We study periodic arrays of impurities that create localized regions of expansion, embedded in two-dimensional crystalline membranes. These arrays provide a simple elastic model of shape memory. As the size of each dilational impurity increases (or the relative cost of bending to stretching decreases), it becomes energetically favorable for each of the $M$ impurities to buckle up or down into the third dimension, thus allowing for of order $2^M$ metastable surface configurations corresponding to different impurity “spin” configurations. With both discrete simulations and the nonlinear continuum theory of elastic plates, we explore the buckling of both isolated dilations and dilation arrays at zero temperature, guided by analogies with Ising antiferromagnets. We conjecture ground states for systems with triangular and square impurity superlattices, and comment briefly on the possible behaviors at finite temperatures.

中文翻译：

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带有扩张阵列的膜的屈曲和亚稳

我们研究了杂质的周期性阵列，这些杂质会在二维晶体膜中嵌入局部膨胀区域。这些阵列提供了简单的形状记忆弹性模型。随着每种膨胀杂质的尺寸增加（或弯曲至拉伸的相对成本降低），对于每种$\mathrm{中号}$ 杂质向上或向下弯曲到第三维，因此可以有序 $2^{\mathrm{中号}}$亚稳态表面构型对应于不同的杂质“自旋”构型。通过离散模拟和弹性板的非线性连续理论，我们在零温度下以伊辛反铁磁体的类比为指导，探索了孤立的扩张和扩张阵列的屈曲。我们推测具有三角形和正方形杂质超晶格的系统的基态，并简要评论有限温度下的可能行为。