Configurations evolution of a buckled ribbon in response to out-of-plane loading

Abstract Classical Euler buckled ribbon has become the cornerstone of many unprecedented applications such as the recently developed stretchable electronics and mechanical-guided 3D assembly. In practical applications, such Euler buckled ribbon may be subjected to out-of-plane loading. However, previous studies mostly concern about the buckling and post-buckling configurations of ribbons under in-plane compression, but the further mechanical behavior of buckled ribbons in response to out-of-plane loading is rarely involved. In this paper, the mechanical behaviors of Euler buckling ribbons under out-of-plane loading are systemically investigated, and distinct configurations evolution paths were observed, which are dependent on both axial in-plane compressive strain of Euler buckling and out-of-plane loading. An analytical model based on geometrical decomposition method and finite deformation beam theory is proposed to clarify the mechanism of the competitions between different configurations evolution paths, to construct phase diagrams to distinguish different deformation modes, and to draw the corresponding deformed profiles, which agree well with the experiments and finite elemental stimulations. Furthermore, the out-of-plane stiffness of Euler buckled ribbons with unusual characteristics (e.g., negative stiffness and stiffness reduction) is also quantified, which has general utility as an engineering design rule for exploiting Euler buckled ribbons in stretchable electronics, 3D bio-interfaces, mechanical-guided 3D assembly and mechanical metamaterials.