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Journal ArticleDOI

Dynamic problems for metamaterials: Review of existing models and ideas for further research

TL;DR: In this article, the authors focus on the design of wave-guides aimed to control wave propagation in micro-structured continua, with particular attention to piezoelectromechanical structures, having a strong coupling between macroscopic motion and some internal degrees of freedom.
About: This article is published in International Journal of Engineering Science.The article was published on 2014-07-01 and is currently open access. It has received 199 citations till now. The article focuses on the topics: Context (language use).
Citations
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Journal ArticleDOI
TL;DR: In this article, a clear classification of mechanical metamaterials have been established based on the fundamental material mechanics, which can be divided into strong-lightweight (E/ρ), pattern transformation with tunable stiffness, negative compressibility (−4G/3), and strong light-weight (S/ρ).

554 citations

Journal ArticleDOI
TL;DR: In this article, the authors considered a discrete spring model for extensible beams and proposed a heuristic homogenization technique of the kind first used by Piola to formulate a continuum fully nonlinear beam model.
Abstract: 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.

333 citations

Journal ArticleDOI
TL;DR: A review of the state of the art in the study of mechanical metamaterials is given in this article, where the very attractive property of having a microstructure capable of determining exotic and specific properties is discussed.
Abstract: In this paper, we give a review of the state of the art in the study of mechanical metamaterials. The very attractive property of having a microstructure capable of determining exotic and specifica...

266 citations


Cites result from "Dynamic problems for metamaterials:..."

  • ...It is also worth mentioning the work of Greco et al. [129] for a first-gradient model with inextensible fibres and the work of Cuomo et al. [130] for a comparison with a model enhanced with second-gradient energetic terms; Turco et al. [131] report on pantographic sheets with nonorthogonal fibres, Scerrato et al. [132] and Giorgio et al. [133] report on pantographic sheets with fibres that are not straight and Turco et al. [134] report on a pantographic sheet discrete model....

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  • ...In the spirit of Piola’s heuristic homogenization, dell’Isola et al. [124], Carcaterra et al. [126] Seppecher et al. [127] and Giorgio et al. [128] present an identification of macroscopic deformation energy (i.e. a macroscopic Lagrangian density) in terms of constitutive parameters of the micro-model....

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  • ...It is worth mentioning also two already existing reviews on metamaterials; those of del Vescovo and Giorgio [33] and dell’Isola et al. [34]....

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  • ...It is worth mentioning also two already existing reviews on metamaterials; those of del Vescovo and Giorgio [33] and dell’Isola et al....

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Journal ArticleDOI
TL;DR: P pantographic metamaterials undergo very large deformations while remaining in the elastic regime, are very tough in resisting to damage phenomena, and exhibit robust macroscopic mechanical behavior with respect to minor changes in their microstructure and micromechanical properties.
Abstract: 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.

264 citations


Cites background from "Dynamic problems for metamaterials:..."

  • ...1) have been proposed as a metamaterial [9], which is well described by second gradient continuum theories [10–12]....

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Journal ArticleDOI
TL;DR: In this paper, the authors present a 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.
Abstract: Hencky (Uber die angenaherte Losung von Stabilitatsproblemen 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.

224 citations


Cites background from "Dynamic problems for metamaterials:..."

  • ...Metamaterials are obtained by the “architectured” assemblage of multiple individual “microscopic” elements constituted by standard materials, usually arranged in (quasi-)periodic substructures [27]....

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References
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Journal ArticleDOI
TL;DR: The principle of relativity Relativistic mechanics Electromagnetic fields electromagnetic waves as discussed by the authors The propagation of light The field of moving charges Radiation of electromagnetic waves Particle in a gravitational field The gravitational field equation
Abstract: The principle of relativity Relativistic mechanics Electromagnetic fields Electromagnetic waves The propagation of light The field of moving charges Radiation of electromagnetic waves Particle in a gravitational field The gravitational field equation The field of gravitational bodies Gravitational waves Relativistic cosmology Index.

9,047 citations

Book
01 Jan 1965
TL;DR: The basic concepts of quantum mechanics Energy and momentum Schrodinger's equation Angular momentum Perturbation theory Spin The identity of particles The atom The theory of symmetry Polyatomic molecules Motion in a magnetic field Nuclear structure Elastic collisions Mathematical appendices.
Abstract: The basic concepts of quantum mechanics Energy and momentum Schrodinger's equation Angular momentum Perturbation theory Spin The identity of particles The atom The theory of symmetry Polyatomic molecules Motion in a magnetic field Nuclear structure Elastic collisions Mathematical appendices.

5,955 citations

Book
01 Dec 1979
TL;DR: Spivak's comprehensive introduction to differential geometry as discussed by the authors takes as its theme the classical roots of contemporary differential geometry, and explains why it is absurdly inefficient to eschew the modern language of manifolds, bundles, forms, etc., which was developed precisely to rigorize the concepts of classical differential geometry.
Abstract: Spivak's Comprehensive introduction takes as its theme the classical roots of contemporary differential geometry. Spivak explains his Main Premise (my term) as follows: "in order for an introduction to differential geometry to expose the geometric aspect of the subject, an historical approach is necessary; there is no point in introducing the curvature tensor without explaining how it was invented and what it has to do with curvature". His second premise concerns the manner in which the historical material should be presented: "it is absurdly inefficient to eschew the modern language of manifolds, bundles, forms, etc., which was developed precisely in order to rigorize the concepts of classical differential geometry". Here, Spivak is addressing "a dilemma which confronts anyone intent on penetrating the mysteries of differential geometry". On the one hand, the subject is an old one, dating, as we know it, from the works of Gauss and Riemann, and possessing a rich classical literature. On the other hand, the rigorous and systematic formulations in current use were established relatively recently, after topological techniques had been sufficiently well developed to provide a base for an abstract global theory; the coordinate-free geometric methods of E. Cartan were also a major source. Furthermore, the viewpoint of global structure theory now dominates the subject, whereas differential geometers were traditionally more concerned with the local study of geometric objects. Thus it is possible and not uncommon for a modern geometric education to leave the subject's classical origins obscure. Such an approach can offer the great advantages of elegance, efficiency, and direct access to the most active areas of modern research. At the same time, it may strike the student as being frustratingly incomplete. As Spivak remarks, "ignorance of the roots of the subject has its price-no one denies that modern formulations are clear, elegant and precise; it's just that it's impossible to comprehend how any one ever thought of them." While Spivak's impulse to mediate between the past and the present is a natural one and is by no means unique, his undertaking is remarkable for its ambitious scope. Acting on its second premise, the Comprehensive introduction opens with an introduction to differentiable manifolds; the remaining four volumes are devoted to a geometric odyssey which starts with Gauss and Riemann, and ends with the Gauss-Bonnet-Chern Theorem and characteristic classes. A formidable assortment of topics is included along the way, in which we may distinguish several major historical themes: In the first place, the origins of fundamental geometric concepts are investigated carefully. As just one example, Riemannian sectional curvature is introduced by a translation and close exposition of the text of Riemann's remarkable paper, Über die Hypothesen, welche der Geometrie zu Grunde

3,840 citations

Journal ArticleDOI

3,323 citations


"Dynamic problems for metamaterials:..." refers background in this paper

  • ...Hence, piezoelectromechanical structures are particular micro-structured continua of the kind studied e.g. by Toupin, Eringen and Mindlin (Toupin, 1962, 1964; Eringen, 1999, 2001; Mindlin, 1964; Mindlin, Eshel, 1968)....

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  • ...…a group constituted by notable scientific personalities as Mindlin, Green and Rivlin, Toupin, Eringen and Germain (Green, Rivlin, 1964a,b,c, 1965; Mindlin, 1964, 1965; Toupin, 1962, 1964; Eringen, 1999, 2001, 2002; Eringen, Suhubi, 1964a,b; Germain, 1973a,b) managed to establish, yet with some…...

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  • ...In Mindlin (1964) Mindlin excludes from his analysis the metamaterials considered in Ghiba et al. (2013) and Neff et al. (2013) since their deformation energy is not definite positive as a function of the defined deformation measures: with this meaning the word singular used in a previous sentence…...

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  • ...by Toupin, Eringen and Mindlin (Toupin, 1962, 1964; Eringen, 1999, 2001; Mindlin, 1964; Mindlin, Eshel, 1968)....

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  • ...For a more modern interpretation of this subject refer to (Mindlin, 1964, 1965; Mindlin, Eshel, 1968; Toupin, 1962, 1964; Eringen, 1999, 2001, 2002; dell’Isola, Seppecher, 1995; dell’Isola, Sciarra, Vidoli, 2009; Sciarra, dell’Isola, Coussy, 2007; Auffray, Bouchet, Brechet, 2009)....

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Journal ArticleDOI
TL;DR: HAL as discussed by the authors is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not, which may come from teaching and research institutions in France or abroad, or from public or private research centers.
Abstract: HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés. Elastic materials with couple-stresses R. Toupin

2,574 citations


"Dynamic problems for metamaterials:..." refers background or methods in this paper

  • ...In this context we mention the classical works by Eringen (1999, 2001, 2002), Eringen, Suhubi (1964a,b), Sedov (1968, 1972), Toupin (1962, 1964) and Bleustein (1967), among those which merit considerable attention....

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  • ...Hence, piezoelectromechanical structures are particular micro-structured continua of the kind studied e.g. by Toupin, Eringen and Mindlin (Toupin, 1962, 1964; Eringen, 1999, 2001; Mindlin, 1964; Mindlin, Eshel, 1968)....

    [...]

  • ...…principle, as in the references stemming from the fundamental work by Lagrange (Lagrange, 1788), e.g. Auffray et al. (2013), Daher, Maugin (1986), Epstein (2010), Germain (1973a,b), Green, Rivlin (1964a), Kroner (1968), Maugin (2013), Sedov (1968), Toupin (1962, 1964) and Bedford (1985)....

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  • ...The aforementioned preliminary results promise a rich variety of behaviour for Mindlin-type materials which could provide amazing developments....

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  • ...Cosserat contributions (Cosserat brothers, 1909) were underestimated for other fifty years and only starting from 1960 a group constituted by notable scientific personalities as Mindlin, Green and Rivlin, Toupin, Eringen and Germain (Green, Rivlin, 1964a,b,c, 1965; Mindlin, 1964, 1965; Toupin, 1962, 1964; Eringen, 1999, 2001, 2002; Eringen, Suhubi, 1964a,b; Germain, 1973a,b) managed to establish, yet with some resistances, the validity of Cosserat’s point of view....

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Frequently Asked Questions (14)
Q1. What contributions have the authors mentioned in the paper "Dynamic problems for metamaterials: review of existing models and ideas for further research" ?

An interesting class of problems in this context regards the design of wave-guides aimed to control wave propagation. The description of the state of the art is followed by some hints addressed to describe some possible research developments and in particular to design optimal design techniques for bone reconstruction or systems which may block wave propagation in some frequency ranges, in both linear and non-linear fields. 

The assumptions, which usually are considered to be immovable paradigms of continuum mechanics, conversely seem to have a much more restricted applicability and their removal open unexpected scientific possibilities towards technological innovations. Many new phenomena, with very interesting possibilities of applications, simply wait to be observed and understood. 

The structure of these boundary conditions may induce• a time delay in the release of incident energy at the interface,• a concentrated dissipation at the interface,• a considerable increase of reflected energy, also with time delay,• a surface trapping of incident energy with its release in different forms, which could have non-parasitic uses,• the origination of surface waves, involving surface displacement, surface thickness variation or, more generally, surface micro-structural oscillations. 

A class of metamaterials is represented by controlled smart structures, recently introduced to optimise the dissipation of mechanical vibration energy. 

The Cosserat brothers were among the first authors who complemented, with additional independent kinematic fields, the standard kinematics constituted by a placement field. 

The research project presented in this paper has been partially supported by the International Research Centre for the Mathematics and Mechanics of Complex Systems “M&MoCS” of the Università de L’Aquila. 

Some preliminary results on the study of wave propagation in micromorphic media indicate that, for particular sets of the constitutive parameter values, propagation of some types of waves can be inhibited or waves that propagate without carrying energy can be also observed. 

the clinical requirements are clear:• the reconstructed bone needs to resemble the natural bone as close as possible; therefore in principle, the bio-resorbable material should be completely resorbed at the end of the process and, during the resorptionremodelling process, cavities should be avoided since they prevent the formation of new bone tissue;• remodelling and reformation process should last a reasonable period of time and the mechanical stability of the system must be continuously assured. 

In this context, the analysis of involved length scales is crucial, as length scale separation allows simplification in the modelling procedure. 

The Principle of Least Action, when formulated for action functionals admitting first differentials, can be regarded as a particular form of the Principle of Virtual Work. 

In these cases, structural modifications of the principal structures are aimed at the addition of some elements that damp vibrations: these devices may be e.g. arrays of piezoelectric actuators coupled with dissipative electronic circuits (dell’Isola, Vidoli, 1998b). 

He was lead by stringent physical considerations to introduce higher gradients of displacement field, as necessary independent variables, in the constitutive equation for the deformation energy of continuous media. 

A preliminary study, reported in a private communication by A. Luongo, seems to indicate that piezoelectromechanical structures are less stable, in a way not yet rigorously specified, than their original mechanical counterpart. 

Some difficulties arise in this innovative application field of micro-structured continuum mechanics: the authors mention for instance, in fibre-reinforced composite materials, the problem to model the permeability both in the initial undeformed configuration and its modifications during the deformation evolution.