Topologically protected hierarchical buckling modes of compressed thin films on Winkler substrates

Abstract

Patterns created by buckling of thin films on substrates have important applications in many fields such as functional devices and surface engineering. Introducing hierarchy to such patterns may further enhance their functionalities. However, unpredictable geometrical and material defects may disturb the formation of patterns. It is thus attractive to construct robust hierarchical buckling patterns that are insensitive to defects. Here, inspired by the construction of topologically protected localized states insensitive to defects in wave mechanics, a thin film of finite length bonded on a simple Winkler substrate is quantitatively designed by just varying its width continuously. Under the action of specific compressive stresses within such a continuous thin film owing to thermal mismatch between the film and substrate, there really exists a hierarchical film-buckling mode generated by topologically protected localization. Conditions under which the topologically protected localization exists are given explicitly, and the underlying mechanism is intuitively demonstrated from the mechanical point of view. Moreover, an analytical solution for the decay rate of localization is obtained as a satisfactory second-order approximation, from which the associated influencing factors are clearly identified. This work is expected to provide a theoretical guidance for the design and optimization of hierarchical patterns in film/substrate systems via the concept of topology.