Unveiling scaling laws for wrinkling in compressed fiber-reinforced bilayers at any elastic mismatch

Wrinkling is a stress-induced bifurcation commonly observed in both natural and engineered systems, arising from physical or geometric mismatches in thin films, membranes, and various systems adhered to substrates with complex microstructures. In biology, such corrugations underly fundamental processes such as leaf morphogenesis, brain folding, vascular mechanics, swelling and drying, tissue growth or repair. While wrinkling in isotropic bilayers under imposed lateral compression is well characterized through established scaling laws–particularly for large film-to-substrate elastic mismatches–corresponding relations for fiber-reinforced bilayers remain comparatively widely underexplored.

In this work, we address this gap by analytically deriving asymptotic expansions for the onset of wrinkling when bifurcation occurs at both small and finite strains under prescribed compression, thereby capturing the full spectrum of elastic mismatches between the two sides of the bilayer. The film is modeled as an axially deformable elastic plate with bending stiffness, while the substrate is described using the Standard Reinforcing Model within finite elasticity.

Two distinct sets of scaling laws for the critical strain and wavenumber at the wrinkling onset are obtained, revealing two regimes: a film-mediated regime and a substrate-governed one. The transition between such behaviors is characterized by an analytically derived threshold, providing a clear criterion for selecting the appropriate scaling law for the case at hand.

Our framework applies to biological systems, as well as to engineered bilayers. For the latter, symmetry-breaking transition in the dispersion relation arises as the film-substrate mismatch decreases. Overall, this work provides a unified theoretical framework for rigorously obtaining scaling laws for wrinkling in fiber-reinforced bilayers across the full range of elastic mismatches, offering new insights into potentially analogous bifurcations in anisotropic compounds.