Design of slender artificial materials and morphogenesis of thin biological tissues typically involve stimulation of isolated regions (inclusions) in the growing body. These inclusions apply internal stresses on their surrounding areas that are ultimately relaxed by out-of-plane deformation (buckling). We utilize the Föppl-von Kármán model to analyze the interaction between two circular inclusions in an infinite plate that their centers are separated a distance of $2\ell$. In particular, we investigate a region in phase space where buckling occurs at a narrow transition layer of length ${\ell }_D$ around the radius of the inclusion, $R$ (${\ell }_D\ll R$). We show that the latter length scale defines two regions within the system, the close separation region, $\ell -R\sim {\ell }_D$, where the transition layers of the two inclusions approximately coalesce, and the far separation region, $\ell -R\gg {\ell }_D$. While the interaction energy decays exponentially in the latter region, $E_{\text{int}}\propto e^{-(\ell -R)/{\ell }_D}$, it presents nonmonotonic behavior in the former region. While this exponential decay is predicted by our analytical analysis and agrees with the numerical observations, the close separation region is treated only numerically. In particular, we utilize the numerical investigation to explore two different scenarios within the final configuration: The first where the two inclusions buckle in the same direction (up-up solution) and the second where the two inclusions buckle in opposite directions (up-down solution). We show that the up-down solution is always energetically favorable over the up-up solution. In addition, we point to a curious symmetry breaking within the up-down scenario; we show that this solution becomes asymmetric in the close separation region.

中文翻译：

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无限薄板中圆形夹杂物之间的屈曲诱导相互作用

细长的人造材料的设计和薄薄的生物组织的形态发生通常涉及刺激生长体中的孤立区域（包含物）。这些夹杂物在其周围区域上施加内部应力，这些应力最终会因平面外变形（屈曲）而松弛。我们利用Föppl-vonKármán模型分析无限圆板中两个圆形夹杂物之间的相互作用，其中心距离为$2\ell$。尤其是，我们研究了相空间中在长度狭窄的过渡层处发生屈曲的区域${\ell }_d$ 围绕夹杂物的半径， $\mathrm{[R}$ （${\ell }_d\ll \mathrm{[R}$）。我们证明了后者的长度尺度定义了系统内的两个区域，即紧密分离区域，$\ell -\mathrm{[R}〜{\ell }_d$，其中两个夹杂物的过渡层大致合并，并且相隔很远， $\ell -\mathrm{[R}\gg {\ell }_d$。尽管相互作用能量在后一个区域呈指数衰减，${Ë}_{\text{整型}}\propto {Ë}^{-（\ell -\mathrm{[R}）/{\ell }_d}$，它在前一个区域呈现非单调行为。尽管这种指数衰减是通过我们的分析分析预测的，并且与数值观测结果一致，但仅对数值进行了离散处理。特别是，我们利用数值研究来探索最终配置中的两种不同情况：第一种情况是两个夹杂物沿相同方向弯曲（向上-向上解决方案），第二种情况是两个夹杂物沿相反方向弯曲（向上-向下）解）。我们表明，上下解决方案始终在能量上优于向上解决方案。此外，我们指出了在上下场景中出现的奇怪对称性。我们表明，该解决方案在紧密分离区域变得不对称。