Modeling the behavior of inclusions in circular plates undergoing 2D-to-3D shape changes

Growth of biological tissues and shape changes in thin synthetic sheets are commonly driven by stimulation of isolated regions (inclusions) in the growing body. These inclusions apply internal forces on their surroundings that, in turn, promote 2D layers to acquire complex 3D structures. We focus on a fundamental building block of these systems, and consider a circular plate that contains an inclusion with dilative strains. We derive an analytical model that predicts the 2D-to-3D shape transitions in the system. The solution of our model reveals two distinct patterns in the post-buckling region of the plate. One is an extensive pattern that holds close to the threshold of the instability, and the second is a localized pattern, which preempts the extensive solution beyond the buckling threshold. Then, we utilize these findings to analyze the interaction between two circular inclusions that are embedded in an infinite plate and undergo a buckling instability into a localized state. We show that when the two inclusions are separated far apart the interaction energy between the inclusions decays exponentially.