Large deformations of planar extensible beams and pantographic lattices: Heuristic homogenisation, experimental and numerical examples of equilibrium

International audienceThe aim of this paper is to find a computationally efficient and predictive model for the class of systems that we call " pantographic structures ". The interest in these materials was increased by the possibilities opened by the diffusion of technology of 3D printing. They can be regarded, once choosing a suitable length scale, as families of beams (also called fibres) interconnected each other by pivots and undergoing large displacements and large deformations. There are, however, relatively few " ready-to-use " results in the literature of non-linear beam theory. In this paper, we consider a discretised springs model for extensible beams and propose a heuristic homogenisation technique of the kind first used by Piola to formulate a continuum fully non-linear beam model. The homogenised energy which we obtain has some peculiar and interesting features which we start to describe by solving numerically some exemplary deformation problems. Furthermore, we consider pantographic structures, find the corresponding homogenised second gradient deformation energies and study some planar problems. Numerical solutions for these 2D problems are obtained via minimisation of energy and are compared with some experimental measurements, in which elongation phenomena cannot be neglected