Equilibrium of Two-Dimensional Cycloidal Pantographic Metamaterials in Three-Dimensional Deformations

A particular pantographic sheet, modeled as a two-dimensional elastic continuum consisting of an orthogonal lattice of continuously distributed fibers with a cycloidal texture, is introduced and investigated. These fibers conceived as embedded beams on the surface are allowed to be deformed in a three-dimensional space and are endowed with resistance to stretching, shearing, bending, and twisting. A finite element analysis directly derived from a variational formulation was performed for some explanatory tests to illustrate the behavior of the newly introduced material. Specifically, we considered tests on: (1) bias extension; (2) compressive; (3) shear; and (4) torsion. The numerical results are discussed to some extent. Finally, attention is drawn to a comparison with other kinds of orthogonal lattices, namely straight, parabolic, and oscillatory, to show the differences in the behavior of the samples due to the diverse arrangements of the fibers.