Nonlinear Analysis of 2D Topological Maxwell Lattices

The relatively recent discovery of novel topological phases of matter has quickly become of primary importance in the study of condensed matter physics, photonics, and more recently mechanics. However, most research in the field of topological physics takes place in the linear regime. Nonlinear topological materials could combine known effects from nonlinear systems such as amplitude dependent responses, harmonic generation, frequency conversion, and solitons, with known topological effects such as robust localized edge modes. In this study, the kinematics of a topological Maxwell lattice is analyzed including all nonlinear geometric effects. Zero energy floppy modes are known to exist in critically coordinated lattices from the Maxwell-Calladine counting theorem. The topological nature of these floppy modes is described by the polarization vector and is found from the Winding Numbers in the phase space of the Connectivity matrix C(q). In this project, the lattice is shown to behave as the linear theory predicts at small perturbation amplitudes. At large amplitudes we successfully demonstrate spatial harmonic generation of the topological floppy modes and show the possibility of local topological polarization switching that creates tunable domain walls in the lattice. An experimental setup using a 3D printed lattice is performed to verify the numerical results. We anticipate particular technological applications may stem from this research in the areas of adaptive and smart materials.