Hierarchical folding of elastic membranes under biaxial compressive stress

摘要

Mechanical instabilities that cause periodic wrinkling during compression of layered materials find applications in stretch-able electronics1"3 and microfabrication, but can also limit an application's performance owing to delamination or cracking under loading8 and surface inhomogeneities during swelling9. In particular, because of curvature localization, finite deformations can cause wrinkles to evolve into folds. The wrinkle-to-fold transition has been documented in several systems, mostly under uniaxial stress10'13. However, the nucleation, the spatial structure and the dynamics of the invasion of folds in two-dimensional stress configurations remain elusive. Here, using a two-layer polymeric system under biaxial compressive stress, we show that a repetitive wrinkle-to-fold transition generates a hierarchical network of folds during reorganization of the stress field. The folds delineate individual domains, and each domain subdivides into smaller ones over multiple generations. By modifying the boundary conditions and geometry, we demonstrate control over the final network morphology. The ideas introduced here should find application in the many situations where stress impacts two-dimensional pattern formation.