Two-dimensional continua capable of large elastic extension in two independent directions: asymptotic homogenization, numerical simulations and experimental evidence

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)) 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) and dell’ Isola et al. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences (2016). it is considered a three-length-scale microstructure 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 elongations in some specific directions while remaining in the elastic regime. However, in pantographic sheets, the deformation energy only depends on the geodesic bending of the actual configuration 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), 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 microstructure 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 first 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.