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Geometry and Anomalous Dynamic Growth of Elastic instabiliTies

Periodic Reporting for period 4 - GADGET (Geometry and Anomalous Dynamic Growth of Elastic instabiliTies)

Reporting period: 2019-10-01 to 2020-06-30

"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."
"The project was divided into two work packages: ""Geometry"" and ""Anomalous Dynamics"". Below we summarise the main results of the action according to the different work packages:

Geometry of Elastic Instability:

Here, our main result has been the description of a new class of elastic deformation, which we call ""Wrinkly isometry"". Wrinkly isometry has been exhibited in detail in a number of particular settings including the indentation of a floating elastic sheet (published in Paulsen et al., PNAS 2016 and Vella & Davidovitch, Phys Rev. E 2018) and a pressurised elastic shell, analogous to the poked beach ball (published in Vella et al. EPL, 2015 and Taffetani & Vella Phil Trans R Soc A 2017).

Part of this work has also involved the clear elucidation of the previous heuristic for understanding such deformations (isometric shapes) and has been published in two papers (Gomez et al. Proc. R. Soc. A 2016 and Taffetani et al. Proc. R Soc. A 2018). Here, part of the question was to understand when a spherical cap, such as a contact lens, can remain stable when turned 'inside out' and when it immediately 'snaps back' to its natural state.

A further strand of this work has been understanding how intrinsic or imposed curvature can increase the rigidity of structures: this 'geometrical rigidity' is familiar from the way in which a piece of pizza can be rigidified against gravity by bending the crust. We quantified this effect through theory, experiment and simulation (see Taffetani et al. EPL, 2019).

Anomalous Dynamics:

Here, we have two main results.

Firstly, on the dynamics of snap-through, we showed that the mathematical structure of elastic snap-through can give rise to surprisingly slow dynamics, even without any dissipative effects (published in Gomez et al. Nat. Phys. 2017). We also investigated when (and how) this geometrical slowing down interacts with the effects of material viscoelasticity (Gomez et al. J. Mech. Phys. Solids 2019). We also investigated the relevance of this mathematical structure for the pull-in instability that is used in accelerometers (work published in Gomez et al. J. Micromech. Microengng. 2018a,b).

The second major result is that the geometrical confinement that leads to wrinkling can also cause the dynamic evolution of wrinkling patterns to happen more slowly than expected. This was shown in two different situations described in Kodio et al. Phys Rev Fluids (2017) and O'Kiely et al. Phys Rev Fluids (2020). Videos of some of the experiments that support this latter work can also be viewed."
The main contributions of the project beyond the state of the art are three-fold.

Firstly, we have introduced the concept of 'wrinkly isometries'. These are deformations that have vanishing elastic energy, but that are facilitated by the wrinkling of very thin sheets. Wrinkly isometries supplement the class of isometric deformations that are already known for thin sheets (and previously represented the state of the art). Crucially, however, because they involving wrinkling, they are not subject to the same restrictions as isometries. In particular, while isometries must preserve the Gaussian curvature of a surface, wrinkly isometries may involve changes in the 'Apparent' Gaussian Curvature of an object because excess length can be buffered by buckling.

Secondly, we have shown how to quantitatively describe the snapping motion of structures such as elastic shells. The mathematical procedure that we have developed allows reduced models of this motion to be derived formally, and reveals new information about how quickly they occur. Crucially, these analyses show that the distance beyond the snapping threshold is a key parameter that determines the speed of this motion. This goes beyond the state of the art in two ways: (i) it shows that slow motion can occur when the system is very close to the snap-through transition (whereas previously it had been assumed slow motion must be a result of more complicated material behaviours) and (ii) it allows us to derive new analytical results for how long snap-through takes (which gives new insight beyond the calibration simulations that previously represented the state of the art).

The third significant advance represented by the project concerns the dynamics of wrinkling instabilities: using a combination of theory and experiment, we have shown that the geometry of wrinkling affects how wrinkles change in time. This means that the time history of wrinkling becomes indelibly stamped on the wrinkle pattern, for example through the way in which the size of wrinkles changes with time.
Wrinkling induced by a droplet on a tensed membrane
Dynamic wrinkling of a thin sheet induced by impact