Two-dimensional continua capable of large elastic extension in two independent directions: asymptotic homogenization, numerical simulations and experimental evidence
Résumé
The synthesis of a 1D full second gradient continuum was obtained by the design of so-called pantographic beam (see Alibert et al. Mathematics and Mechanics of Solids (2003) [2]) and the problem of the synthesis of planar second gradient continua has been faced in several subsequent papers: in dell'Isola et al. Zeitschrift für angewandte Mathematik und Physik (2015) [5] and dell'Isola et al. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences (2016). [4] it is considered a three-length-scale mi-crostructure in which two initially orthogonal families of long Euler beams (i.e. beams much longer than the size of the homogenization cell but much slenderer than it) are interconnected by perfect or elastic pivots (hinges). The corresponding homogenized two-dimensional continuum (which was called pantographic sheet) has a D4 orthotropic symmetry. It has been proven to have a deformation energy depending on the second gradient of in-plane displacements and to allow for large elon-gations in some specic directions while remaining in the elastic regime. However, in pantographic sheets, the deformation energy only depends on the geodesic bending of the actual conguration of its symmetry directions (see for more details Steigmann et al. Acta Mechanica Sinica (2015) [3] and Placidi et al. Journal of Engineering Mathematics (2017) [6]). On the other hand, in Seppecher et al. J. of Physics: Conference Series vol. 319 (2011) [1], it was designed a bi-pantographic architectured sheet where the previously considered Euler beams were replaced by pantographic beams to form a more complex three-length-scale mi-crostructure and it was proven that, once homogenized, such a bi-pantographic sheet, in planar and linearized deformation states, produces a more complete second gradient two-dimensional continuum. Derivatives of elongations along the two symmetry directions now appear in the deformation energy. The aim of the present paper is the experimental validation of the second gradient behavior of such bi-pantographic sheets. As their intrinsic mechanical structure produces a geometrically non-linear behavior for relatively small total deformation , we rst need to extend the homogenization result to the regime of large deformations. Subsequently we compare the predictions obtained using such second gradient model with experimental evidence, as elaborated by local Digital Image Correlation (DIC) focused on the discrete kinematics of the hinges. 1 Introduction In physical literature the problem of synthesis has been confronted in many contexts. Namely, given a La-grangian potential (describing the conservative part of considered phenomena) and a Rayleigh dissipation potential , specied in terms of suitable kinematic descrip-tors, one has to nd a physical system, belonging to a class specied a priori, whose evolution is governed by the corresponding Hamilton-Rayleigh principle. In the period (1930-1970) in which the prevalence of digital computers was not yet achieved, the problem of synthesis of electric circuits was confronted in order to design suitable, and dedicated, analog computers (see, e.g., Kron [13, 14]). When the a priori class of physical systems is constituted by electric circuits with only passive elements, the previous general problem was par-ticularized as follows: given quadratic Lagrangian and Rayleigh potentials, in terms of a nite number of degrees of freedom, one has to nd the graph of a circuit and the interconnecting (linear and passive) electric elements such that it is governed by the corresponding Hamilton-Rayleigh principle. The available results found in the literature for the synthesis of passive electric circuits were recently used to synthesize piezoelectromechanical metamateri-als, suitably tailored to dampen out mechanical vibrations (see dell'Isola et al. [15, 16, 17] and [18, 25]). The synthesis problem for mechanical metamaterials reads as: given any choice of the continuous elds describing a kinematics, given functionals expressing the deformation energy, the kinetic energy and the dissipa-tion potential in terms of these elds, nd the architec-tured mechanical structure (possibly multi-scale) such that, in the homogenization limit, the obtained continuous model is exactly the one chosen a priori. Therefore , the qualitative behaviour of a metamaterial shall be given by its multi-scale architecture rather than by the constituting base materials [31].
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