Towards the Galerkin approximation of tetraskelion metamaterials

The connection of two orthogonal families of parallel equispaced duoskelion beams results in a 2D microstructure characterizing so-called tetraskelion metamaterials. In this paper, based on the homogenization results already obtained for duoskelion beams, we retrieve the internally-constrained two-dimensional nonlinear Cosserat continuum describing the in-plane mechanical behaviour of tetraskelion metamaterials when rigid connection is considered among the two families of duoskelion beams. Contrarily to duoskelion beams, due to the dependence of the deformation energy upon partial derivatives of kinematic quantities along both space directions, the limit model of tetraskelion metamaterials cannot be reduced to an initial value problem describing the motion of an unconstrained particle subjected to a potential. This calls for the development of a finite element formulation taking into account the internal constraint. In this contribution, after introducing the continuum describing tetraskelion metamaterials in terms of its deformation energy, we exploit the Virtual Work Principle to get governing equations in weak form. These equations are then localised to get the equilibrium equations and the associated natural boundary conditions. The feasibility of a Galerkin approach to the approximation of tretraskelion metamaterials is tested on duoskelion beams by defining two different equivalent weak formulations that are discretised and then solved by a Newton-Rhapson scheme for clamped-clamped pulling/pushing tests. It is concluded that, given the high nonlinearity of the problem, the choice of the initial guess is crucial to get a solution and, particularly, a desired one among the several bifurcated ones.