We resort to variational methods to evaluate the asymptotic behavior of fine metamaterials as a function of cell size. To zeroth order, the metamaterial behaves as a micropolar continuum with both displacement and rotation degrees of freedom, but exhibits linear-elastic fracture mechanics scaling and therefore no size effect. To higher order, the overall energetics of the metastructure can be characterized explicitly in terms of the solution of the zeroth-order continuum problem by the method of Γ-expansion. We present explicit expressions of the second-order correction for octet frames. As an application, we evaluate the compliance of double-cantilever octet specimens to second order and use the result to elucidate the dependence of the apparent toughness of the specimen on cell size. The analysis predicts the discreteness of the metamaterial lattice to effectively shield the crack-tip, a mechanism that we term lattice shielding. The theory specifically predicts anti-shielding, i. e., coarser is weaker, in agreement with recent experimental observations.

中文翻译：

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机械超材料中的断裂和尺寸效应


我们采用变分方法来评估精细超材料的渐近行为与单元大小的函数关系。到零阶，超材料表现为具有位移和旋转自由度的微极连续体，但表现出线弹性断裂力学缩放，因此没有尺寸效应。对于更高阶，超结构的整体能量可以用Γ展开方法求解零阶连续体问题来明确表征。我们提出了八位字节帧的二阶校正的显式表达式。作为一项应用，我们评估了双悬臂八位字节试样对二阶的顺应性，并使用结果来阐明试样的表观韧性对泡孔大小的依赖性。该分析预测了超材料晶格的离散性，以有效地屏蔽裂纹尖端，我们称之为晶格屏蔽的机制。该理论特别预测了反屏蔽，即较粗较弱，这与最近的实验观察结果一致。