The classical Cauchy continuum theory is suitable to model highly homogeneous materials. However, many materials, such as porous media or metamaterials, exhibit a pronounced microstructure. As a result, the classical continuum theory cannot capture their mechanical behaviour without fully resolving the underlying microstructure. In terms of finite element computations, this can be done by modelling the entire body, including every interior cell. The relaxed micromorphic continuum offers an alternative method by instead enriching the kinematics of the mathematical model. The theory introduces a microdistortion field, encompassing nine extra degrees of freedom for each material point. The corresponding elastic energy functional contains the gradient of the displacement field, the microdistortion field and its Curl (the micro-dislocation). Therefore, the natural spaces of the fields are ${\left[H^1\right]}^3$ for the displacement and ${\left[\mathit{H}\left(\mathrm{curl}\right)\right]}^3$ for the microdistortion, leading to unusual finite element formulations. In this work we describe the construction of appropriate finite elements using Nédélec and Raviart–Thomas subspaces, encompassing solutions to the orientation problem and the discrete consistent coupling condition. Further, we explore the numerical behaviour of the relaxed micromorphic model for both a primal and a mixed formulation. The focus of our benchmarks lies in the influence of the characteristic length $L_{\mathrm{c}}$ and the correlation to the classical Cauchy continuum theory.

经典的柯西连续统理论适用于模拟高度均匀的材料。然而，许多材料，如多孔介质或超材料，表现出明显的微观结构。因此，如果不完全解析潜在的微观结构，经典连续统理论就无法捕捉它们的机械行为。就有限元计算而言，这可以通过对整个身体（包括每个内部细胞）进行建模来完成。松弛的微形态连续体通过丰富数学模型的运动学提供了一种替代方法。该理论引入了一个微畸变场，每个材料点包含九个额外的自由度。相应的弹性能泛函包含位移场的梯度、微畸变场及其卷曲（微位错）。因此，田野的自然空间是${\left[H^1\right]}^3$对于位移和${\left[\mathit{H}\left(\mathrm{卷曲}\right)\right]}^3$对于微失真，导致不寻常的有限元公式。在这项工作中，我们描述了使用 Nédélec 和 Raviart-Thomas子空间构建适当的有限元，包括定向问题和离散一致耦合条件的解决方案。此外，我们探索了原始和混合配方的松弛微形态模型的数值行为。我们的基准测试的重点在于特征长度的影响${\mathrm{大号}}_{\mathrm{C}}$以及与经典柯西连续统理论的相关性。