Sulcus formation in a compressed elastic half space

When a block of rubber, biological tissue or other soft material is subject to substantial compression, its surfaces undergo a folding instability. Rather than having a smooth profile, these folds contain cusps and hence have been called creases or sulcii rather than wrinkles. The stability of a compressed surface was first investigated by Biot (1965), assuming the strains associated with the instability were small. However, the compression threshold predicted with this approach is substantially too high. I will introduce a family of analytic area preserving maps that contain cusps (and hence points of infinite strain) that save energy before the linear stability threshold even at vanishing amplitude. This establishes that there is a region before the linear stability threshold is reached where the system is unstable to infinitesimal perturbations, but that this instability is quintessentially non-linear and cannot be found with linear strain elasticity.