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The main characters of this chapter will be Yeroen, Nikkie, Luit, Mama and Kanzi, chimpanzees’and bonobos' dictators whose behaviour has been described by Frans De Waal. One could believe that what chimpanzees and bonobos are doing is not at all interesting for us. Instead, as De Waal is explaining so clearly in all his books, the observation of their behaviour is extremely useful to unmask the true nature of human behaviour and in particular to show the true nature of human dictators.

In this chapter we show how Arrow's Theorem can be applied to understand the structure of social groups and many biological, psycological and social phenomena.

In this chapter the main ideas underlying Condorcet's Conjecture and Arrow's Theorem are presented and some first applications to social sciences are discussed. The mathematical results are presented in an informal way to make the discussion accessible to the layman.

International audience; A three-dimensional continuum theory for fibrous mechanical metamaterials is proposed, in which the fibers are assumed to be spatial Kirchhoff rods whose mechanical response is controlled by a deformation field and a rotation field, the former accounting for strain of the rod and the latter for flexure and twist of the rod as it deforms. This leads naturally to a model based on Cosserat elasticity. Rigidity constraints are introduced that effectively reduce the model to a variant of second-gradient elasticity theory.

In this chapter it is presented the translation of the opera “Sull’applicazione de’ principj della Meccanica Analitica del Lagrange ai principali problemi” by Gabrio Piola. It was presented for the contest for the prize and crowned by the I.R. Istituto di Scienze the 4th of October 1824.

In this Volume II of the translations into English of the works by Gabrio Piola, we begin from the true first work written by the young Gabrio before 1824, when he was less than 30 years old. The content of the work Sull’applicazione de’ principj della meccanica analitica del Lagrange ai principali problemi. Memoria di Gabrio Piola presentata al concorso del premio e coronata dall’I.R. Istituto di Scienze, ecc. nella solennita del giorno 4 ottobre 1824, Milano, Imp. Regia stamperia, 1825, submitted to respond to a research program proposed by the I.R. Istituto on 4th October 1822, is in some aspects very modern and astonishingly topical.

International audience; We consider two ‘comprehensive’ modelling approaches for engineering fabrics. We distinguish the two approaches using the terms ‘semi-discrete’ and ‘continuum’, reflecting their natures. We demonstrate a fitting procedure, used to identify the constitutive parameters of the continuum model from predictions of the semi-discrete model, the parameters of which are in turn fitted to experimental data. We, then, check the effectiveness of the continuum model by verifying the correspondence between semi-discrete and continuum model predictions using test cases not previously used in the identification process. Predictions of both modelling approaches are compared against full-field experimental kinematic data, obtained using stereoscopic digital image correlation techniques, and also with measured force data. Being a reduced order model and being implemented in an implicit rather than an explicit finite-element code, the continuum model requires significantly less computational power than the semi-discrete model and could therefore be used to more efficiently explore the mechanical response of engineering fabrics.

International audience; We discuss the particular class of strain-gradient elastic material models which we called the reduced or degenerated strain-gradient elasticity. For this class the strain energy density depends on functions which have different differential properties in different spatial directions. As an example of such media we consider the continual models of pantographic beam lattices and smectic and columnar liquid crystals.

International audience; Here we present few remarks on the development of the models of microstuctured media and the generalized continua.

In an analysis presented in 1909, Poynting (page 546 of [1] ) had shown that for finite elastic deformations, the lines of greatest extension and contraction are inclined to the diagonals of the rhombus into which a square is sheared. The implication of this finding was that when slender structures are twisted, they undergo elongation as experimentally verified by Poynting's measurements with wires [1] . While many theoretical analyses have shown the possibility of Poynting and reverse (inverse) Poynting effect [2] , [3] , [4] , [5] , [6] , [7] , measurements of such effects are rather sparse [8] , [9] , [10] , [11] . We present here measurements of a highly nonlinear Poynting effect, including its reversal from positive to negative (elongation to compression) direction during torsion. Such atypical behavior is exhibited by a rather exceptional material system that has a pantographic internal structure. For this material system, the classical Cauchy-type continuum model fails and a 2nd gradient continuum model is necessary to describe many of its deformation behaviors.

International audience; In this paper, the phenomena of resorption and growth of bone tissue and resorp-tion of the bio-material inside a bi-component system are studied by means of a numerical method based on finite elements. The material behavior is described by a poro-viscoelastic model with infiltrated voids. The mechanical stimulus that drives these processes is a linear combination of density of strain energy and viscous dissipation. The external excitation is represented by a bending load slowly variable with sinusoidal law characterized by different frequencies. Investigated aspects are the influence of the load frequency, of type of the stimulus and of the effective porosity on the time evolution of the mass densities of considered system.

International audience; We claim that the models needed to describe scientific creativity (and, in particular, suitable to effectively detect it) have to be very sophisticated. Indeed, the process of creating original and predictive scientific theories is manifestly the most complex ever investigated by the human mind (There are also some paradoxical aspects in the action of a mind that is investigating its own way of functioning, but we are confident that it will be possible to avoid them). In mathematical physics one of the most complex state space structures is given by Fréchet Metric Manifolds. We conjecture that they will be needed to model the state of complexity of the mind of a scientist (However, we will not be surprised if an even more complex structure could be needed). The obtained models have a very important application: they are essential to design and rule the selection process that assigns a university chair (or a research grant). Recently some algorithms have been introduced to calculate some bibliometric indices. We claim that it is not reasonable to use them to evaluate the scientific quality of researchers, chair or grant holders, departments or whole universities. Instead, the only presently viable process must involve carefully designed procedures, similar to those used for forming juries. These procedures must be enforced to rule the formation and functioning of ad hoc peer committees entrusted to evaluate academic institutions and nominate professors, chairs or research grant holders. Bibliometrics and Scientometrics are too young as disciplines and therefore it is not possible yet, by means of the theoretical insight gained thanks to them, to design a more effective evaluation process. Only when game and artificial intelligence theories become sufficiently advanced will it become possible to efficiently replace selection peers committees (i.e. academic juries)

International audience; Second gradient theories are nowadays used in many studies in order to describe in detail some transition layers which may occur in micro-structured materials and in which physical properties are sharply varying. Sometimes higher order theories are also evoked. Up to now these models have not been based on a construction of stresses similar to the one due to Cauchy, which has been applied only for simple materials. It has been widely recognized that the fundamental assumption by Cauchy that the traction depends only on the normal of the dividing surface cannot be maintained for higher gradient theories. However, this observation did not urge any author, to our knowledge, to revisit the Cauchy construction in order to adapt it to a more general conceptual framework. This is what we do in this paper for a continuum of grade N (also called N-th gradient continuum). Our construction is very similar to the one due to Cauchy; based on the tetrahedron argument, it does not introduce any argument of a different nature. In particular, we avoid invoking the principle of virtual work. As one should expect, the balance assumption and the regularity hypotheses have to be adapted to the new framework and tensorial computations become more complex.

Numerical simulations of the axisymmetric response of a 2nd grade, fiber-reinforced, elastic cylinder are presented and discussed. The underlying model accounts for surface strain and for bending and twist, as in conventional theories, and also includes the non-standard effect of geodesic fiber bending at the constitutive level.

International audience

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.

One of the most interesting challenges in the modern theory of materials consists in the determination of those microstructures which produce, at the macro-level, a class of metamaterials whose elastic range is many orders of magnitude wider than the one exhibited by ‘standard’ materials. In dell’Isolaet al.(2015Zeitschrift für angewandte Mathematik und Physik66, 3473–3498. (doi:10.1007/s00033-015-0556-4)), it was proved that, with a pantographic microstructure constituted by ‘long’ micro-beams it is possible to obtain metamaterials whose elastic range spans up to an elongation exceeding 30%. In this paper, we demonstrate that the same behaviour can be obtained by means of an internal microstructure based on a king post motif. This solution shows many advantages: it involves only microbeams; all constituting beams are undergoing only extension or compression; all internal constraints are terminal pivots. While the elastic deformation energy can be determined as easily as in the case of long-beam microstructure, the proposed design seems to have obvious remarkable advantages: it seems to be more damage resistant and therefore to be able to have a wider elastic range; it can be realized with the same three-dimensional printing technology; it seems to be less subject to compression buckling. The analysis which we present here includes: (i) the determination of Hencky-type discrete models for king post trusses, (ii) the application of an effective integration scheme to a class of relevant deformation tests for the proposed metamaterial and (iii) the numerical determination of an equivalent second gradient continuum model. The numerical tools which we have developed and which are presented here can be readily used to develop an extensive measurement campaign for the proposed metamaterial.

International audience; It is well known that size effects play an important role in the mechanical behavior of bone tissues at different scales. In this paper we propose a second gradient model for accounting these effects in a visco-poro-elastic material and present some sample applications where bone is coupled with bioresorbable artificial materials of the kind used in reconstructing surgery.

In this paper, it is proposed a model for deformable porous media saturated by compressible nematic liquid crystal subjected to slowly varying electric fields. from a mechanical point of view, we assume that such a system can be described by means of a Biot-type model and that the mechanical action of the NLC on the solid matrix can be modeled by means of a suitable modification of Biot constitutive equations for pore pressure only. The nonlinear nature of NLCs and the presence of bifurcations make the analysis particularly challenging. We prove that suitable electrical stimulus applied on the NLC specimen may induce both type of Biot waves, fast and slow, along with shear waves in the porous matrix. This effect may be of use when one may wish to damp mechanically induced pressure waves using Darcy dissipation.

International audience; We use a simple discrete system in order to model deformation and fracture within the same theoretical and numerical framework. The model displays a rich behavior, accounting for different fracture phenomena, and in particular for crack formation and growth. A comparison with standard Finite Element simulations and with the basic Griffith theory of fracture is provided. Moreover, an ‘almost steady’ state, i.e. a long apparent equilibrium followed by an abrupt crack growth, is obtained by suitably parameterizing the system. The model can be easily generalized to higher order interactions corresponding, in the homogenized limit, to higher gradient continuum theories.

International audience; The synthesis of a 1D full second gradient continuum was obtained by the design of so-called pantographic beam (see Alibert et al. Mathematics and Mechanics of Solids (2003) [2]) and the problem of the synthesis of planar second gradient continua has been faced in several subsequent papers: in dell'Isola et al. Zeitschrift für angewandte Mathematik und Physik (2015) [5] and dell'Isola et al. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences (2016). [4] it is considered a three-length-scale mi-crostructure in which two initially orthogonal families of long Euler beams (i.e. beams much longer than the size of the homogenization cell but much slenderer than it) are interconnected by perfect or elastic pivots (hinges). The corresponding homogenized two-dimensional continuum (which was called pantographic sheet) has a D4 orthotropic symmetry. It has been proven to have a deformation energy depending on the second gradient of in-plane displacements and to allow for large elon-gations in some specic directions while remaining in the elastic regime. However, in pantographic sheets, the deformation energy only depends on the geodesic bending of the actual conguration of its symmetry directions (see for more details Steigmann et al. Acta Mechanica Sinica (2015) [3] and Placidi et al. Journal of Engineering Mathematics (2017) [6]). On the other hand, in Seppecher et al. J. of Physics: Conference Series vol. 319 (2011) [1], it was designed a bi-pantographic architectured sheet where the previously considered Euler beams were replaced by pantographic beams to form a more complex three-length-scale mi-crostructure and it was proven that, once homogenized, such a bi-pantographic sheet, in planar and linearized deformation states, produces a more complete second gradient two-dimensional continuum. Derivatives of elongations along the two symmetry directions now appear in the deformation energy. The aim of the present paper is the experimental validation of the second gradient behavior of such bi-pantographic sheets. As their intrinsic mechanical structure produces a geometrically non-linear behavior for relatively small total deformation , we rst need to extend the homogenization result to the regime of large deformations. Subsequently we compare the predictions obtained using such second gradient model with experimental evidence, as elaborated by local Digital Image Correlation (DIC) focused on the discrete kinematics of the hinges. 1 Introduction In physical literature the problem of synthesis has been confronted in many contexts. Namely, given a La-grangian potential (describing the conservative part of considered phenomena) and a Rayleigh dissipation potential , specied in terms of suitable kinematic descrip-tors, one has to nd a physical system, belonging to a class specied a priori, whose evolution is governed by the corresponding Hamilton-Rayleigh principle. In the period (1930-1970) in which the prevalence of digital computers was not yet achieved, the problem of synthesis of electric circuits was confronted in order to design suitable, and dedicated, analog computers (see, e.g., Kron [13, 14]). When the a priori class of physical systems is constituted by electric circuits with only passive elements, the previous general problem was par-ticularized as follows: given quadratic Lagrangian and Rayleigh potentials, in terms of a nite number of degrees of freedom, one has to nd the graph of a circuit and the interconnecting (linear and passive) electric elements such that it is governed by the corresponding Hamilton-Rayleigh principle. The available results found in the literature for the synthesis of passive electric circuits were recently used to synthesize piezoelectromechanical metamateri-als, suitably tailored to dampen out mechanical vibrations (see dell'Isola et al. [15, 16, 17] and [18, 25]). The synthesis problem for mechanical metamaterials reads as: given any choice of the continuous elds describing a kinematics, given functionals expressing the deformation energy, the kinetic energy and the dissipa-tion potential in terms of these elds, nd the architec-tured mechanical structure (possibly multi-scale) such that, in the homogenization limit, the obtained continuous model is exactly the one chosen a priori. Therefore , the qualitative behaviour of a metamaterial shall be given by its multi-scale architecture rather than by the constituting base materials [31].

A gradient elasticity theory is proposed for the mechanics of woven fabrics. This is motivated by a series of recent experiments in which strongly localized deformation features are observed in the so-called bias test. Such features are reminiscent of solutions to problems posed in the setting of gradient elasticity. In turn, gradient effects in woven fabrics may be motivated by the presence of a local length scale in the pattern of the weave. The presumed influence of this scale on macroscopic constitutive response leads naturally to a special gradient theory.

International audience; Since the first studies dedicated to the mechanics of deformable bodies (by Euler, D’Alembert, Lagrange) the principle of virtual work (or virtual velocities) has been used to provide firm guidance to the formulation of novel theories. Gabrio Piola dedicated his scientific life to formulating a continuum theory in order to encompass a large class of deformation phenomena and was the first author to consider continua with non-local internal interactions and, as a particular case, higher-gradient continua. More recent followers of Piola (Mindlin, Sedov and then Richard Toupin) recognized the principle of virtual work (and its particular case, the principle of least action) as the (only!) firm foundation of continuum mechanics. Mindlin and Toupin managed to formulate a conceptual frame for continuum mechanics which is able to effectively model the complex behaviour of so-called architectured, advanced, multiscale or microstructured (meta)materials. Other postulation schemes, in contrast, do not seem able to be equally efficient. The present work aims to provide a historical and theoretical overview of the subject. Some research perspectives concerning this theoretical approach are outlined in the final section.

The well-posedness of the boundary value problems for second gradient elasticity has been studied under the assumption of strong ellipticity of the dependence on the second placement gradients (see, e.g., Chambon and Moullet in Comput. Methods Appl. Mech. Eng. 193:2771–2796, 2004 and Mareno and Healey in SIAM J. Math. Anal. 38:103–115, 2006). The study of the equilibrium of planar pantographic lattices has been approached in two different ways: in dell’Isola et al. (Proc. R. Soc. Lond. Ser. A 472:20150, 2016) a discrete model was introduced involving extensional and rotational springs which is also valid in large deformations regimes while in Boutin et al. (Math. Mech. Complex Syst. 5:127–162, 2017) the lattice has been modelled as a set of beam elements interconnected by internal pivots, but the analysis was restricted to the linear case. In both papers a homogenized second gradient deformation energy, quadratic in the neighbourhood of non deformed configuration, is obtained via perturbative methods and the predictions obtained with the obtained continuum model are successfully compared with experiments. This energy is not strongly elliptic in its dependence on second gradients. We consider in this paper also the important particular case of pantographic lattices whose first gradient energy does not depend on shear deformation: this could be considered either a pathological case or an important exceptional case (see Stillwell et al. in Am. Math. Mon. 105:850–858, 1998 and Turro in Angew. Chem., Int. Ed. Engl. 39:2255–2259, 2000). In both cases we believe that such a particular case deserves some attention because of what we can understand by studying it (see Dyson in Science 200:677–678, 1978). This circumstance motivates the present paper, where we address the well-posedness of the planar linearized equilibrium problem for homogenized pantographic lattices. To do so: (i) we introduce a class of subsets of anisotropic Sobolev’s space as the most suitable energy space $E$ relative to assigned boundary conditions; (ii) we prove that the considered strain energy density is coercive and positive definite in $E$ ; (iii) we prove that the set of placements for which the strain energy is vanishing (the so-called floppy modes) must strictly include rigid motions; (iv) we determine the restrictions on displacement boundary conditions which assure existence and uniqueness of linear static problems. The presented results represent one of the first mechanical applications of the concept of Anisotropic Sobolev space, initially introduced only on the basis of purely abstract mathematical considerations.

With the advancements in 3D printing technology, rapid manufacturing of fabric materials with complex geometries became possible. By exploiting this technique, different materials with different structures have been developed in the recent past with the objective of making generalized continuum theories useful for technological applications. So-called pantographic structures are introduced: Inextensible fibers are printed in two arrays orthogonal to each other in parallel planes. These superimposed planes are inter-connected by elastic cylinders. Five differently-sized samples were subjected to shear-like loading while their deformation response was analyzed. Results show that deformation behavior is strong non-linear for all samples. Furthermore, all samples were capable to resist considerable external shear loads without leading to complete failure of the whole structure. This extraordinary behavior makes these structures attractive to serve as an extremely tough metamaterial.

International audience; The bias-extension test is a rather simple experiment aiming to determine in-plane shear properties of textile composite reinforcements. However the mechanics during the test involves fibrous material at large shear strains and large rotations of the fibres. Several aspects are still being studied and are not yet modeled in a consensual manner. The standard analysis of the test is based on two assumptions: inextensibility of the fibers and rotations at the yarn crossovers without slippage. They lead to the development of zones with constant fibre orientations proper to the bias-extension test. Beyond the analysis of the test within these basic assumptions, the paper presents studies that have been carried out on the lack of verification of these hypothesis (slippage, tension in the yarns, effects of fibre bending). The effects of temperature, mesoscopic modeling and tension locking are also considered in the case of the bias-extension test.

We present a homogenized model for the analysis of a 2D continuum with two straight families of inextensible fibres embedded in it. The kinematics of the continuum is analyzed and, motivated by phenomenological observations, it is assumed that the strain energy depends on the shear deformation of the fibres and on their bending curvature. It is shown that in order to account for the latter deformation it is necessary to introduce second gradient strains. The problem is formulated as a nonlinear constrained minimization, after introducing a suitable discretization of the domain. Some deformation processes are simulated using different constitutive hypotheses, comparing the predictions obtained assuming the presence of only first gradient or second gradient deformations, or a combination of both. It is found that the first gradient model leads to the presence of discontinuities in the rotation of the fibres, while the second gradient model regularizes these discontinuities by means of boundary layers. In particular in some deformation processes an instability of geometrical nature is observed when the second gradient model is used, that can be suppressed by the first gradient contribution.

International audience; In continuum mechanics, there exists a unique theory for elasticity, which includes the first gradient of displacement. The corresponding generalization of elasticity is referred to as strain gradient elasticity or higher gradient theories, where the second and higher gradients of displacement are involved. Unfortunately, there is a lack of consensus among scientists how to achieve the generalization. Various suggestions were made, in order to compare or even verify these, we need a generic computational tool. In this paper, we follow an unusual but quite convenient way of formulation based on action principles. First, in order to present its benefits, we start with the action principle leading to the well-known form of elasticity theory and present a variational formulation in order to obtain a weak form. Second, we generalize elasticity and point out, in which term the suggested formalism differs. By using the same approach, we obtain a weak form for strain gradient elasticity. The weak forms for elasticity and for strain gradient elasticity are solved numerically by using open-source packages—by using the finite element method in space and finite difference method in time. We present some applications from elasticity as well as strain gradient elasticity and simulate the so-called size effect.

In this paper, we consider linear pantographic sheets which in their natural configuration are con- stituted by two orthogonal arrays of straight fibers interconnected by internal pivots. We introduce a continuous model by means of a micro-macro identification procedure based on the asymptotic homogenization method of discrete media. The rescaling of the mechanical properties and of the deformation measures is calibrated in order to complies with the specific kinematics imposed by the quasi-inextensibility of the fibers together with the large pantographic deformability. The ob- tained high order continuum model shows interesting and exotic features, related to its extreme anisotropy but also to the sub-coercivity of its deformation energy. Some first numerical simula- tions are presented, showing that the model can account for experimental uncommon phenomena occuring in pantographic sheets. The paper focuses on the precise analysis and the understanding of the effective behaviour based on a well-calibration of the extension and bending phenomena arising at the local scale. In an upcoming work the analysis will be extended to oblique arrays, to some analytical solutions to proposed equations and to some further applications.

International audience; In this paper we present a model in which the motions of two families of beams constituting a pantographic sheet, when described with two independent placement fields, allows to correctly simulate the points of fracture onset in axial tension test. The deformation energy is postulated in function of these independent displacements as involving an additive quadratic term of deformation energy depending on the squared norm of their difference. The first performed numerical simulations establish that the most strained pivots are exactly the ones where one observes the first rupture.

International audience; Hencky (Über die angenäherte Lösung von Stabilitätsproblemen im Raum mittels der elastischen Gelenkkette. Ph.D. thesis, Engelmann, 1921) proposed a discrete model for elasticae by introducing rigid bars and rotational springs. Hencky (Proc R Soc Lond A Math Phys Eng Sci 472(2185), 2016) approach has been introduced to heuristically motivate the need of second gradient continua. Here, we present a novel numerical code implementing directly the discrete Hencky-type model which is robust enough to solve the problem of the determination of equilibrium configurations in the large deformation and displacement regimes. We apply this model to study some potentially applicable problems, and we compare its performances with those of the second gradient continuum model. The numerical evidence presented supports the conjecture that Hencky-type converges to second gradient model.

Two tests for woven fabrics with orthogonal fibres are examined using simplified kinematic assumptions. The aim is to analyse how different constitutive assumptions may affect the response of the specimen. The fibres are considered inextensible, and the kinematics of 2D continua with inextensible chords due to Rivlin is adopted. In addition to two forms of strain energy depending on the shear deformation, also two forms of energy depending on the gradient of shear are examined. It is shown that this energy can account for the bending of the fibres. In addition to the standard bias extension test, a modified test has been examined, in which the head of the specimen is rotated rather than translated. In this case more bending occurs, so that the results of the simulation carried out with the different energy models adopted differ more that what has been found for the BE test.

International audience; The aim of this paper is to show some typical mechanisms in the transmission of scientific knowledge through the study of some examples. We will start by considering some ancient examples concerning Democritus, Heron, Galileo and the history of the theory of tides. Then we will mainly focus on the works of the Italian scientist Gabrio Piola (1794–1850). In particular: 1. we show clear similarities between Noll’s postulation of mechanics and the ‘ancient’ presentation by Piola of the ideas needed to found Analytical Continuum Mechanics; 2. we prove that non-local and higher gradient continuum mechanics were conceived (and clearly formulated) already in Piola’s works; 3. we explain the reasons of the unfortunate circumstances which caused the (temporary) erasure of the memory of many among Piola’s contributions to mechanical sciences. Moreover, we discuss how the theory which has recently been called peridynamics, i.e. a mechanical theory which assumes that the force applied on a material particle of a continuum depends on the deformation state of a neighbourhood of the particle, was first formulated in Piola’s works. In this way we argue that in the passage from one a cultural tradition to another the content of scientific texts may often be lost, and it is possible to find more recent sources which are scientifically more primitive than some more ancient ones.

Navier–Cauchy format for Continuum Mechanics is based on the concept of contact interaction between sub-bodies of a given continuous body. In this paper, it is shown how—by means of the Principle of Virtual Powers—it is possible to generalize Cauchy representation formulas for contact interactions to the case of Nth gradient continua, that is, continua in which the deformation energy depends on the deformation Green–Saint-Venant tensor and all its N − 1 order gradients. In particular, in this paper, the explicit representation formulas to be used in Nth gradient continua to determine contact interactions as functions of the shape of Cauchy cuts are derived. It is therefore shown that (i) these interactions must include edge (i.e., concentrated on curves) and wedge (i.e., concentrated on points) interactions, and (ii) these interactions cannot reduce simply to forces: indeed, the concept of K-forces (generalizing similar concepts introduced by Rivlin, Mindlin, Green, and Germain) is fundamental and unavoidable in the theory of Nth gradient continua.

International audience; In 1913, Ernst Hellinger finished his remarkable article DIE ALLGEMEINEN ANSÄTZE DER MECHANIK DER KON-TINUA, which appeared in German language in the " Encyklopädie der mathematischen Wissenschaften mit Einschluss ihrer Anwendungen ". Following his predecessors J.-L. Lagrange, G. Piola and the Cosserat brothers, Hellinger presents in this paper a nonlinear field theory of continuum physics entirely based on a variational principle " the principle of virtual displacements ". Even though Hellinger focuses on the fundamentals of continuum mechanics, he presents within the very same variational framework the physics of optics, electrodynamics, thermodynamics and the theory of relativity. Accordingly, Hellinger's paper can be understood as a contribution to continuum physics in general. However, due to the establishment of English as the upcoming scientific language and due to the refusal of a variational formulation of continuum mechanics in the subsequent period, Hellinger's contribution to the foundations of continuum physics has been ignored for decades and has almost fallen into oblivion.

International audience; We study plane waves in second gradient solids and their reflection and transmission at plane displacement discontinuity surfaces. The needed extension of the treatment adopted to study plane wave propagation in Cauchy continua is not straightforward and is developed here. In particular, the balance of mechanical energy valid for second gradient continua is deduced. The presented results may be of interest as many boundary layer phenomena can be accounted for by second gradient models. We prove that second gradient elastic moduli may influence, in a measurable manner, how planar waves behave at discontinuity surfaces: the novel results presented here can be the basis of experimental procedures to estimate some among these moduli. We explicitly remark that reflection and transmission coefficients which we have estimated show a significant dependence on frequency, which indeed makes easier to conceive effective measurement methods.

In this commentary, we try to make clearer the state of the art concerning the relation between mechanical contact interactions and the different notions of stresses. We emphasize the importance of the concept of virtual displacements. Its role has been recognized in Mechanics and in Continuum Mechanics long ago (see e.g., Vailati in Il principio dei lavori virtuali da Aristotele a Erone d’Alessandria, 113–128, 1987; Russo in The forgotten revolution, Springer, Berlin, 2003, or Cosserat and Cosserat in Sur la Theorie des Corps Deformables, Herman, Paris, 1909; Cosserat and Cosserat in Note sur la theorie de l.action euclidienne, Gauthier-Villars, Paris, 1908), and it is central as well when starting with an expression of the power expended by internal stresses and deducing the form of contact interactions as when starting with some form of the contact interactions and developing a representation theorem for these contact interactions based on the Cauchy tetrahedron construction.

The aim of this paper is to find a computationally efficient and predictive model for the class of systems that we call ‘pantographic structures’. The interest in these materials was increased by the possibilities opened by the diffusion of technology of three-dimensional printing. They can be regarded, once choosing a suitable length scale, as families of beams (also called fibres) interconnected to each other by pivots and undergoing large displacements and large deformations. There are, however, relatively few ‘ready-to-use’ results in the literature of nonlinear beam theory. In this paper, we consider a discrete spring model for extensible beams and propose a heuristic homogenization technique of the kind first used by Piola to formulate a continuum fully nonlinear beam model. The homogenized energy which we obtain has some peculiar and interesting features which we start to describe by solving numerically some exemplary deformation problems. Furthermore, we consider pantographic structures, find the corresponding homogenized second gradient deformation energies and study some planar problems. Numerical solutions for these two-dimensional problems are obtained via minimization of energy and are compared with some experimental measurements, in which elongation phenomena cannot be neglected.

In this paper we consider Pipkin-type bi-dimensional continua with two orthogonal families of inextensible fibers. We generalize the representation formula due to Rivlin (J. Ration. Mech. Anal. 4(6):951–974, 1955) valid for planar placement fields. They are the sum of two vector functions each of which depends on one real variable only, are piece-wise $\mathscr{C}^{2}$ and may exhibit jumps of their first gradients on some inextensible fibers. Subsequently we consider a deformation energy depending only on the shear deformation relative to the inextensible families of fibers. In the suitably introduced space of configurations representing considered constrained kinematics, we formulate the relative energy minimization problem for the standard bias extension test problem, i.e., elongation of specimens which (i) have the shape of a rectangle with one side exactly three times longer than the other; (ii) are subject to a relative displacement of shorter sides in the direction parallel to the longer one. By exploiting the material and geometric symmetries, we reduce the aforementioned minimization problem to the determination of a piece-wise real function defined in a real interval. A delicate calculation of the energy first variation produces a necessary stationarity condition: it consists of an integral equation which is to be satisfied by the unknown function. The crucial points of this deduction are represented by (i) the reduction of two-dimensional integrals to one-dimensional integrals by the Fubini Theorem and (ii) the determination of the set of admissible kinematical functions on the basis of imposed boundary conditions, which implies a further integral constraint condition on the unknown function. Therefore the energy minimization problem requires the introduction of a global Lagrange multiplier. The established integral equations are solved numerically with a scheme based on a contraction type iterative process. Finally, the equilibrium shapes of the specimen undergoing large deformations, as determined by the presented model, are shown and briefly discussed.