Mechanics of bistable cross-shaped structures through loading-path controlled 3D assembly

•This paper presents a theoretical study of the different buckling configurations in cross-shaped ribbon structures with various numbers of creases.•The developed model can predict the stability of different buckling modes of cross-shaped ribbon structures induced during loading-path controlled 3D assembly.•Design diagrams for typical cross-shaped ribbon structures were obtained and validated by experiments and FEA.•This study provides theoretical guidelines for future designs of reconfigurable ribbon-shaped mesostructures. Morphable three-dimensional (3D) structures capable of reversible shape changes between distinct geometric configurations have widespread applications in many important engineering areas. Recent advances in mechanically-guided 3D assembly provided a powerful approach to achieve morphable 3D mesostructures in a broad set of high-performance materials, over length scales from several micrometers to tens of centimeters. This approach relies on prestrained elastomer substrates released with different sequences to trigger the stabilization of distinct 3D buckling modes in specially engineered 2D precursor structures. Many of the reported 2D precursor structures are constructed with ribbon components that incorporate creases with reduced stiffness at strategic locations. The design of crease geometries and crease locations is essential to the structural bistability through this loading-path approach, which requires the development of a theory to serve as the design basis. This paper presents a finite-deformation model to analyze the stability of different buckling modes induced during the simultaneous and sequential compressions of structures with cross-shaped ribbon geometries, a very representative class of precursor patterns. By introducing a perturbation method to obtain an analytic solution to the deformed configuration of a uniform beam, a theoretical model is developed to predict the postbuckling deformations in cross-shaped ribbon structures with a prescribed number of creases. Based on the analyses of the strain-energy landscape, a stability coefficient is proposed to evaluate the bistability of cross-shaped ribbon structures. The developed model is validated by finite element analyses (FEA) and experimental measurements of structures with a broad range of cross-shaped geometries. This model allows the construction of design diagrams to identify the bistability for a variety of geometric parameters, including the normalized width and