Curvature delays growth-induced wrinkling

Abstract

Wrinkling patterns can be induced by the growth of a thin elastic film over a soft elastic substrate. While there is a good understanding of how this pattern is initiated on a flat geometry, wrinkling patterns over a curved surface are more complicated. Here, we consider this phenomenon within the framework of large deformation morphoelasticity by investigating surface wrinkling of a growing thin elastic film bonded to a large elastic cylinder. The system has two important dimensionless parameters: the ratio $\ensuremath{\eta}$ of the film thickness by the cylinder radius and the relative stiffness of the two layers $\ensuremath{\xi}$. Depending on the values of $\ensuremath{\xi}$ and $\ensuremath{\eta}$ we identify four different regimes for which we find the critical growth and wrinkling mode number. By combining asymptotic methods with numerical computations we determine the effect of the curvature on the bifurcation and establish that it always induces a delay at the bifurcation: Larger growth is needed on a curved surface to induce the same wrinkling instability. These results are crucial to understand pattern formation on surface with varying curvatures.