Buckling and post-buckling of an elastic rod embedded in a bilayer matrix

Many biological and engineered systems can be modeled as buckled thin rods with constraints. Examples include microtubules in cytoskeleton, plant roots in soil, and oil pipes within a wellbore. However, most previous studies focused on the buckling of a rod in a homogeneous environment, an idealization which is often not realistic. Here, we study the buckling behaviors of an elastic rod embedded in a bilayer elastic matrix using a combined experimental, theoretical, and computational method. Our experiments showed, for the first time, that the buckling amplitude can increase from the end where the compressive load is applied. To interpret this new phenomenon, we built a theoretical model and identified an ansatz for the transverse displacement. Our numerical results showed that material inhomogeneity, geometry, and loading all have significant influences on the post-buckling behaviors of the rod. Moreover, our study indicated that the stiffer layer of the elastic medium can be treated as a clamped boundary. These results could find applications ranging from the penetration of needles through biological tissues to the development of underground structures.