Pattern Formation in Confined Core-Shell Structures: Stiffness, Curvature, and Hierarchical Wrinkling

Adapting sheets to doubly curved surfaces is a longstanding challenge in various engineering disciplines, from flexible and stretchable electronics to the automotive industry. However, current understanding often focuses on specific scenarios and neglects the diverse range of substrate conditions encountered in nature. This study investigates the pattern formation in confined core-shell structures by modulating the levels of stiffness and curvature of the substrate. Beginning with the Föppl–von Kármán equation, this theory uncovers how the degree of confinement determines the location of wrinkles within confined sheets. Furthermore, the synchronization of patterns is observed: the formation of dimple or buckyball patterns on curved surfaces simultaneously with hierarchical wrinkles due to boundary constraints. The analytical models elucidate these phenomena, and both macroscopic and microscopic features can be systematically engineered to align with quantitative predictions. This research expands the current understanding of the sheet conformation problem and paves the way for pattern engineering across a range of curved structures.