In this paper we model the size-effects of metamaterial beams under bending with the aid of the relaxed micromorphic continuum. We analyze first the size-dependent bending stiffness of heterogeneous fully discretized metamaterial beams subjected to pure bending loads. Two equivalent loading schemes are introduced which lead to a constant moment along the beam length with no shear force. The relaxed micromorphic model is employed then to retrieve the size-effects. We present a procedure for the determination of the material parameters of the relaxed micromorphic model based on the fact that the model operates between two well-defined scales. These scales are given by linear elasticity with micro and macro elasticity tensors which bound the relaxed micromorphic continuum from above and below, respectively. The micro elasticity tensor is specified as the maximum possible stiffness that is exhibited by the assumed metamaterial while the macro elasticity tensor is given by standard periodic first-order homogenization. For the identification of the micro elasticity tensor, two different approaches are shown which rely on affine and non-affine Dirichlet boundary conditions of candidate unit cell variants with the possible stiffest response. The consistent coupling condition is shown to allow the model to act on the whole intended range between macro and micro elasticity tensors for both loading cases. We fit the relaxed micromorphic model against the fully resolved metamaterial solution by controlling the curvature magnitude after linking it with the specimen's size. The obtained parameters of the relaxed micromorphic model are tested for two additional loading scenarios.

翻译：本文借助松弛微极连续介质对超材料梁在弯曲下的尺寸效应进行建模。首先分析了异质全离散化超材料梁在纯弯曲载荷下随尺寸变化的弯曲刚度。引入两种等效加载方案，使梁沿长度方向产生恒定弯矩且无剪力作用。随后采用松弛微极模型来获取尺寸效应。基于该模型在两个明确标定的尺度之间运行的特点，提出了松弛微极模型材料参数的确定方法。这两个尺度分别由微弹性张量和宏弹性张量定义的线弹性理论给出，它们分别从上方和下方约束松弛微极连续介质。微弹性张量被指定为假定超材料所能表现的最大可能刚度，而宏弹性张量则由标准周期一阶均匀化方法给出。针对微弹性张量的识别，展示了两种不同方法，这些方法依赖于具有最刚响应候选单胞变体的仿射与非仿射狄利克雷边界条件。一致耦合条件使模型能够在两种加载情况下都能在宏微弹性张量之间的整个预定范围内起作用。通过将曲率幅值与试样尺寸关联，我们利用完全解析的超材料解拟合松弛微极模型。所得松弛微极模型参数在两个额外加载场景中进行了测试。