Truss Modular Beams with Deformation Energy Depending on Higher Displacement Gradients

Alibert, Jean-Jacques; Seppecher, Pierre; Dell’Isola, Francesco

doi:10.1177/1081286503008001658

Cited by 465 publications

(363 citation statements)

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“…A review of results on the theoretical foundation of a variational approach for higher gradient theories is [71]. Besides, the use of the methods of metamaterials [72][73][74] like those for pantographic structures [75][76][77] could be considered even to take into account the effects of damage [78][79][80][81][82]. Another possible extension of this model is to include surface effects [83][84][85][86][87][88][89] and the anisotropy induced by the orientation of the NLC directors, as in [90] , or as it is done in granular materials [91].…”

Section: Numerical Simulationsmentioning

confidence: 99%

“Fast” and “slow” pressure waves electrically induced by nonlinear coupling in Biot-type porous medium saturated by a nematic liquid crystal

Rosi

Placidi

Dell’Isola

2017

Z. Angew. Math. Phys.

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Section: Numerical Simulationsmentioning

confidence: 99%

“Fast” and “slow” pressure waves electrically induced by nonlinear coupling in Biot-type porous medium saturated by a nematic liquid crystal

Rosi

Placidi

Dell’Isola

2017

Z. Angew. Math. Phys.

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“…We then give some examples of different possible floppy modes and the associated continuous models. More details about the presented ideas can be found in [1].…”

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confidence: 99%

Second Gradient and Generalized Continua. A workshop held on 12–16 March 2012 in Cisterna di Latina

Dell’Isola

Forest

2014

Z Angew Math Mech

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This editorial intends to complement the scientific content of the papers which follow in this special issue. The workshop "Second Gradient and Generalized Continua" held on 12-16 March 2012 at Palazzo Caetani in Cisterna di Latina, Italy was organized by Francesco dell'Isola and Samuel Forest. Very interesting, exciting and creative discussions started nearly immediately during the seminars and the round tables: the spirit of all participants was very productive and open minded and nobody felt necessary to hide his views, also when they were in contrast with those of other participants, so that a truly frank and direct scientific discussion was possible. The video-recorded seminars are available at the website http://memocs.univaq.it/?page id=2858, http://memocs.univaq.it/?page id=3071.In the following paragraphs we will reproduce, following the seminar order, only the abstracts of those presentations which did not produce a paper published in the present issue of ZAMM.Francesco dell'Isola: "Contact interactions in N-th gradient continua". Cauchy format for continuum mechanics is not general enough to include the description of some important phenomena occurring in a large class of "microscopically" inhomogeneous bodies. This circumstance has been recognized already by E. and F. Cosserat and later by Mindlin, Rivlin, Toupin, and Green. Particularly detailed is the development of second gradient model due to Paul Germain, which is the simplest continuum model which cannot be included in the Cauchy format. Actually Cauchy Continuum model has a wide but limited scope of predictivity: therefore the logical possibility of generalizing it truly corresponds to the need of describing some new and interesting phenomena, which include, but are not limited to, the formation of many different kinds of boundary layers. The class of generalized (N -th gradient) continua which is presented here is constituted by those models in which deformation energy may depend on N + 1 gradients of displacement field. The precise definition of N -th gradient continua and the nature of contact interactions which may arise in them can be mathematically formulated and characterized only by using suitable concepts from the theory of distribution and differential geometry, but it is definitively not so difficult to generalize -in the considered context -the Cauchy concept of stress state. The mathematical abstraction required does not seem inappropriate as it actually supplies a useful description of some phenomena of relevance in the mechanics of growing tissues and opens interesting perspectives in the design of artificial materials showing "exotic" mechanical behavior. More details about the presented ideas can be found in [17,18].Pierre Seppecher: "Linear elastic trusses leading to continua with exotic mechanical interactions". When studying the statics of a truss made of a discrete network of nodes joined by linear elastic springs, the kernel of the quadratic potential energy (space of floppy modes) plays an essential role. In particular w...

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“…Using the method proposed in [20,11], this decomposition is easily obtained, and the determination of the symmetry classes is completely solved in [?,23]. We believe that these simple formulas can be of great help to develop, for example, higher-order constitutive laws [26,8] and to design micro-macro homogenization procedure for anisotropic materials [2]. …”

Section: Resultsmentioning

confidence: 99%

“…If in the case of classical elasticity these results are well-known [5,6], their extension to other situations is not straightforward. However, due to a growing interest in the modeling of size and non-local effects in materials and structures, the need for generalized continuum theories becomes evident [7,8,9]. This note, and following a path already developed in some previous contributions [10,11,12], aims at providing some tools that can help the modeling of these non-conventional behaviors.…”

Section: Introductionmentioning

confidence: 99%

Analytical expressions for odd-order anisotropic tensor dimension

Auffray

2014

Comptes Rendus. Mécanique

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According to the symmetries of the matter the number of coefficients needed to define a tensorial relation varies. It is well-known that in linear elasticity the number of generic coefficients varies from 21 for a complete anisotropic material to 2 in case of isotropy. In a previous contribution, we provided analytical expressions that give the number of generic anisotropic coefficients in any anisotropic system for an even-order tensor. In the present note, we aim at extending the previous results to the case of odd-order tensors. As an illustration, the dimension of any anisotropic system for third-order piezoelectricity tensors and of the fifth-order coupling tensors of Mindlin's strain-gradient elasticity are determined. To cite this article: N. Auffray, C. R. Mécanique 341 (2013). RésuméExpressions analytiques donnant la dimension d'un tenseur anisotrope d'ordre impair. En fonction des symétries qu'un milieu possède, le nombre de coefficients génériques nécessaireà la définition d'une loi tensorielle varie. Dans le contexte de l'élasticité linéaire, si le milieu ne presente aucune symmétrie 21 coefficientś elastiques sont génériquement nécessaire, tandis que dans le cas de l'isotropie ce nombre se réduità 2. Dans une précédente note, nous avions dérivé des formules analytiques donnant, dans le cas d'un tenseur pair, le nombre de coefficients génériques nécessaire pour chaque type d'anisotropie. Le but de cette nouvelle contribution, est de compléter ces formules en lesétendant au cas des tenseurs impairs. En guise d'illustration, nous calculerons, pour l'ensemble des systèmes d'anisotropie possibles, la dimension des tenseurs piezoélectriques (ordre 3) ainsi que des tenseurs de couplage de la théorie de l'élasticitéà gradient (ordre 5). Pour citer cet article : N. Auffray, C. R. Mécanique 341 (2013).

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Truss Modular Beams with Deformation Energy Depending on Higher Displacement Gradients

Cited by 465 publications

References 30 publications

“Fast” and “slow” pressure waves electrically induced by nonlinear coupling in Biot-type porous medium saturated by a nematic liquid crystal

“Fast” and “slow” pressure waves electrically induced by nonlinear coupling in Biot-type porous medium saturated by a nematic liquid crystal

Second Gradient and Generalized Continua. A workshop held on 12–16 March 2012 in Cisterna di Latina

Analytical expressions for odd-order anisotropic tensor dimension

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