"Elastic instabilities are ubiquitous, from the wrinkles that form on skin to the ‘snap-through’ of an umbrella on a windy day. The complex patterns such instabilities make, and the great speed with which they develop, have led to a host of technological and scientific applications. However, recent experiments have revealed significant gaps in our theoretical understanding of such instabilities, particularly in the roles played by geometry and dynamics. In this project, I established a group to develop and validate new theoretical frameworks within which these results can be understood.
Central to our approach was an appreciation of the crucial role of geometry in the pattern formation and dynamics of elastic instabilities. The project was therefore divided into two inter-linked work packages: geometry and dynamics.
Under ""geometry"", we showed that the ability to wrinkle allows thin objects to deform in new ways. This can be seen by poking a beach ball: with sufficient poking, the beach ball wrinkles. Our results show that the shape that the beach ball forms after wrinkling is a new deformation mode counter to what is expected based on the standard theory. This is an example of a new deformation mode, called ""wrinkly isometries"", which are different to the shapes that would otherwise be possible. As well as creating new shapes, wrinkly isometries change how stiff the object is: the force required to poke the object depends not on the properties of the object (like its material) but instead on the other features of the system. In the example of the poked beach ball, this is the internal pressure and radius of the ball. However, the principle is much more general and has implications for measurements of the properties of thin objects from Graphene and viruses at the small scale to vessels holding pressurised gas at larger scales.
Under ""dynamics"", we focussed on understanding how fast elastic instabilities occurs, focussing in particular on the 'snap-through' instability seen in umbrellas and in plants like the Venus flytrap. Previous experiments on such snap-through had shown that this motion, while fast, is not as fast as it should be. We were able to show that the mathematical structure of this snap-through transition is one cause of this slow motion - this mathematical structure in turn relates to the geometry of the object that is undergoing snap-through, and provides an important link to the geometry considered in the first part of the project. The understanding and analytical results obtained in this part of the project have possible applications including to the accelerometers used in smart phones and soft robotic devices.
Taking the two parts of the project together, we have shown that the geometry of elastic objects is crucial to understanding some of the unusual behaviour that they exhibit."
