Two layers pantographs: A 2D continuum model accounting for the beams’ offset and relative rotations as averages in SO(3) Lie groups

In the problem of synthesis metamaterials, pantographic architecture revealed remarkable potentialities. Indeed it allowed for second gradient 2D (nonlinear) continua: i.e. shells whose deformation energy depends also on derivatives displacements in tangent directions to reference configuration. Moreover, seems be able produce metamaterials macroscopic elongations are large, albeit remaining elastic regime. The theoretically shown potentialities have started become ?practical? interest thanks a series experiments, which were made possible by recent 3D additive manufacturing. actual construction has been based design two arrays beams interconnected small cylinders, behavior can modeled different ways: if they very short regarded as clamps, while enough (or inelastic large rotations) cylindrical hinges connecting arrays. Otherwise, must inelastic) elements allowing relative rotations and displacements. this paper, we focus particular case introduce, after homogenization heuristic arguments, generalized continuum model kinematics is characterized placement rotation fields (one each array beams) rotations. offset between proven an essential tool defining effective invariant kinematical measures. facts, one wants postulate introduced gives predictions agreement with those given more refined where described its maximum geometric complexity constituting material assumed modelable standard first continuum. present order arrive at correct conjecture postulated energy, consider concept averages SO(3) Lie group. used enriched seen alternative adoption models. Some rather surprising processes studied, interesting non-symmetric post-buckling phenomena observed both models used. Mentioned experimentally.