Vol. 5, No. 2, 2017

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Linear pantographic sheets: Asymptotic micro-macro models identification

Claude Boutin, Francesco dell’Isola, Ivan Giorgio and Luca Placidi

Vol. 5 (2017), No. 2, 127–162
DOI: 10.2140/memocs.2017.5.127

In this paper we consider linear pantographic sheets, which in their natural configuration are constituted by two orthogonal arrays of straight fibers interconnected by internal pivots. We introduce a continuous model by means of a micro-macro identification procedure based on the asymptotic homogenization method of discrete media. The rescaling of the mechanical properties and of the deformation measures is calibrated in order to comply with the specific kinematics imposed by the quasi-inextensibility of the fibers together with the large pantographic deformability. The obtained high-order continuum model shows interesting and exotic features related to its extreme anisotropy and also to the subcoercivity of its deformation energy. Some initial numerical simulations are presented, showing that the model can account for experimental uncommon phenomena occurring in pantographic sheets. The paper focuses on the precise analysis and the understanding of the effective behavior based on a well-calibration of the extension and bending phenomena arising at the local scale. In an upcoming work, the analysis will be extended to oblique arrays, some analytical solutions to proposed equations and some further applications.

pantographic structures, second gradient elasticity, woven fabrics
Mathematical Subject Classification 2010
Primary: 74KXX, 74QXX, 76AXX
Received: 9 March 2016
Revised: 26 October 2016
Accepted: 11 January 2017
Published: 13 May 2017

Communicated by Pierre Seppecher
Claude Boutin
Département Génie Civil et Bâtiment - URA CNRS 1652
Ecole Nationale des Travaux Publics de l’Etat - Université de Lyon
rue Maurice Audin
69518 Vaulx-en-Velin
Francesco dell’Isola
Dept. di Ingegneria Strutturale e Geotecnica
Università di Roma “La Sapienza”
Via Eudossiana 18
I-00184 Roma
Ivan Giorgio
Università di Roma “La Sapienza”
Via Eudossiana 18
I-00184 Roma
Luca Placidi
International Telematic University Uninettuno
C.so Vittorio Emanuele II, 39
I-00186 Roma