# Andre Galligo - UNS, LJAD - Keywords: Elasticity, Surprising non linear behavior

We consider a polycarbonate honeycomb with circular close-packed cells, a widely used material, that we submitted to a in-plane quasi-static uniaxial compression. Since the common tangents to the circles of the plane initial configuration form an (abstract) hexagonal mesh, there are two different natural directions of compression, namely when the vertical compression axis is either parallel or perpendicular to a tangent. We observed that in the first case the deformation was homogeneous, compressing all the rows. However, in the second case, we observed a localization along few horizontal rows. More surprisingly the second kind of deformations were also reversible, contradicting usual expectations. We will briefly review related works on this subject.

In order to explain these observations, we analyze the deformations of the intercellular curved triangles made by 3 portions of circles (instead of the circles as borders of discs). We call them triangular elastica, in reference to the classical Euler's elastica, and represent them by graphical spline approximations. Their compressions can give rise to interesting deformations: they can combine buckling, rotation and straightening, that we further study, in particular the loss and recovery of symmetries.

We also developed a "mesoscopic" approximation of our setting by a simpler structure, with few degrees

of freedom, for which we provide a complete mathematical analysis. It involves a compressed rod jointed to a moving platform attached by two springs which harden in compression.

More generally for honeycombs and foams, our approach allows to compare and discuss, the global vs local elasticity behaviors of the material as well as the abstract vs effective crash of the structure

and its surprising recovery.

We relied on computer simulations and the talk will be illustrated with graphs, photos and animations.