Wrinkling of a bilayer with spatially varying stiffness: from wrinkle branching to cascades

We consider the problem of elastic wrinkling for the case where the energetically favourable wrinkle number (or wavelength) varies in space. Ordinarily, such changes are believed to take place via a 'wrinkle cascade' in which individual wrinkles break into two (or more) wrinkles all at the same location: many wrinkles are introduced at a single location, and away from this location the wrinkle pattern appears mono-chromatic, with a single mode dominating the pattern at each location. We show that it is also possible for wrinkles to be added gradually, via a small subset of wrinkles branching repeatedly. In this 'wrinkle branching' regime, the wrinkle pattern is polychromatic, with several modes dominating the pattern at each location. Using a combination of numerical simulations, analysis and experiments, we show that this wrinkle branching mode is observed when the rate of change of preferred wrinkle number is sufficiently slow. We note that in this regime the branching occurs where the wrinkle amplitude is smallest, making it difficult to observe experimentally, and speculate that this may explain why it appears not to have been often reported before.