Mechanics of Fractal-Inspired Horseshoe Microstructures for Applications in Stretchable Electronics

Abstract

Fractal-inspired designs represent an emerging class of strategy for stretchable electronics, which have been demonstrated to be particularly useful for various applications, such as stretchable batteries and biointegrated electrophysiological electrodes. The fractal-inspired constructs usually undergo complicated, nonlinear deformations under mechanical loading, because of the highly complex and diverse microstructures inherent in high-order fractal patterns. The underlying relations between the nonlinear mechanical responses and microstructure geometry are essential in practical applications, which require a relevant mechanics theory to serve as the basis of a design approach. Here, a theoretical model inspired by the mechanism of ordered unraveling is developed to study the nonlinear stress–strain curves and elastic stretchability for a class of fractal-inspired horseshoe microstructures. Analytic solutions were obtained for some key mechanical quantities, such as the elastic modulus and the tangent modulus at the beginning of each deformation stage. Both the finite-element analyses (FEA) and experiments were carried out to validate the model. Systematic analyses of the microstructure–property relationship dictate how to leverage the various geometric parameters to tune the multistage, J-shaped stress–strain curves. Moreover, a demonstrative example shows the utility of the theoretical model in design optimization of fractal-inspired microstructures used as electrophysiological electrodes, aiming to achieve maximum elastic stretchability for prescribed filling ratios. The results indicate a substantial enhancement (e.g., >4 times) of elastic stretchability by using fractal designs, as compared to traditional horseshoe designs. This study can serve as design guidelines of fractal-inspired microstructures in different stretchable electronic systems.