Computational identification of double-bending stiffness: from Zigzagged Articulated Parallelograms with Articulated Braces (ZAPAB) structures to pure-curvature gradient planar inextensible 1D continua

Larry Murcia Terranova; Emilio Turco; Anil Misra; Francesco dell’Isola

doi:10.5802/crmeca.300

Article de recherche

Computational identification of double-bending stiffness: from Zigzagged Articulated Parallelograms with Articulated Braces (ZAPAB) structures to pure-curvature gradient planar inextensible 1D continua
[Identification computationnelle de la rigidité de la double flexion : des parallélogrammes articulés en zigzag aux structures de bras articulés aux continuums 1D inextensibles planaires à gradient de courbure pur]

Larry Murcia Terranova ¹ ; Emilio Turco ² ; Anil Misra ³ ; Francesco dell’Isola ^4,^5,⁶

¹ Department of Information Engineering, Computer Science and Mathematics, University of L’Aquila, L’Aquila, Italy
² Department of Architecture, Design and Urban Planning, University of Sassari, Alghero, Sassari, Italy
³ Department of Civil and Environmental Engineering, Florida International University, Miami, Florida, USA
⁴ Department of Civil, Construction-Architecture and Environmental Engineering, University of L’Aquila, L’Aquila, Italy
⁵ Warsaw University of Technology, Poland
⁶ ENS Paris Saclay, France

Comptes Rendus. Mécanique, Volume 353 (2025), pp. 647-672.

Abstract (VO)
Résumé

An inextensible 1D continuum whose deformation energy purely depends on the gradient of the associated curvature is introduced to describe the behavior of Zigzagged Articulated Parallelograms with Articulated Braces truss structures (ZAPAB structures) after homogenization. We choose a particular ZAPAB structure in which all but one of the constituting bars of the basic module do not change their length under applied loads. This judicious choice allows us to verify, through numerical simulations, that the corresponding 1D continuum indeed has a deformation energy that depends solely on the derivative of curvature. Thus, by employing a best-fitting approach based on the least squares method, we numerically identify the best stiffness coefficient (in the least squared sense) associated with the energy contribution due to the gradient of curvature, termed as double-bending stiffness. The presented simulations consider the case of uniformly distributed applied dead loads, and reveal a strong match between the current configurations of the proposed 1D continuum model, obtained numerically through the Finite Element Method, and the current configurations of the ZAPAB structure (for a selected number of basic modules), obtained through a discrete numerical approach, with the curves coinciding up to certain intrinsic error. These results require the development of an analytical micro–macro identification procedure. ZAPAB structures facilitate advances in the synthesis of tailored materials and the n-th gradient theory. We adopt a theory-driven approach with the expectation of devising materials with exotic behaviors. Specifically, we anticipate that material lines capable of not storing deformation energy under uniform bending (constant curvature) will be obtained after homogenization, thereby paving the way for future work that introduces complex materials built upon them. Our discussion is inspired by well-known pantographic structures, which serve as archetypes of second gradient materials designed in such a way that no deformation energy is stored under uniform extension.

Un continuum 1D inextensible dont l’énergie de déformation dépend purement du gradient de la courbure associée est introduit pour décrire le comportement des structures en treillis Zigzagged Articulated Parallelograms with Articulated Braces (structures ZAPAB) après homogénéisation. Nous choisissons une structure ZAPAB particulière dans laquelle toutes les barres constitutives du module de base, sauf une, ne changent pas de longueur sous l’effet des charges appliquées. Ce choix judicieux nous permet de vérifier, par des simulations numériques, que le continuum 1D correspondant a effectivement une énergie de déformation qui dépend uniquement de la dérivée de la courbure. Ainsi, en employant une approche de meilleur ajustement basée sur la méthode des moindres carrés, nous identifions numériquement le meilleur coefficient de rigidité (au sens des moindres carrés) associé à la contribution énergétique due au gradient de courbure, appelé rigidité de double courbure. Les simulations présentées considèrent le cas de charges permanentes uniformément réparties et révèlent une forte correspondance entre les configurations actuelles du modèle continu 1D proposé, obtenues numériquement par la méthode des éléments finis, et les configurations actuelles de la structure ZAPAB (pour un nombre sélectionné de modules de base), obtenues par une approche numérique discrète, les courbes coïncidant jusqu’à une certaine marge d’erreur intrinsèque. Ces résultats nécessitent le développement d’une procédure analytique d’identification micro–macro.

Reçu le : 2025-02-28
Révisé le : 2025-04-29
Accepté le : 2025-05-04
Publié le : 2025-05-20

DOI : 10.5802/crmeca.300

Keywords: Metamaterials, 1D continuum model, Third gradient continua, Gradient of curvature, Computational identification, ZAPAB structure
Mots-clés : Métamatériaux, Modèle de continuum 1D, Continuum de troisième gradient, Gradient de courbure, Identification computationnelle, Structure ZAPAB

Affiliations des auteurs :

Larry Murcia Terranova ¹ ; Emilio Turco ² ; Anil Misra ³ ; Francesco dell’Isola ^{4, 5, 6}

¹ Department of Information Engineering, Computer Science and Mathematics, University of L’Aquila, L’Aquila, Italy
² Department of Architecture, Design and Urban Planning, University of Sassari, Alghero, Sassari, Italy
³ Department of Civil and Environmental Engineering, Florida International University, Miami, Florida, USA
⁴ Department of Civil, Construction-Architecture and Environmental Engineering, University of L’Aquila, L’Aquila, Italy
⁵ Warsaw University of Technology, Poland
⁶ ENS Paris Saclay, France

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     title = {Computational identification of double-bending stiffness: from {Zigzagged} {Articulated} {Parallelograms} with {Articulated} {Braces} {(ZAPAB)} structures to pure-curvature gradient planar inextensible {1D} continua},
     journal = {Comptes Rendus. M\'ecanique},
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Larry Murcia Terranova; Emilio Turco; Anil Misra; Francesco dell’Isola. Computational identification of double-bending stiffness: from Zigzagged Articulated Parallelograms with Articulated Braces (ZAPAB) structures to pure-curvature gradient planar inextensible 1D continua. Comptes Rendus. Mécanique, Volume 353 (2025), pp. 647-672. doi : 10.5802/crmeca.300. https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.5802/crmeca.300/

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