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We report a continuum theory for 2D strain gradient materials accounting for a class of dissipation phenomena. The continuum description is constructed by means of a (reversible) placement function and by (irreversible) damage and plastic functions. Besides, expressions of elastic and dissipation energies have been assumed as well as the postulation of a hemi-variational principle. No flow rules have been assumed and plastic deformation is also compatible, that means it can be derived by a placement function. Strain gradient Partial Differential Equations (PDEs), boundary conditions (BCs) and Karush-Kuhn-Tucker (KKT) type conditions are derived by a hemi variational principle. PDEs and BCs govern the evolution of the placement descriptor and KKT conditions that of damage and plastic variables. Numerical experiments for the investigated homogeneous cases do not need the use of Finite Element simulations and have been performed to show the applicability of the model. In particular, the induced anisotropy of the response has been investigated and the coupling between damage and plasticity evolution has been shown.

Duoskelion structures have been recently introduced by Barchiesi et al. (2021) as a proof-of-concept motif for a new class of metamaterials. The properties of these periodic beam-like chiral structural elements have been investigated, up to now, by means of a discrete model formulation whose predictions are obtained by numerical methods. In this paper we select a specific scaling law for micro stiffnesses aimed at deriving, via asymptotic homogenization, an internally-constrained Cosserat one-dimensional planar continuum model as the limit of a duoskelion structure. We analyze qualitatively and quantitatively the family of equilibrium configurations of the homogenized continuum when subjected to axial loading and compare the results of the analysis with those obtained by means of the discrete model formulation.

Numerical simulations of several planar failure modes of masonry structures are presented, based on the model and solving code from a recent hemivariational block-based model inspired from granular micromechanics (Tran et al. [33]). The numerical tests include a comparison with literature results for a constant shearing load, a parametric study of the influence of mortar thickness, and the simulation of common failures such as bending, shear sliding, and rocking. The results show that the studied model is able to capture common failure modes. Conclusions and insights towards the extension of the studied model are then given.

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.

Dynamics of pantographic sheets presents exotic aspects that deserve investigation. In this paper, we focus the attention on some possible modalities of non-linear wave propagation in planar pantographic sheets. We use a smaller length-scale lattice model, in which the beams and pivots constituting the sheet are described by constrained Euler- Bernoulli beams together with a meso-reduced-order model which belongs to the class of second-gradient elastic materials and with a macro multi-field 1D continuum model, whose displacement is augmented by a specific class of cross-section deformations. Such a three-step reduction process is developed to allow for fast computational analysis of symmetric wave propagation patterns with respect to the longitudinal axis of the sheet. It is conceived by using suitable kinematical hypotheses for the 1D continuum descriptors referring to pantographic sheet sections which are inspired by the numerical evidence obtained performing simulations based on the smaller scale lattice model. The deformation energy of the pantographic sheet, successfully postulated in dell'Isola et al. (2016) for a meso-reduced-order second gradient model, is pivotal in the whole model reduction process: It allows for the determination of generalized 1D deformation energy in terms of the mechanical properties of the micro-lattice model. Performed numerical simulations prove that several waveforms propagate in planar pantographic sheets with low dispersion and motivate further investigations in the subject.(c) 2022 Elsevier B.V. All rights reserved.

We postulate a deformation energy for describing the mechanical behavior of so called pantographic blocks, that is bodies constituted by stacking of layers of pantographic sheets. We remark that the pantographic effect is limited in the plane of pantographic sheets and therefore only the second derivatives of transverse displacements along the pantographic fibers appear in the chosen deformation energy. We use this novel energy to predict the behavior of pantographic blocks when subjected to : (i) compression and traction test, (ii) torsion, (iii) shear and (iv) bending. A linearization of the energy shows one floppy mode in addition to the rigid body motions, assuming perfect pivots, similar to pantographic sheets.

We present and discuss the results of some numerical simulations dealing with wave motion in one-dimensional pantographic media, also known as pantographic beams. Specifically, the analysis is carried out in large displacements regime and pantographic beams are modeled by using a completely discrete approach. Nonlinear vibrations induced by a transversal impulse are studied. A simple yet efficient time-stepping scheme was used to reconstruct the motion and, consequently, to explore the propagation of strain waves in the pantographic medium. The numerical study concludes that the main features of the observed strain wave propagation are extremely useful toward the understanding of the dynamic behavior of the pantographic micro-structure and its continuum counterparts.

In this paper, we represent second-gradient internal work functionals in Lagrangian (referential) and Eulerian (spatial) descriptions, and we deduce the corresponding expressions for the Piola transformations of stress and double-stress tensors and of external forces and double-forces. We also derive, in both the Eulerian and Lagrangian description, the expression of surface and edge contact interactions (which include forces and double-forces) for second-gradient continua in terms of the normal and the curvature of contact boundary surfaces and edge shapes.

Starting from a discrete model of a pantographic beam based on beam elements with 6 DOF in 2D space, 2 translational and 1 rotational for each end node, and proposing an energy quadratic in elongation, shear, bending and torsion deformation measures, we find by asymptotic homogenization a model of a complete 1D second gradient continuum. Compared to previous models, the discrete model of the present work includes shear deformation by considering the beam element’s end nodes relative orientation. The resulting continuum model differs in a stiffness coefficient stemming from the shear deformation, but it coincides in the limit of shear stiff elements.

In this chapter, we present the translation of the main excerpts of Heiberg’s Prolegomena to his Archimedes Edition. This text was originally written in Latin [Heiberg, J. L. (1910). Archimedis Opera omnia cum commentariis Eutocii: Vol. 1-3. In aedibus BG Teubneri.] and contains the evidence of interesting phenomena in the transmission of ancient scientific texts. Considering the nature of the present work, which is rather interested in problems concerning the transmission and degradation of scientific knowledge through the centuries, the following translation concerns mainly the pages in which issues related to this type of problem are addressed. We have, however, omitted the translation of extremely technical parts of the philologist's work, which are beyond the scope of the present work.

In this paper, we discuss well-posedness of the boundary-value problems arising in some “gradient-incomplete” strain-gradient elasticity models, which appear in the study of homogenized models for a large class of metamaterials whose microstructures can be regarded as beam lattices constrained with internal pivots. We use the attribute “gradient-incomplete” strain-gradient elasticity for a model in which the considered strain energy density depends on displacements and only on some specific partial derivatives among those constituting displacements first and second gradients. So, unlike to the models of strain-gradient elasticity considered up-to-now, the strain energy density which we consider here is in a sense degenerated, since it does not contain the full set of second derivatives of the displacement field. Such mathematical problem was motivated by a recently introduced new class of metamaterials (whose microstructure is constituted by the so-called pantographic beam lattices) and by woven fabrics. Indeed, as from the physical point of view such materials are strongly anisotropic, it is not surprising that the mathematical models to be introduced must reflect such property also by considering an expression for deformation energy involving only some among the higher partial derivatives of displacement fields. As a consequence, the differential operators considered here, in the framework of introduced models, are neither elliptic nor strong elliptic as, in general, they belong to the class so-called hypoelliptic operators. Following (Eremeyev et al. in J Elast 132:175–196, 2018.10.1007/s10659-017-9660-3) we present well-posedness results in the case of the boundary-value problems for small (linearized) spatial deformations of pantographic sheets, i.e., 2D continua, when deforming in 3D space. In order to prove the existence and uniqueness of weak solutions, we introduce a class of subsets of anisotropic Sobolev’s space defined as the energy space E relative to specifically assigned boundary conditions. As introduced by Sergey M. Nikolskii, an anisotropic Sobolev space consists of functions having different differential properties in different coordinate directions.

International audience; In this paper, we reveal that the mathematical discrete model of Hencky type, introduced in [1], is appropriate for describing the mechanical behavior of micro-metric pantographic elementary modules. This behavior does not differ remarkably from what has been observed for milli-metric modules, as we prove with suitably designed experiments. Therefore, we conclude that the concept of pantographic microstructure seems feasible for micro-metrically architected microstructured (meta)materials as well. These results are particularly indicative of the possibility of fabricating materials that can have an underlying pantographic microstructure at micrometric scale, so that its unique behavior can be exploited in a larger range of technological applications.

International audience; The overall behavior of an articulated beam structure constituted by elements arranged according to a specific chirality is studied. The structure as a whole, due to its slenderness and geometry, is called duoskelion beam. The name duoskelion is a neologism which is inspired by the Greek word δύοσκέλιον (two-legged). A discrete model for shearable beams, formulated recently, is exploited to investigate its mechanics. A purposely designed numerical scheme, adapting the Riks rationale, is used to calculate large displacement and deformation equilibria of duoskelion beams. Aimed at computing the current step correction, the Riks arc-length method is modified and made more efficient by applying a specific orthogonality condition, defined via the stiffness matrix, to an adapted extrapolation step. The robustness of the resulting scheme and its capability to follow equilibrium branches allows, in principle, for the exploration of the whole set of local energy minima in the introduced space of configurations, by using suitably modulated perturbative external loads. The developed numerical tool can be used to understand the mechanics of duoskelion beams. It is proved that there exists a stable principal equilibrium branch in which only compression is observed for any compression load. Additional stable equilibrium branches are found in compression such that the clamped–clamped compressed beam assumes a characteristic S shape which, upon reaching a critical load, is significantly amplified. A mechanically relevant stable equilibrium is also found in extension, being observed the S-shaped configuration experimentally found in Misra et al. (2020).

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.

Indexado en Scopus Describing the emerging macro-scale behavior by accounting for the micro-scale phenomena calls for microstructure-informed continuum models accounting properly for the deformation mechanisms identifiable at the micro-scale. Classical continuum theory, in contrast to the micromorphic continuum theory, is unable to take into account the effects of complex kinematics and distribution of elastic energy in internal deformation modes within the continuum material point. In this paper, we derive a geometrically nonlinear micromorphic continuum theory on the basis of granular mechanics, utilizing grain-scale deformation as the fundamental building block. The definition of objective kinematic descriptors for relative motion is followed by Piola’s ansatz for micro–macro-kinematic bridging and, finally, by a limit process leading to the identification of the continuum stiffness parameters in terms of few micro-scale constitutive quantities. A key aspect of the presented approach is the identification of relevant kinematic measures that describe the deformation of the continuum body and link it to the micro-scale deformation. The methodology, therefore, has the ability to reveal the connections between the micro-scale mechanisms that store elastic energy and lead to particular emergent behavior at the macro-scale. info:eu-repo/semantics/publishedVersion Revisión por pares

International audience; This paper shows how the classical representation techniques for the solution of elasticity problems, based on the Green's functions, can be generalized to second-gradient continua focusing on the specific case of pantographic lattices. As these last are strongly anisotropic, the fundamental solutions of isotropic second-gradient continua involving bi-Helmholtztype operators are not applicable. More specifically we establish the analytical fundamental solution for the linearized equations governing the equilibrium of pantographic 2D continua in the neighbourhood of the reference configuration. Moreover, by means of found novel Green's functions, it is shown that it is possible to solve aforesaid equilibrium equations by using Fredholm integral equations. It is seen that an approximated analytical solution for the standard bias test for pantographic 2D continua can be found by using judiciously the found analytical fundamental solutions. The micro-macroasymptotic identification allows for a clear and satisfactory physical interpretation of the obtained analytical results.

International audience; This contribution presents the results of a campaign of numerical simulations aimed at better understanding the propagation of longitudinal waves in pantographic beams within the large-deformation regime. Initially, we recall the key features of a Lagrangian discrete spring model, which was introduced in previous works and that was tested extensively as capable of accurately forecasting the mechanical response of structures based on the pantographic motif, both in statics and dynamics. Successively, a stepwise integration scheme used to solve equations of motions is briefly discussed. The key content of the present contribution concerns the thorough presentation of some selected numerical simulations, which focus in particular on the propagation of stretch profiles induced by impulsive loads. The study takes into account different tests, by varying the number of unit cells, i.e., the total length of the system, spring stiffnesses, the shape of the impulse, as well as its properties such as duration and peak amplitude, and boundary conditions. Some conjectures about the form of traveling waves are formulated, to be confirmed by both further numerical simulations and analytical investigations.

International audience; In this paper, the method of electric analog synthesis is applied to design a piezo-electro-mechanical arch able to show the capacity of multimodal damping. An electric-analog circuit is designed by using a finite number of lumped elements representing the equivalent of a curved beam. Spatial and frequency coherence conditions are proven to be verified for the modes to be damped: in fact, lumped-element circuit can damp only a finite number of vibration modes. Analogous boundary conditions are ensured, so that natural frequencies and mode shapes of both the curved beam and the analog circuit are equal. The instance considered here is the vibration mitigation of a piezo-electro-mechanical arch. Having a view towards prototypical applications, all simulations consider values of physically feasible passive circuital elements. It is believed that the present results may represent a step towards the design of multi-physics metamaterials based on micro-structures exploiting the principle of multimodal damping.

Dieser Beitrag ist mit Zustimmung des Rechteinhabers aufgrund einer (DFG geförderten) Allianz- bzw. Nationallizenz frei zugänglich., This publication is with permission of the rights owner freely accessible due to an Alliance licence and a national licence (funded by the DFG, German Research Foundation) respectively., The paper ‘Zur Dynamik elastisch gekoppelter Punktsysteme’ by Schrödinger (1914) does not seem to have attracted the attention that it deserves. We translate it into English here and we discuss its results in detail, with a view to its possible influence in the modern theories of generalised continua. The clever solution found, in terms of Bessel functions, by Schrödinger of the problem of the vibrations of a linear infinite chain of molecules gives some important methodological hints for contemporary research.

International audience; Background: Since the mechanical behavior of pantographic metamaterials depends upon the properties of their microstructure, accurate descriptions of unit cells are needed. Objective: The present effort is motivated by this requirement to characterize the detailed deformation of unit cells formed of two orthogonal sets of 3 beams. Methods: Their deformations in a bias extension test were measured via digital image correlation performed at different scales. Results: Thanks to the gray level residuals, the microscale results were found in better agreement with the experiment than mesoscale and macroscale analyses. Fine analyses around the hinges showed that relative displacements occurred between the two beam layers. Conclusions: Such experimental analyses supply full-field data to validate models (e.g., as a starting point in homogenization procedures) for describing the mechanical behavior of pantographic metamaterials.

The concept of Industry 4.0 is defined as a common term for technology and the concept of new digital tools to optimize the manufacturing process. Within this framework of modular smart factories, cyber-physical systems monitor physical processes creating a virtual copy of the physical world and making decentralized decisions. This article presents a review of the literature on virtual methods of computer-aided manufacturing processes. Numerical modeling is used to predict stress and temperature distribution, springback, material flow, and prediction of phase transformations, as well as for determining forming forces and the locations of potential wrinkling and cracking. The scope of the review has been limited to the last ten years, with an emphasis on the current state of knowledge. Intelligent production driven by the concept of Industry 4.0 and the demand for high-quality equipment in the aerospace and automotive industries forces the development of manufacturing techniques to progress towards intelligent manufacturing and ecological production. Multi-scale approaches that tend to move from macro- to micro- parameters become very important in numerical optimization programs. The software requirements for optimizing a fully coupled thermo-mechanical microstructure then increase rapidly. The highly advanced simulation programs based on our knowledge of physical and mechanical phenomena occurring in non-homogeneous materials allow a significant acceleration of the introduction of new products and the optimization of existing processes.

In the problem of the synthesis of metamaterials, the pantographic architecture revealed remarkable potentialities. Indeed it allowed for the synthesis of second gradient 2D (nonlinear) continua: i.e. 2D shells whose deformation energy depends also on the second derivatives of displacements in the tangent directions to the reference configuration. Moreover, pantographic architecture seems to be able to produce metamaterials whose macroscopic elongations are large, albeit remaining in the elastic regime. The theoretically shown potentialities have started to become of ≫practical≪ interest thanks to a series of experiments, which were made possible by the recent 3D additive manufacturing. The actual construction of pantographic architecture has been based on the design of two arrays of beams interconnected by small cylinders, whose behavior can be modeled in different ways: if they are very short they can be regarded as clamps, while if they are short enough as elastic (or inelastic for large rotations) cylindrical hinges connecting the beams of different arrays. Otherwise, they must be modeled as elastic (or inelastic) elements allowing for relative rotations and displacements. In this paper, we focus on this particular case and we introduce, after a homogenization based on heuristic arguments, a 2D generalized continuum model whose kinematics is characterized by two placement and rotation fields (one for each array of beams) and whose deformation energy depends on relative displacements and rotations. The offset between the two beams arrays is proven to be an essential tool for defining effective invariant kinematical deformation measures. In facts, one wants to postulate a deformation energy for the introduced 2D generalized continuum which gives predictions in agreement with those given by the more refined 3D model where the pantographic architecture is described with its maximum geometric complexity and where the constituting material is assumed to be modelable as a standard 3D first gradient continuum. In the present paper, in order to arrive at the correct conjecture for the postulated energy, we consider the concept of averages of rotations in SO(3) Lie group. The used enriched kinematics is seen to be a possible alternative to the adoption of second gradient 2D models. Some rather surprising deformation processes are studied, where interesting non-symmetric post-buckling phenomena are observed in both the models used. Mentioned post-buckling has been observed experimentally.

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.

The paper presents the results of dynamic testing of two wood species: lime-tree (Tilia europoea) and pine (Pinaceae). The dynamic compressive tests were carried out using the traditional Kolsky method in compression tests. The Kolsky method was modified for testing the specimen in a rigid limiting holder. In the first case, stress–strain diagrams for uniaxial stress state were obtained, while in the second, for uniaxial deformation. To create the load a gas gun was used. According to the results of the experiments, dynamic stress–strain diagrams were obtained. The limiting strength and deformation characteristics were determined. The fracture energy of lime and pine depending on the type of test was also obtained. The strain rates and stress growth rates were determined. The influence of the cutting angle of the specimens relative to the grain was noted. Based on the results obtained, the necessary parameters of the wood model were determined and their adequacy was assessed by using a special verification experiment.

In present paper wave propagation poroviscoelastic solids is studied. Study of wave propagation in saturated porous media is an important issue of engineering sciences. The poroelasticity theory was developed and nowadays is an important to engineering applications. Also research is dedicated to modeling of a slow compressional wave in poroviscoelastic media by means of boundary-element method. Poroviscoelastic formulation is based on Biot's model of fully saturated poroelastic media with a correspondence principal usage. Standard linear solid model is employed in order to describe viscoelastic behavior of the skeleton in porous medium. The boundary-value problem of the three-dimensional dynamic poroviscoelasticity is written in terms of Laplace transforms. Direct approach of the boundary integral equation method is employed. The boundary-element approach is based on the mixed boundary-element discretization of surface with generalized quadrangular elements. Subsequent application of collocation method leads to the system of linear equations, and then to the solution in Laplace domain. Numerical inversion of Laplace transform is used to obtain time-domain solution. The problem of the load acting on a poroelastic prismatic solid is solved by means of developed software based on boundary element approach. An influence of permeability of porous material on dynamic responses is studied. Slow wave phenomena appearance is demonstrated. Viscosity parameter influence on dynamic responses of displacements and pore pressure is studied.