Nonlinear dynamics of origami metamaterials: energetic discrete approach accounting for bending and in-plane deformation of facets

In this paper, we start the analysis of the nonlinear dynamics of structural elements having an origami-type microstructure and micro-kinematics, also known as origami metamaterials. We use a finite-dimensional Lagrangian system to explore, via numerical simulations, the overall behaviour of an origami beam. This provides some significant hints about the structure of an effective homogenized continuum model for such a beam. We introduce a geometrically exact two-dimensional triangular discrete element, whose kinematics is given in terms of three-dimensional nodal displacements. Inertial terms are taken into account. Facets—which in this paper are quadrilateral—are modelled as the union of several triangles, each triangle deforming affinely in plane. Facet bending and sheet folding are taken into account by constraining through cylindrical hinges adjacent triangles and placing in-between torsional springs between them. In-plane and bending/folding strain energies are estimated from the elongation of the triangles’ sides and from the relative rotations of adjacent triangles, respectively. The actual reconstruction of the equilibrium path is performed numerically through a stepwise time integration scheme that can handle large displacements.