Exact solutions for the wrinkle patterns of confined elastic shells

Complex textured surfaces occur in nature and industry, from fingerprints to lithography-based micropatterns. Wrinkling by confinement an incompatible substrate is attractive way of generating reconfigurable patterned topographies, but controlling the often asymmetric apparently stochastic wrinkles that result remains elusive goal. Here, we describe a new approach understanding confined elastic shells, using Lagrange multiplier place stress. Our theory reveals simple set geometric rules predicting emergence layout orderly wrinkles, explaining surprisingly generic co-existence ordered disordered wrinkle domains. The results agree with numerous test cases across simulation experiment represent elementary toolkit for designing complex patterns.