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.

两个平行等距双轴梁的正交系列的连接产生了表征所谓的四轴超材料的 2D 微观结构。在本文中，基于已经获得的双骷髅梁的均质化结果，我们检索了内部约束的二维非线性 Cosserat 连续体，该连续体描述了在两类双骷髅梁中考虑刚性连接时，四骷髅超材料的面内机械行为。与双轴梁相反，由于变形能依赖于沿两个空间方向的运动学量的偏导数，因此四轴超材料的极限模型不能简化为描述受势影响的无约束粒子运动的初值问题。这需要在考虑内部约束的情况下开发有限元公式。在这篇文章中，在介绍了根据变形能描述四分体超材料的连续体之后，我们利用虚拟工作原理来获得弱形式的控制方程。然后将这些方程进行局部化，以获得平衡方程和相关的自然边界条件。通过定义两个不同的等效弱公式，在双轴梁上测试伽辽金方法近似 tretraskelion 超材料的可行性，这些公式被离散化，然后通过用于夹紧-夹紧拉/推测试的 Newton-Rhapson 方案求解。结论是，鉴于问题的高度非线性，初始估计值的选择对于获得解决方案至关重要，尤其是几个分叉解决方案中的所需解决方案。