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Thin wires or films can be
spontaneously folded into different shapes, and such phenomena hold promising
applications in engineering, especially at micro and nanoscales. Based upon the
established potential energy functional, we derived the governing equation and
adhesive boundary condition for a self-folding system. Considering the inextensible
condition of the structure, a closed-form solution for the deflection of a racket-like
structure was obtained in terms of elliptical integrals, which applies for both macro
and nanodimensions. We then determined the critical adhesive length under specified
geometric and energetic constraints. The results show that self-folding is energetically
favorable and thermodynamically stable with the cohesive work being strong
enough and the structure being sufficiently flexible. As soon as the self-folding
configuration is formed, the slender structure must possess an initial adhesive
length. These conclusions are beneficial for the design of nanostructures,
and the enhancement of their mechanical, chemical, optical, and electronic
properties.
