Elasticity Solutions to Nonbuckling Serpentine Ribbons

Stretchable electronics have found wide applications in bio-mimetic and bio-integrated electronics attributing to their softness, stretchability, and conformability. Although conventional electronic materials are intrinsically stiff and brittle, silicon and metal membranes can be patterned into in-plane serpentine ribbons for enhanced stretchability and compliance. While freestanding thin serpentine ribbons may easily buckle out-of-plane, thick serpentine ribbons may remain unbuckled upon stretching. Curved beam (CB) theory has been applied to analytically solve the strain field and the stiffness of freestanding, nonbuckling serpentine ribbons. While being able to fully capture the strain and stiffness of narrow serpentines, the theory cannot provide accurate solutions to serpentine ribbons whose widths are comparable to the arc radius. Here we report elasticity solutions to accurately capture nonbuckling, wide serpentine ribbons. We have demonstrated that weak boundary conditions are sufficient for solving Airy stress functions except when the serpentine’s total curve length approaches the ribbon width. Slightly modified weak boundary conditions are proposed to resolve this difficulty. Final elasticity solutions are fully validated by finite element models (FEM) and are compared with results obtained by the curved beam theory. When the serpentine ribbons are embedded in polymer matrices, their stretchability may be compromised due to the fact that the matrix can constrain the in-plane rotation of the serpentine. Comparison between the analytical solutions for freestanding serpentines and the FEM solutions for matrix-embedded serpentines reveals that matrix constraint remains trivial until the matrix modulus approaches that of the serpentine ribbon.