Lattice shells composed of two families of curved Kirchhoff rods: an archetypal example, topology optimization of a cycloidal metamaterial

A nonlinear elastic model for nets made up of two families of curved fibers is proposed. The net is planar prior to the deformation, but the equilibrium configuration that minimizes the total potential energy can be a surface in the three-dimensional space. This elastic surface accounts for the stretching, bending, and torsion of the constituent fibers regarded as a continuous distribution of Kirchhoff rods. A specific example of fiber arrangement, namely a cycloidal orthogonal pattern, is examined to illustrate the predictive abilities of the model and assess the limit of applicability of it. A numerical micro–macro-identification is performed with a model adopting a standard continuum deformable body at the level of scale of the fibers. A few finite element simulations are carried out for comparison purposes in statics and dynamics, performing modal analysis. Finally, a topology optimization problem has been carried out to change the macroscopic shear stiffness to enlarge the elastic regime and reduce the risk of damage without excessively losing bearing capacity.