Buckling shape transition of an embedded thin elastic rod after failure of surrounding elastic medium

When the compressive load to a thin elastic rod embedded in an elastic medium exceeds a threshold, the thin rod buckles into an exponentially decaying short wavelength profile to minimize the total energy of the system. As the compressive load continues to increase, the buckling amplitude increases correspondingly, until the rod/medium interface fractures and the short wavelength buckling profile morphs into a different shape as fracture propagates into the surrounding medium. In this study, such shape transition in the presence of surrounding medium failure is investigated using a combined experimental and theoretical approach. We identify the ansatz that can be used to describe the post-fracture buckling profile, and then develop a forward scheme using the energy principle to predict the buckling profile of the thin rod when fracture happens in the medium. We also develop a backward scheme where we use the post-fracture buckling profile to estimate the buckling profile before fracture of the surrounding medium. Comparison of experimental and theoretical results indicates that the modeling framework can be used to characterize the buckling profile transition of a thin elastic rod embedded in a fractured elastic medium.