The classic picture of how spheres deform (e.g. when poked) is that they adopt something called 'mirror buckling' - this is a special deformation (an isometry) that is geometrically very elegant. This deformation is also very cheap (in terms of the elastic energy) and so it has long been assumed that this is what a physical shell (e.g. a ping pong ball or beach ball) will do when poked. However, experience shows that actually many shells don’t adopt this state - instead, beach balls wrinkle and ping pong balls crumple. Why is this wrinkled or crumpled state preferred to the ‘free lunch’ offered by mirror buckling? In a series of papers Oxford Mathematician Dominic Vella and colleagues address this question for the case of the beach ball: an elastic shell with an internal pressure.
Wrinkling is caused by compressive forces within the shell (just as a piece of paper buckles when you compress it at its edges). The key insight is that wrinkling allows the shell to relax the compressive stress so that there is essentially no compression in the direction perpendicular to the wrinkles, and a very high tension along the wrinkles. This change in the stress causes the shell to adopt a new kind of shape, that is qualitatively different to mirror buckling. To determine the energetic cost of this new shape requires a detailed calculation of how wrinkles behave - we find that the wrinkle pattern is intricate, changing spatially (see picture) and also evolving as the degree of poking changes. However, we also show that despite this, the energetic cost of the wrinkling is relatively small, and so this wrinkly shape is an approximate isometry (a ‘ wrinkly isometry’).
The shell now has two choices of cheap deformations to adopt: the wrinkly isometry or mirror buckling. The final piece of the puzzle is to realize that elastic energy is not the only energy that matters in this system - since the shell has an internal pressure, the gas within it must also be compressed. In this case, the wrinkly isometry displaces less gas and hence costs less energy; this is why it is the preferred state.
For more information about the research funded by the European Research Council (ERC):
https://arxiv.org/abs/1508.06146
https://arxiv.org/abs/1612.06834