Enhanced Piola–Hencky discrete models for pantographic sheets with pivots without deformation energy: numerics and experiments

Emilio Turco (1) , Anil Misra (2) , Marek Pawlikowski (3) , Francesco Dell 'Isola (4) , François Hild (5)

1 UNISS, Dipartimento di Architettura, Design e Urbanistica
2 CEAE - Department of Civil, Environmental and Architectural Engineering
3 Warsaw University of Technology [Warsaw]
4 M&MoCS - International Research Centre on Mathematics & Mechanics of Complex Systems
5 LMT - Laboratoire de Mécanique et Technologie

The problem of the synthesis of second gradient (meta)materials, via architectured microstructures constituted by beams micro lattices, has been solved by choosing ideal pivots as preferred constraints. The obtained homogenised macro equationsshow some pathologies which actually reflect the exotic behaviour of considered metamaterials, even if they are of interest by themselves. Indeed the theoretical issues that they provoke not only represent an intellectual challenge but also present a mean for disclosing potentially interesting new phenomena. Obviously, to make such disclosure evident, the related technological demand arose: to find an innovative design and production process to construct, by using additive manufacturing, some pantographic sheets (made in this instance in polyamide but hopefully later also in metal) whose pivots do twist practically without deformation and with negligible dissipation. Remarkably the specimens could be printed in a monolith and required no post assembly but only an easily standardised run-in procedure. In this paper, in order to introduce a mathematical description for pantographic sheets with perfect pivots and to avoid to face the aforementioned pathologies, we formulate a discrete, finite dimensional, Lagrangian model. Moreover in order to include the case in which the beams interconnecting the pivots are long enough to store non negligible bending energy between the closest pairs of pivots we formulate an enhanced Piola–Hencky discrete model. In it we distinguish two kinds of nodes: the first one interconnects the two pantographic fibers, the second one simply interconnects two different segments of the same fiber. The Vietnam long neck peculiar deformed shape experimentally observed in standard extension bias test, is obtained with very short computing time, so that the innovative code which has been elaborated can be used as subroutine in more complex computation schemes. A preliminary digital image correlation analysis is performed and shows that a remarkable agreement between theoretical predictions and experimental evidence can be obtained. This circumstance is easily explained by observing that said numerical code is based on a discrete model directly inspired by the mechanical properties of pantographic sheets and that, therefore, the passages to a continuum model via homogenisation and then to the subsequent re-discretization, via the introduction of more or less suitable finite elements, are avoided. In our opinion a theory driven formulation of a directly discrete numerical model presents many advantages and it seems suitable for attacking future structural optimisation problems.

Mots clés: pantographic sheets; Mechanics of materials; discrete models; nonlinear analysis

Domaines: Mécanique des matériaux [physics.class-ph]

Emilio Turco, Anil Misra, Marek Pawlikowski, Francesco Dell 'Isola, François Hild. Enhanced Piola–Hencky discrete models for pantographic sheets with pivots without deformation energy: numerics and experiments. International Journal of Solids and Structures, 2018, 147, pp.94-109. ⟨10.1016/j.ijsolstr.2018.05.015⟩. ⟨hal-01791894⟩