Exact analytical solutions in two dimensional plate-like mechanical metamaterials: State of free deformation in a topological cylinder

Abstract

Nodal displacements for two dimensional plate-like lattices under static loading are solved in exact analytical form, when periodic boundary conditions are applied to the top and bottom surfaces of the lattice forming a topological cylinder. Discrete Fourier transform converts the governing equation of static equilibrium into a set of one dimensional wavenumber dependent problems. Characteristic solutions are developed for each Fourier mode of deformation. Inverse discrete Fourier transform converts the wavenumber domain function back to the spatial domain by taking a linear combination of the one dimensional harmonic solutions. As an example, the point force is decomposed into a set of wavenumber dependent loading profiles. A linear combination of the displacement solutions for the individual harmonic loading profiles is sufficient to reproduce the overall displacement solution for the point load. This property holds for any type of the load. Nodal displacements, expressed with analytical dependence on the nodal indices n and m, match the results of commercial finite element analysis software. Overall, this paper demonstrates that the static response of discrete lattices is equivalently represented as a superposition of the wavenumber domain solutions, which is analogous to frequency domain analysis in acoustic metamaterials and phononics.