[De bi-parallélogrammes articulés à des milieux continus 1D de troisième gradient]
Recently, it has been conjectured that a specific modular articulated bi-parallelogram microstructure is a solution for the problem of synthesis for a 1D continuum whose deformation elastic energy depends on the curvature gradient (dell’Isola et al. [Math. Mech. Complex Syst. 12 (2024)] and Terranova et al. [Comptes Rendus. Mécanique 353 (2025)]). In this paper, we present an asymptotic procedure for getting the homogenized description of that microstructured system under large in-plane deformation. It is shown that such a periodic structure behaves macroscopically as a third gradient 1D continuum.
The elastic energy stored in a module of such a microstructured system deformed by a gradient of curvature combined with extension is first established. This leads to the energy density of the effective 1D continuum. The latter involves three elastic stiffness coefficients related to the curvature gradient, the elongation, and the coupling of both, whose expressions are explicitly related to the morphology of the module, the stiffnesses of the micro bars, and the curvature.
The strong formulation of the equilibrium condition of the microstructured system is then deduced following the Euler–Lagrange method of minimization of energy. The calculation of the first variation of the deformation energy allows for the determination of the generalized external forces which can be applied to the equivalent 1D continuum whose deformation energy depends on the gradient of curvature and extension: that is, normal and transverse force together with couple and double couple. Consequently, we determine the corresponding balance equations and the constitutive equations for forces and both couple and double couples.
Some numerical examples are given in the case of newly introduced 1D continua loaded at their extremity by couples and double couples.
Il a récemment été conjecturé qu’une microstructure périodique spécifique dont le module est formé de deux parallélogrammes articulés était une solution au problème de la synthèse d’un milieu continu 1D dont l’énergie élastique de déformation dépend du gradient de courbure (dell’Isola et al. [Math. Mech. Complex Syst. 12 (2024)] et Terranova et al. [Comptes Rendus. Mécanique 353 (2025)]). Dans cet article, nous présentons une procédure asymptotique permettant d’obtenir la description homogénéisée de ce système microstructuré soumis à de grandes déformations dans son plan. Il est démontré qu’une telle structure périodique se comporte macroscopiquement comme un continu 1D de triple gradient.
L’énergie élastique stockée dans le module d’un tel système déformé par un gradient de courbure combiné à une extension est d’abord établie par homogénéisation. Ceci conduit à la densité d’énergie du continu 1D effectif. Cette dernière implique trois coefficients de rigidité élastique liés au gradient de courbure, à l’allongement et au couplage des deux, dont les expressions sont explicitement liées à la morphologie du module, aux rigidités des microbarres et à la courbure. La formulation forte de l’équilibre du système est ensuite obtenue par minimization de l’énergie suivant la méthode d’Euler–Lagrange. Le calcul de la première variation de l’énergie de déformation permet de déterminer les forces externes généralisées qui peuvent être appliquées au continu 1D équivalent, à savoir des force normale et transversale ainsi qu’un couple et un double couple. Les équations d’équilibre correspondantes et les équations constitutives pour les forces et les couples et double couples sont également établies. Quelques illustrations numériques sont présentées pour ces nouveaux continus 1D chargés à leurs extrémités par des couples et des doubles couples.
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Mots-clés : Métamatériaux, mécanisme, milieu continu de troisième gradient, gradient de courbure, homogénéisation
Claude Boutin 1 , 2 ; Simir Moschini 2 , 3 ; Francesco d’Annibale 2 , 3 ; Francesco dell’Isola 2 , 3
CC-BY 4.0
Claude Boutin; Simir Moschini; Francesco d’Annibale; Francesco dell’Isola. From articulated-bi-parallelograms to third gradient 1D continua. Comptes Rendus. Mécanique, Volume 354 (2026), pp. 593-619. doi: 10.5802/crmeca.367
@article{CRMECA_2026__354_G1_593_0,
author = {Claude Boutin and Simir Moschini and Francesco d{\textquoteright}Annibale and Francesco dell{\textquoteright}Isola},
title = {From articulated-bi-parallelograms to third gradient {1D} continua},
journal = {Comptes Rendus. M\'ecanique},
pages = {593--619},
year = {2026},
publisher = {Acad\'emie des sciences, Paris},
volume = {354},
doi = {10.5802/crmeca.367},
language = {en},
}
TY - JOUR AU - Claude Boutin AU - Simir Moschini AU - Francesco d’Annibale AU - Francesco dell’Isola TI - From articulated-bi-parallelograms to third gradient 1D continua JO - Comptes Rendus. Mécanique PY - 2026 SP - 593 EP - 619 VL - 354 PB - Académie des sciences, Paris DO - 10.5802/crmeca.367 LA - en ID - CRMECA_2026__354_G1_593_0 ER -
%0 Journal Article %A Claude Boutin %A Simir Moschini %A Francesco d’Annibale %A Francesco dell’Isola %T From articulated-bi-parallelograms to third gradient 1D continua %J Comptes Rendus. Mécanique %D 2026 %P 593-619 %V 354 %I Académie des sciences, Paris %R 10.5802/crmeca.367 %G en %F CRMECA_2026__354_G1_593_0
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