Torsional deformation dominated buckling of serpentine structures to form three-dimensional architectures with ultra-low rigidity

Buckling of a single ribbon is the simplest type of instability in mechanics. Recently postbuckling has been adopted in three-dimensional (3D) assembly technique to form diverse 3D architectures from planar precursors. amongst various structures, 3D serpentine structures are remarkable, especially in stretchable electronics and micro-electro-mechanical systems, as they provide ultra-low rigidity and high stretchability. Previous studies on those structures are mainly based on finite element analysis and individually focused on features such as the resonant frequency, critical compression for buckling and stretchability. A comprehensive theoretical model that fully reveals the mechanical behaviour of such structures and provides detailed information such as the displacement/rotation, curvature and strain distribution, is essential for applications that require a tailored structural design. Here an analytical model in very concise and explicit form is developed, showing that the postbuckling of serpentine structures obeys similar rules as that of a straight ribbon, with the effective torsional rigidity playing the role of the bending rigidity. The effective torsional rigidity involves not only the cross-sectional profile (ribbon width and thickness) but also the geometry parameters unique in the serpentine structures, which significantly enlarges the design space. Using serpentine structures of unidentical ribbons, an inverse design strategy is proposed to form various architectures buckled from planar precursors, without adopting spatial-varying cross-sectional thicknesses or widths as in previous works. A multi-directional strain sensor with ultra-low rigidity is also presented to demonstrate the application of the theoretical model. The above results also indicate that complexity (rich design space) and conciseness (simple rules) could coexist in 3D serpentine structures, which is rare in other structures formed by the mechanics-guided 3D assembly.