  Wrinkling, creasing and folding are frequent phenomena encountered in
biological and man-made bilayers made by thin films bonded to thicker and
softer substrates often containing fibers. Paradigmatic examples of the latter
are the skin, the brain, and arterial walls, for which wiggly cross-sections
are detected. Although experimental investigations on corrugation of these and
analog bilayers would greatly benefit from scaling laws for prompt comparison
of the wrinkling features, neither are they available nor have systematic
approaches yielding to such laws ever been provided before. This gap is filled
in this paper, where a uniaxially compressed bilayer formed by a thin elastic
film bonded on a hyperelastic fiber-reinforced substrate is considered. The
force balance at the film-substrate interface is here analytically and
numerically investigated for highly mismatched film-substrates. The onset of
wrinkling is then characterized in terms of both the critical strain and its
corresponding wavenumber. Inspired by the asymptotic laws available for
neo-Hookean bilayers, the paper then provides a systematic way to achieve novel
scaling laws for the wrinkling features for fiber-reinforced highly mismatched
hyperelastic bilayers. Such novel scaling laws shed light on the key
contributions defining the response of the bilayer, as it is characterized by a
fiber-induced complex anisotropy. Results are compared with Finite Element
Analyses and also with outcomes of both existing linear models and available
adhoc scalings. Furthermore, the amplitude, the global maximum and minimum of
ruga occurring under increasing compression spanning the wrinkling, period
doubling and folding regimes are also obtained.
