Fiber-reinforcement is a universal feature of many biological tissues. It involves the interplay between fiber stiffness, fiber orientation, and the elastic properties of the matrix, influencing pattern formation and evolution in layered tissues. Here, we investigate the deformation of a compressed film bonded to a half-space, where either the film or the substrate exhibits anisotropy. Within the framework of finite elasticity, we formulate nonlinear incremental equations, enabling linear and weakly nonlinear analyses. These analyses yield exact bifurcation conditions and an amplitude equation for surface wrinkling. In particular, for a simple fiber-reinforced model, we show that the bifurcation can be supercritical or subcritical depending on the ratio between the substrate and the film moduli. These findings underscore the pivotal role of fiber-reinforcement in shaping pattern formation in anisotropic tissues and provide insights into the morphological evolution of biological tissues.

37 pages, 17 figures