Snap-buckling in asymmetrically constrained elastic strips

When a flat elastic strip is compressed along its axis, it is bent in one of two possible directions via spontaneous symmetry breaking and forms a cylindrical arc, a phenomenon well known as Euler buckling. When this cylindrical section is pushed in the other direction, the bending direction can suddenly reverse. This instability is called snap-through buckling and is one of the elementary shape transitions in a prestressed thin structure. Combining experiments and theory, we study snap-buckling of an elastic strip with one end hinged and the other end clamped. These asymmetric boundary constraints break the intrinsic symmetry of the strip, generating rich exotic mechanical behaviors including largely hysteretic but reproducible force responses and switch-like discontinuous shape changes. We establish the set of exact analytical solutions that fully explain all of our major experimental and numerical findings. Asymmetric boundary conditions arise naturally in diverse situations when a thin object is in contact with a solid surface at one end, but their profound consequences for the buckling mechanics have been largely overlooked to date. The idea of introducing asymmetry through boundary conditions would yield new insight into complex and programmable functionalities in material and industrial design.