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

Guoquan Luo, Haoran Fu, Xu Cheng, Ke Bai, Liping Shi, Xiaodong He, John A. Rogers, Yonggang Huang, Yihui Zhang*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

33 Scopus citations


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 location of each crease in the cross-shaped patterns. These results elucidate how to select the various design parameters to achieve bistable 3D structures through the loading-path approach. Furthermore, the developed model is extended to the bistability analyses of precursor structures with generalized cross patterns and cross-shaped patterns consisting of locally stiffened elements. This study offers a theoretical reference for the design of ribbon-shaped mesostructures to achieve diverse morphable microelectronic devices.

Original languageEnglish (US)
Pages (from-to)261-277
Number of pages17
JournalJournal of the Mechanics and Physics of Solids
StatePublished - Aug 2019


  • 3D assembly
  • Bistable structure
  • Loading path
  • Postbuckling

ASJC Scopus subject areas

  • Condensed Matter Physics
  • Mechanics of Materials
  • Mechanical Engineering


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