  In this article, we discuss new models for static nonlinear deformations via
scale-invariant conformal energy functionals based on the linear distortion. In
particular, we give examples to show that, despite equicontinuity estimates
giving compactness, minimising sequences will have strictly lower energy than
their limit, and that this energy gap can be quite large. We do this by showing
that Iwaniec's theorem on the failure of rank-one convexity for the linear
distortion of a specific family of linear mappings, is actually generic and we
subsequently identify the optimal rank-one direction to deform a linear map to
maximally decrease its distortion.
