopted is Poisson white noise, which is a process having Dirac delta occurrences with random intensity distributed in time according to Poisson's law. The response probability density function can be obtained by solving the related Kolmogorov-Feller integro-differential equation. An approximate solution is derived by transforming this equation to a first-order partial differential equation. The method of characteristics is then applied to obtain an explicit solution. The theory has been validated through numerical simulations.

International audience; The effects of a tuned added mass on the aeroelastic stability of a single degree of freedom bluff body exposed to a steady flow are investigated. The model captures the essential aspects of the behaviour of flexible structures equipped with Tuned Mass Dampers undergoing galloping oscillations. The system exhibits simple as well double Hopf bifurcations, of non-resonant and 1:1 resonant type. Postcritical behaviour of the system in the neighbourhood of the 1:1 resonant type bifurcation is investigated. Employing the Multiple Scale Method, a second order bifurcation equation in the complex amplitude of motion is obtained. Analytical solutions are used to describe the bifurcation scenario in the cases of both undercritical and supercritical aerodynamic behaviour of the bluff body. The effectiveness of the Tuned Mass Damper even in the postcritical range is proved.

International audience; General bi-coupled periodic systems are dealt with by means of transfer matrices of single units. The solutions of the associated characteristic equation are discussed in terms of invariant quantities by exploiting the well-known reversibility of its coefficients. An exhaustive description of the free wave propagation patterns is given on the invariant plane where propagation domains with qualitatively different character are identified. The asymptotic behavior of the roots of the characteristic equation when the invariants tend to infinity is analyzed. The contour plot of the real part of the propagation constants, responsible for the amount of attenuation of the characteristic waves, is illustrated on the invariants' plane. Next, several models of bi-coupled periodic structures made up of beams resting on elastic supports are considered. A non-linear mapping from the invariants' plane to the physical parameters plane provides a concise representation of the pattern of the propagation domains. A mechanical interpretation associated with the boundaries of these regions is given. Finally, the proper'selection of the physical parameters governing the propagation modes is discussed.

International audience; It has been known since the pioneering works by Piola, Cosserat, Mindlin, Toupin, Eringen, Green, Rivlin and Germain that many micro-structural effects in mechanical systems can be still modeled by means of continuum theories. When needed, the displacement field must be complemented by additional kinematical descriptors, called sometimes microstructural fields. In this paper, a technologically important class of fibrous composite reinforcements is considered and their mechanical behavior is described at finite strains by means of a second gradient, hyperelastic, orthotropic continuum theory which is obtained as the limit case of a micromorphic theory. Following Mindlin and Eringen, we consider a micromorphic continuum theory based on an enriched kinematics constituted by the displacement field u and a second order tensor field ψ describing microscopic deformations. The governing equations in weak form are used to perform numerical simulations in which a bias extension test is reproduced. We show that second gradient energy terms allow for an effective prediction of the onset of internal shear boundary layers which are transition zones between two different shear deformation modes. The existence of these boundary layers cannot be described by a simple first gradient model and its features are related to second gradient material coefficients. The obtained numerical results, together with the available experimental evidences, allow us to estimate the order of magnitude of the introduced second gradient coefficients by inverse approach. This justifies the need of a novel measurement campaign aimed to estimate the value of the introduced second gradient parameters for a wide class of fibrous materials.

International audience; A general, multiparameter system admitting a double-zero eigenvalue at a critical equilibrium point is considered. A sensitivity analysis of the critical eigenvalues is performed to explore the neighborhood of the critical point in the parameter space. Because the coalescence of the eigenvalues implies that the Jacobian matrix is defective (or nilpotent), well-suited techniques of perturbation analysis must be employed to evaluate the eigenvalues and the eigenvector sensitivities. Different asymptoticmethods are used, based on perturbations both of the eigenvalue problem and the characteristic equation. The analysis reveals the existence of a generic (nonsingular) case and of a nongeneric (singular) case. However, even in the generic case, a codimension-1 subspace exists in the parameter space on which a singularity occurs. By the use of the relevant asymptotic expansions, linear stability diagrams are built up, and different bifurcationmechanisms (divergence-Hopf, double divergence, double divergence-Hopf, degenerate Hopf) are highlighted.The problem of Ž nding a unique expression uniformly valid in the whole space is then addressed. It is found that a second-degree algebraic equation governs the behavior of the critical eigenvalues. It also permits clariŽ cation of the geometrical meaning of the unfolding parameters, which has been discussed in literature for the Takens-Bogdanova bifurcation. Finally, a mechanical system loaded by nonconservative forces and exhibiting a double-zero bifurcation is studied as an example.

International audience; The multiple time-scale method is adapted to study the post-critical behavior of general non-conservative symmetric systems, possibly affected by imperfections, for which divergence and Hopf bifurcations interact. The procedure illustrated makes it possible to elude the computational burden related to the application of the center manifold reduction. It also furnishes explicit expressions of the coefficients of the standard normal form bifurcation equations in terms of the coefficients of the original system. As an example, the method is applied to a two-degree-of-freedom rigid bar subjected to axial load (Augusti's model) and transversal flow. The critical and post-critical scenarios are analyzed in detail, for both the perfect and imperfect systems.

The issue to passively control aeroelastic instability of general nonlinear multi-d.o.f. systems, suffer- ing Hopf bifurcation, is addressed. The passive device consists of an essentially nonlinear oscillator (Nonlinear Energy Sink), having the task to transfer energy from the main to the secondary structure. The mathematical problem is attacked by a new algorithm, based on a suitable combination of the Multiple Scale and the Harmonic Balance Methods. The procedure is able to furnish the codimension-2 invariant manifold on which the motion occurs, thus revealing the passive character of the oscillations of the NES. The algorithm also provides the bifur- cation equations, which govern the slow flow on the manifold, expressed in terms of the main structure amplitude and phase of motion. It is shown that NES, under suitable conditions, can shift forward the bifurcation point, and, moreover, it can reduce the amplitude of the limit cycles. Theory is applied to a sample structure already studied in literature, consisting of a two-d.o.f. rigid wing under steady wind. Relevant asymptotic results are compared, for validation purposes, with numerical simulations.

Several algorithms consisting in ‘non-standard’ versions of the Multiple Scale Method are illustrated for ‘difficult’ bifurcation problems. Preliminary, the ‘easy’ case of bifurcation from a cluster of distinct eigenvalues is addressed, which requires using integer power expansions, and it leads to bifurcation equations all of the same order. Then, more complex problems are studied. The first class concerns bifurcation from a defective eigenvalue, which calls for using fractional power expansions and fractional time-scales, as well as Jordan or Keldysh chains. The second class regards the interaction between defective and non-defective eigenvalues. This problem also requires fractional powers, but it leads to differential equations which are of a different order for the involved amplitudes. Both autonomous and parametrically excited non-autonomous systems are studied. Moreover, the transition from a codimension-3 to a codimension-2 bifurcation is explained. As a third class of problems, singular systems possessing an evanescent mass, as Nonlinear Energy Sinks, are considered, and both autonomous systems undergoing Hopf bifurcation and non-autonomous systems under external resonant excitation, are studied. The algorithm calls for a suitable combination of the Multiple Scale Method and the Harmonic Balance Method, the latter is applied exclusively to the singular equations. Several applications are shown, to test the effectiveness of the proposed methods. They include discrete and continuous systems, autonomous, parametrically and externally excited systems.

International audience; A nonlinear model of visco-elastic balloon, interposed between a couple of moving rigid bodies, is formulated. The pneumatic structure is modeled as a thin, infinitely long cylindrical membrane, pre-stressed by an internal pressure. External pressure is assumed to be zero. The pushing bodies are modeled as a couple of frictionless rigid parallel plates, approaching each other normally, and causing squeezing of the pneumatic body. Motion is assumed to be slow, in such a way that any inertial effects are negligible. Both cases of long and short plates are considered, the latter entailing the possibility of puncture of the deformable body. A thermoplastic polyurethane material behavior is considered, and proper constitutive relationships adopted. Several models are formulated, which differ for constitutive laws (inextensible, elastic, linearly or non linearly visco-elastic) and/or for kinematic description (small strains and displacements or finite kinematics). The governing mixed differential-algebraic equations are analytically or numerically integrated for several impressed motion time-histories and the main features of the phenomenon are investigated.

This paper discusses the computational problems that arise in the application of higher-order multiple scale methods to general multiresonant multiparameter systems. Complex amplitude equations are first analytically derived and their structure analyzed solely in the light of existing resonance conditions. A qualitative study of the complex amplitude equations is performed, using both an analytical and a geometrical approach able to furnish a complete description of the classes of motion admitted by the system and of the structure of the variational matrix governing their stability. The problem of representing the amplitude equations using real quantities is then addressed. It is shown that the use of a mixed polar-Cartesian form leads to a standard form of equation if suitable hypotheses are satisfied. This form makes it possible to overcome problems arising in the stability analysis of (incomplete) classes of motion when the most popular polar form is used. The final section discusses the reconstitution method, consisting in alternative techniques that can be used to combine the amplitude equations on different scales into a unique equation on a single scale. Four classes of method are considered, based on the consistency or inconsistency of the approach and on the completeness or incompleteness of the terms retained in the analysis. The four methods are critically compared and general conclusions drawn. Examples are given to clarify all the procedures discussed.

International audience; The paper discusses the nonlinear free dynamics of an orbiting string satellite system. The focus is on the transversal oscillations, which are governed by two partial integro-differential equations in two transversal displacement components with quadratic nonlinearities. The system is weakly nonlinear but in practice works in conditions of simultaneous internal resonance. The investigation focuses on nonstationary motions arising from perturbed steady-state nonplanar oscillations. A four-mode model is used to study the problem: two modes are necessary to describe the basic oscillation and at least two other modes are involved in the resonance phenomena when the motion is perturbed. The multiple time scales method is used to obtain the equations that govern the amplitude and phase modulations. For increasing levels of system energy, fundamental and bifurcated paths of fixed points of the seven first-order differential equations are determined and their stability is investigated. The trajectories of motion of periodically modulated amplitude solutions and their stability are also studied. A model with a higher number of modes is used to evaluate the accuracy of the stability analysis of two-mode nonplanar oscillations perturbed by a two-mode disturbance.

International audience; Several new constitutive models are formulated for the planar interface of a soft body sliding on a rigid soil, describing stick-slip phenomena due to friction, and wear due to abrasion. Attention is focused on damage at the interface, by neglecting any interaction with damage of body and any propagation of damage inside the body. Models are formulated in the general framework of the Thermodynamics of the irreversible processes and account for suitable defined internal variables of phenomenological type, namely gap, isotropic friction hardening and wear. The main feature of the new presented models is that the formulation of the wear process at the interface is obtained in the contest of Damage Mechanics, and it is based on the formal analogy between abrasion of a soft body and ductile damage of an elastic-plastic material. By following this approach, a scalar wear field, an effective stress and appropriate state and dissipation potentials are defined, able to describe a tangential isotropic wear process due to stick-slip and to hardening mechanism. Both cases of linear and nonlinear friction hardening are formulated; moreover, wearable and no-wearable bodies are considered. Numerical results relevant to one-dimensional problems are illustrated for monotonic, forward-backward and cyclic displacement time-histories, showing evolution for stress, gap and wear. Results furnished by different models are compared and discussed.

International audience; An asymptotically exact method for static and dynamic analysis of geometrically nonlinear planar frames is illustrated. The method is based on an integration of the nonlinear equations for the beam, carried out via a perturbation method, aiming to express the forces at the ends as series expansion of the displacements at the ends and of the distributed loads. Since the beams are assumed to be inextensible and unshearable, also reactive stresses appear among the unknowns, while compatibility conditions must be appended to the equilibrium equations. The element state-relations are assembled for the frame, and discrete, nonlinear perturbation equations are derived. Examples are worked out and relevant results compared with purely numerical solutions.

International audience; A general, nonlinear, multi-d.o.f. structure, excited by harmonic external force in 1:1 resonance with one of the modes of the system, is considered. The structure is attached to an essentially nonlinear oscillator, with small mass and damping (Nonlinear Energy Sink, NES). The scope of the NES is to passively control the amplitude of vibrations of the main structure. A mixed Multiple Scale/Harmonic Balance Method (MSHBM) is proposed to get the differential equations describing the slow- and fast-flow dynamics of the whole structure. The main advantage of the procedure is that no complexification-averaging is required, so that the analysis is reconducted in the framework of the classical perturbation techniques.

International audience; A two-d.o.f. system experiencing codimension-three double-zero/Hopf bifurcation is considered. This is a special bifurcation which simultaneously involves a defective and a nondefective pair of critical eigenvalues, therefore, requiring a perturbation method specifically tailored on it. A nonstandard version of the multiple scale method is implemented, in which fractional power expansions, both for state-variables and time are used, and high-order arbitrary amplitudes are introduced. Bifurcation equations are obtained, governing the slow flow on the center manifold, which turns out to be tangent to the space spanned by the four critical eigenvectors. These are used to analyze the transition from codimension-three to codimension-two single-zero/Hopf bifurcations, occurring when the modulus of the damping is increased from small to order-one values. Bifurcation charts are obtained, displaying the role of quasi-periodic motions in the transition.

International audience; A two-degree of freedom nonlinear system, describing the dynamics of two towers exposed to turbulent wind flow and linked by a nonlinear viscous device, is considered. The steady component of the wind is responsible for self-excitation, while the turbulent part causes both parametric and external excitations, considered in a specific resonance condition. The possible occurrence of multiple Hopf bifurcations is taken into account. The periodic and quasi-periodic solutions are studied after applying a perturbation scheme. The effect of the viscous device on the dynamics of the structure is analyzed, in order to investigate its effectiveness in mitigating the oscillations of the two independent towers.

International audience; The mechanical behavior of a non-conservative non-linear beam, internally and externally damped, undergoing codimension-1 (static or dynamic) and codimension-2 (double-zero) bifurcations, is analyzed. The system consists of a purely flexible, planar, visco-elastic beam, fixed at one end, loaded at the tip by a follower force and a dead load, acting simultaneously. An integro-differential equation of motion in the transversal displacement, with relevant boundary conditions, is derived. Then, the linear stability diagram of the trivial rectilinear configuration is built-up in the space of the two loading parameters. Attention is then focused on the double-zero bifurcation, for which a post-critical analysis is carried out without any a-priori discretization. An adapted version of the Multiple Scale Method, based on a fractional series expansion in the perturbation parameter, is employed to derive the bifurcation equations. Finally, bifurcation charts are evaluated, able to illustrate the system behavior around the codimension-2 bifurcation point.