One early theme was the study of elastic patterns and defects in thin plates and shells. We provided a theoretical and experimental description of the fundamental singularity in thin elastic sheets, that is ubiquitous in crumpled paper, origami, drapery, etc. This has helped resolve how approximately isometric deformations of a two-dimensional surface help pack it in three-dimensions. We also derived a general nonlinear theory of how fine scales such as wrinkling arise in slender structures combining geometry and physics, with predictions for the wavelength and amplitude that were verified by our experiments and data that range over 16 orders of magnitude in scale, from nanotube wrinkling to tectonic subduction.
We have also explored the implications of these instabilities in different physical situations by quantitatively explaining how nested self-similar wrinkles arise in polymer skins, and provided the outline of a theory of self -organized origami to create foldable structures such as the Miura-ori and related fine-scale instabilities in supported and unsupported thin films. Our work on how singularities and fine scales form has led to substantial interest in these problems from the engineering community who now use the design principles we uncovered to build surfaces with unusual frictional, optical and other properties, and in the mathematical community interested in a rigorous analysis of the scaling laws.
A second theme is the study of soft material interfaces. A unifying theme is the use of simple geometric and scaling arguments combined with approximate analysis and experiments, complementing the large-scale computational approaches that are commonly used. We uncovered a fundamentally new instability in the simplest of these systems, a half-space that is compressed uniformly (in-plane strain) can fold into a scale-free sulcus akin to the structures in the brain – unexplained for more than a century – and yet as ubiquitous and general as buckling and cracking. Our work used experiment, theory, and computations to show that it is akin to a first-order transition, but without a barrier, with a number of physical and mathematical implications that are only just beginning to be explored. In studying soft interfaces and thin solid films, we showed how a confined elastic meniscus can lose stability to the formation of fingers (digits) in a manner that is similar to the classical Saffman-Taylor instability in hydrodynamics, but via a sub-critical instability that opens the way for patterning thin films digitally.