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

An inextensible 1D continuum whose deformation energy purely depends on the gradient of the 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 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. R & eacute;sum & eacute;. Un continuum 1D inextensible dont l'& eacute;nergie de d & eacute;formation d & eacute;pend purement du gradient de la courbure associ & eacute;e est introduit pour d & eacute;crire le comportement des structures en treillis Zigzagged Articulated Parallelograms with Articulated Braces (structures ZAPAB) apr & egrave;s homog & eacute;n & eacute;isation. Nous choisissons une structure ZAPAB particuli & egrave;re dans laquelle toutes les barres constitutives du module de base, sauf une, ne changent pas de longueur sous l'effet des charges appliqu & eacute;es. Ce choix judicieux nous permet de v & eacute;rifier, par des simulations num & eacute;riques, que le continuum 1D correspondant a effectivement une & eacute;nergie de d & eacute;formation qui d & eacute;pend uniquement de la d & eacute;riv & eacute;e de la courbure. Ainsi, en employant une approche de meilleur ajustement bas & eacute;e sur la m & eacute;thode des moindres carr & eacute;s, nous identifions num & eacute;riquement le meilleur coefficient de rigidit & eacute; (au sens des moindres carr & eacute;s) associ & eacute; & agrave; la contribution & eacute;nerg & eacute;tique due au gradient de courbure, appel & eacute; rigidit & eacute; de double courbure.Les simulations pr & eacute;sent & eacute;es consid & egrave;rent le cas de charges permanentes uniform & eacute;ment r & eacute;parties et r & eacute;v & egrave;lent une forte correspondance entre les configurations actuelles du mod & egrave;le continu 1D propos & eacute;, obtenues num & eacute;riquement par la m & eacute;thode des & eacute;l & eacute;ments finis, et les configurations actuelles de la structure ZAPAB (pour un nombre s & eacute;lectionn & eacute; de modules debase), obtenues par une approche num & eacute;rique discr & egrave;te, les courbes co & iuml;ncidant jusqu'& agrave; une certaine marge d'erreur intrins & egrave;que. Ces r & eacute;sultats n & eacute;cessitent le d & eacute;veloppement d'une proc & eacute;dure analytique d'identification micro-macro.