Honeycomb-like microstructures have been shown to exhibit local elastic buckling under compression, with three possible geometric buckling modes, or pattern transformations. The individual pattern transformations, and consequently also spatially distributed patterns, can be induced by controlling the applied compression along two orthogonal directions. Exploitation of this property holds great potential in, e.g., soft robotics applications. For fast and optimal design, efficient numerical tools are required, capable of bridging the gap between the microstructural and engineering scale, while capturing all relevant pattern transformations. A micromorphic homogenization framework for materials exhibiting multiple pattern transformations is therefore presented in this paper, which extends the micromorphic scheme of Rokoš et al. (2019) [1], for elastomeric metamaterials exhibiting only a single pattern transformation. The methodology is based on a suitable kinematic ansatz consisting of a smooth part, a set of spatially correlated fluctuating fields, and a remaining, spatially uncorrelated microfluctuation field. Whereas the latter field is neglected or condensed out at the level of each macroscopic material point, the magnitudes of the spatially correlated fluctuating fields emerge at the macroscale as micromorphic fields. We develop the balance equations which these micromorphic fields must satisfy as well as a computational homogenization approach to compute the generalized stresses featuring in these equations. To demonstrate the potential of the methodology, loading cases resulting in mixed modes in both space and time are studied and compared against full-scale numerical simulations. It is shown that the proposed framework is capable of capturing the relevant phenomena, although the inherent multiplicity of solutions entails sensitivity to the initial guess.

蜂窝状的微结构已经显示出在压缩下具有局部弹性屈曲，具有三种可能的几何屈曲模式或模式转换。可以通过控制沿两个正交方向施加的压缩来诱导各个模式转换，进而也可以实现空间分布的模式。利用这种特性在例如软机器人应用中具有巨大的潜力。为了实现快速，最佳的设计，需要有效的数值工具，该工具能够弥合微观结构和工程规模之间的差距，同时捕获所有相关的模式转换。因此，本文提出了一种具有多种模式转变的材料的微形态均质框架，该框架扩展了Rokoš等人的微形态方案。（2019）[1]，用于弹性体超材料的展示仅具有单个图案转变。该方法基于合适的运动学基础，包括一个平滑部分，一组空间相关的波动场和一个剩余的空间不相关的微波动场。尽管后一个场在每个宏观物质点的水平上被忽略或凝聚，但空间相关的波动场的大小在宏观上以微形态场出现。我们开发了这些微晶场必须满足的平衡方程，以及计算均化方法来计算这些方程中的广义应力。为了展示该方法的潜力，研究了在空间和时间上导致混合模式的载荷工况，并将其与全尺寸数值模拟进行了比较。结果表明，所提出的框架能够捕获相关现象，尽管解决方案固有的多样性要求对初始猜测具有敏感性。