Buckling-induced interaction between circular inclusions in an infinite thin plate
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). Here, 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 2ℓ. In particular, we investigate a region in phase space where buckling occurs at a narrow transition layer of length ℓ D around the radius of the inclusion, $R (ℓ_D \ll R)$. We show that the latter length scale defines two regions within the system, the close separation region, $ℓ – R \sim ℓ_D$, where the transition layers of the two inclusions approximately coalesce, and the far separation region, $ℓ – R \gg ℓ_D$. While the interaction energy decays exponentially in the latter region, $E_{\text{int}} ∝ e^{– (ℓ – R)/ℓ_D}$, 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.