  A controlled surface wrinkling pattern has been widely used in diverse
applications, such as stretchable electronics, smart windows, and haptics.
Here, we focus on hexagonal wrinkling patterns because of their great
potentials in realizing anisotropic and tunable friction and serving as a
dynamical template for making non-flat thin films through self-assembling
processes. We employ large-scale finite element simulations of a bilayer
neo-Hookean solid (e.g., a film bonded on a substrate) to explore mechanical
principles that govern the formation of hexagonal wrinkling patterns and
strategies for making nearly perfect hexagonal patterns. In our model, the
wrinkling instabilities are driven by the confined film expansion. Our results
indicate robust hexagonal patterns exist at a relatively small modulus mismatch
(on the order of 10) between the film and substrate. Besides, the film
expansion should not exceed the onset of wrinkling value too much to avoid
post-buckling patterns. By harnessing the imperfection insensitivity of
one-dimension sinusoidal wrinkles, we apply a sequential loading to the bilayer
structure to produce the nearly perfect hexagonal patterns. Lastly, we discuss
the connection between the simple bilayer model and the gradient structures
commonly existed in experiments.
