具有多重几何图形变换的机械超材料的扩展微态计算均匀化
Extended Micromorphic Computational Homogenization for Mechanical Metamaterials Exhibiting Multiple Geometric Pattern Transformations
摘要：
蜂窝状微结构已被证明在压缩下表现出局部弹性屈曲，具有三种可能的几何屈曲模式或模式变换。可以通过控制沿两个正交方向施加的压缩来引起各个图案变换，并且因此也引起空间分布的图案。利用这一特性具有巨大的潜力，例如，软机器人应用为了快速和优化设计，需要高效的数值工具，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图案转换。因此，本文提出了一种用于表现出多种模式转换的材料的微观均匀化框架，该框架扩展了Rokokovic等人的微观方案，J. Mech. Phys. Solids 123，119-137（2019），用于仅表现出单一图案变换的弹性体超材料。该方法是基于一个合适的运动学的anastomosis组成的一个光滑的部分，一组空间相关的波动场，和剩余的，空间不相关的微波动场。而后者场被忽略或浓缩在每个宏观物质点的水平上，空间相关的波动场的幅度出现在宏观尺度上作为微观场。我们开发的平衡方程，这些微观领域必须满足以及计算均匀化的方法来计算这些方程中的广义应力。为了证明该方法的潜力，在空间和时间的混合模式的加载情况下进行了研究，并与全尺寸模拟比较。结果表明，所提出的框架是能够捕捉相关的现象，虽然固有的多重性的解决方案需要敏感的初始猜测。
英文摘要：
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., J. Mech. Phys. Solids 123, 119-137 (2019), 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 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.
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