 I just met Dominic about 10 years ago, and he's working on this very beautiful, hydrodynamic behavior, the dub, the Cheerios effect. He has worked on a number of very interesting problems, and I'm looking forward to hear his presentation on elastic deformation, so take it away, Dominic. Okay, thanks, Mike, and thanks to the organizers of the school for everything they've done and for the invitation to be here. I've really enjoyed it. So what I'm gonna tell you a little bit about is some of the work we've been doing on elastic deformations and really thinking about instabilities. So there's sort of a classic area of research in mechanical engineering and physics and applied math, but generally what people are interested in is understanding when things go wrong. So if you're designing a water tower, then you don't want this to happen, okay? So this is a water tower that was built in Belgium in 1972, and it was a very good water tower until they filled it with water, at which point it promptly collapsed, okay? But you see other examples of elastic instabilities, for example in numbers that you use in your house, as they sort of heat and cool differentially, that you get this delamination instability, and if you have rugs or carpets, then you quite often get this little bump, okay? So these are examples of instabilities that you may want to stop or prevent, and that sort of mindset gives you a sort of a set of questions that you're interested in. So for example, you're interested in understanding when your water tower is gonna fall down, and you don't really care how it falls down, right? You just wanna make sure that it doesn't, so you work out when it will happen, you add some safety factor, and then you're done, and maybe you wanna worry a bit about imperfections. But more recently, people have started to think a little bit about how instability can be a good thing. So there's a paper by Pedro Rice, who did a post-doc with Mark, I think, recently, so from 2015, where he says, okay, actually, there are some features of instability that are very useful, and nature knows about this, okay? So for example, the Venus flytrap uses fast motions associated with snap-through instabilities to catch this poor fly, okay? And similarly, there's a recent paper in PNES a few months back, looking at how a ladybug that deploys its wings, and again there, it's kind of an elastic instability. This is videoed at 4,000 frames a second, and then played back at 25, so you can see that it's very quickly deploying the actual wings underneath the case. So people have started to use these fast motions to generate, for example, here, there's some lenses that are gonna suddenly change their curvature, and so sort of de-focus, and the idea is that you can have very fast motions just by using elasticity. So nature here says, okay, actually, elasticity is a good way to make very slow changes and to store energy slowly, and then to release it very quickly, that's why the Venus flytrap is able to move so quickly, it couldn't do that otherwise. But there are other features of instability, so for example, you can quite often get some very regular patterns, so what I'm gonna talk a lot about here is some wrinkling, and this is useful in photovoltaics because people wanna make larger surface areas, capture more light, there are other applications in flexible electronics and making lock and key colloids. But the point I want to make is that if you're asking these kinds of questions about how do you actually use instability, then the kinds of questions you ask are a little bit different to just understanding when does it happen so that I can avoid it. Okay, so I don't know how much people know about elasticity, so I wanna sort of take a step back and say, well, how do thin elastic objects deform? So what you should be thinking of is something like a piece of paper. A piece of paper is very long and very, very thin. So if I can press it, obviously if I sort of push the two ends together, it does have the option to change its length a little bit, it can all sort of squash up, and you know, hopefully you know a little bit of maybe high school physics that tells you that I can change the length and that's gonna cost me, oh no, sorry, I'll go back a second. So I can accommodate this by just changing my length a little bit. I squash the two ends together by some distance delta L. But you also know that if I do this, in reality, what it's gonna do is not change its length, but instead buckle out of the plane. Okay, so that's what we see, what we call bending. And if you think about the very, even though the thickness is very small, you can think that maybe along the centerline there's no stretching and on the outside there's gonna be a bit of stretching, on the inside there's gonna be a bit of compression. Okay, so the question is when will it do each of these two things? So if we look at the energy, there's basically an elastic energy from changing your length and here there's an elastic energy not from changing the length, but rather from stretching the outside and compressing the inside. And it turns out that the cost of changing the length compared to the cost of bending scales with the length times how far you push the two ends together divided by the thickness squared, okay? So the important thing here is that this is thin, okay? So if this is thin then the T squared on the bottom line is very, very tiny and this energy for changing my length is very, very high compared to the energy of just bending out of the way, okay? So as the thickness goes to zero you expect objects to deform without changing their length. That's a sort of class of deformations that are called isometries, okay? So if you look in a dictionary an isometry is a map that or it means of or having equal dimensions, but in mathematics it means a transformation without changing the size or the shape, okay? So what we're doing when I bend this piece of paper the length of the piece of paper does not change appreciably. Okay, so why is this really useful? Well, it's useful because there's a theorem due to Gauss called Gauss's remarkable theorem that says that the Gaussian curvature of a sheet doesn't change when you do an isometric deformation. So what's the Gaussian curvature? So the Gaussian curvature is if I do this there I can write down two curvatures there's one in this direction and there's one in this direction, okay? And the Gaussian curvature is just the product of those two curvatures. Why is this important? Well, it's important because it tells you that if you take a piece of paper and you try to sort of stick it onto a sphere that's the closest sphere I have then you cannot do it without stretching, okay? So if you do you can't just sort of nicely map it onto that, onto a sphere. And obviously that's a big problem for map makers. You want to make a two-dimensional map of a spherical earth. And so you cannot do that keeping the distances all the same. So you have to stretch some distances and there are different projections depending on which distances you want to stretch. I've got two kids. So the thing I care about is when you try to put a bandaid or a sticky plaster on a knee then you cannot sort of get this two-dimensional plaster to go on a nearly spherical knee because you get, and instead you get these little delamination blisters, okay? So this is really a consequence of Gauss's theorem and the fact that very thin objects don't want to stretch. But it's also quite useful. So in our group the Gauss's theorem is actually called the theorem pizzeria, okay? And the reason for that is that when you eat a slice of pizza you sort of pick it up and you know that it's just gonna droop under its weight, okay? So that's bad because it's very hard to eat this piece of pizza. But we also know that what you do is you say, okay, if I bend the edge so I curve the edge with my hand then this direction stays straight and I can eat it very happily. So why is that? Well, it's because of Gauss's theorem. So Gauss's theorem tells us that this thing is flat. It's got to keep the same Gaussian curvature. The Gaussian curvature here is zero because it's not, it's straight in both directions, okay? And so if I impose a curvature in this direction, in this direction, I know that the product of this curvature and this curvature has got to stay zero and the only way I can do that is to keep this curvature zero, okay? So then this direction stays flat, okay? And I can eat my pizza, okay? And this is the same idea about that's used to make corrugated roofs, okay? So the idea is that if you induce some curvature in one direction, it becomes very difficult to bend in the other direction and that gives you very strong materials with very low weight. Okay, so I've shown you that a flat object can't be mapped to a spherical one, okay? And I've also sort of hinted that if you turn it into a cylinder like this or if I roll it up into a cylinder, that's fine because although I've got one curvature in this direction, I don't have any curvature in this direction, okay? And so the product of the curvature, so the Gaussian curvature is still zero. But that's, okay, so mapping to a cylinder is fine, but that's a bit boring. If that's the only thing we can do with a piece of paper, then life is a bit boring. And the question is, are there any non-trivial isometries, okay? So I'll come back to this later, but one example is what's called the decode, okay? So this is what you get if you take a piece of paper and put it on a cup and then poke. You get this little bump. And what you see is you sort, if you look at this very carefully, you can see that although it's curved in this direction, in the sort of perpendicular direction, it stays straight. So that's why it's called a cone, okay? And the D in decone is because it's developable. It still has no stretching and it still satisfies Gauss's theorem. I've mentioned here a few papers from the late 90s, partly to convince you, well partly to sort of in the interest of academic honesty, but also partly to try and convince you that although this is quite a simple observation, it's actually led to a number of papers in some pretty high profile journals. And so if you have the right question and are able to sort of find a simple demonstration of it, then there is an option for having high impact as well. Okay, so the decone is one example of a non-trivial isometry for a flat sheet. What about a curved object? So if you take a tennis ball, okay? So a tennis ball has non-zero Gaussian curvature, okay? So it's got a curvature in this direction and a curvature in this direction. So the Gaussian curvature is one over R squared. So clearly we know that I can't make it flat because that would mean you'd have to change the Gaussian curvature to zero. But what you can do is you can turn it inside out, okay? And when you turn it inside out, then okay, it's not quite true at the edges, but the main part of the sphere has exactly the same radii of curvature as it did in the other state. And so again, that's an example of an isometry and it's what's called mirror buckling because it's as if you reflect the sphere or the spherical cap in a mirror. Okay, so what I want to sort of spend the rest of the talk on is thinking about some poking induced deformations, okay? So this is how I sort of came into this area and I was originally interested in this because we wanted to use poking to measure things about plant cells. And I was talking to Toyota about it yesterday. So there's definitely potential application but in everyday life, you want to check before you get on your bicycle, you want to check whether the tires have the right pressure. So what you do is you kind of push on it and if it's too soft, you know that you need to inflate the tires a bit more. A bit of Google science revealed that there's something called the finger test for testing whether meat is well done. So the idea is that you cook your meat in the oven and you want to know is it cooked to the way I like it and the idea is that you poke it and see how soft it is and then you need to compare it to something. And the thing you should compare it to is if you take your thumb and your little finger and put them together and then push this bit here, you see that that's quite stiff. Okay, so that's the sort of consistency you should be looking for in well done meat. And if instead you like it to be rare, then you need to take your thumb and your index finger and push that bit there and you see that you feel that it's much softer. Okay, and that's apparently called the finger test. But obviously this is a bit quantitative. So people also want to make this quantitative. So there's some experiments on polymer zones which are basically big, well, 100 micron shells, capsules, okay, filled with some solute and that gives them a sort of turgor pressure. And the question is how can you measure that turgor pressure without poking a hole in it? And the idea is that you basically come along with an AFM tip and you just kind of push it and you measure the indentation for a given force or the force for a given indentation. And then if you do the calculations properly, you can sort of work out what the pressure is from that. So we started off doing this and the basic idea is that when you push a shell like this, there's this deformation that I told you about that's isometric and it's called mirror buckling. So what you read in the textbooks is that if you take, I won't be able to do this now. So if you take a sphere and you push on it, then at some point it is able to deform as if, so not exactly like this, but as if I turned a bit of it inside out and then glued it back on. So if I poke this, it accommodates it by kind of turning a bit of it inside out and being glued back on. And that's what's called, so that's mirror buckling. So you can just do some simple geometry that says, if I poke by a distance delta, then because it's a parabolic geometry, the radius of the cap that I need to turn inside out is a square root of delta times the radius. And the point is what I told you earlier that this mirror buckling is an isometry. It doesn't cost you any elastic energy to do that. There's a bit of energy where I have to do the gluing because there I've kind of changing the curvature very quickly, but everywhere else, it's basically isometric and so it's free. So if you look in the textbooks in Landau and so on, it tells you that a shell will try to do this. Okay? And the point is that as the thickness of the shell goes to zero, this, the Pogorelov ridge, that's a name for the bit where you do the gluing, becomes very small and the energy in it becomes very small as well. And so this poking is really a kind of free lunch. It doesn't cost you any energy. Okay? So the idea is that mirror buckling is a free lunch. So, you know, we know that in physics, everything likes to go to the smallest possible energy. So it should do this, right? The problem is that when we started doing simulations of this, this is an abacus simulation of a pressurized shell. So just like the polymer zone, you basically poke on this shell and everything proceeds very nicely. You can convince yourself that it's gonna do this mirror buckling. And then at some point, everything goes horribly wrong. Okay? And it starts to wrinkle. So we had a long back and forth. So this was done by some colleagues in the US, the simulations. And I said, oh, well, maybe it's wrinkling. And they said, oh, no, I don't think so. And I think the thing that convinced them in the end was to take this sort of packaging that you get in your Amazon boxes or whatever. So it's a kind of some plastic inflated. And then if you poke on it, then you do indeed see, the native guys in the front row, maybe you can see that there is indeed some wrinkling. Okay? So, okay, we did an experiment. We sort of characterized some of the properties of this and we used a beach ball just to make a pretty picture. But it's basically the same mechanics. The same thing happens if you don't have an internal pressure. So if you take a water bottle and then just poke on the edge where it's duck curved in two directions, then you again see that instead of being axi-symmetric and doing the mirror buckling thing, it prefers to make a triangle than a square and then this crazy pentagon star thing, okay? And so this poses the question of, well, you know, it might be, it's all very well to say that mirror buckling is a solution, okay? But it's not the solution that you see when you do an experiment. And so the question then is, well, why? We said that mirror buckling is a free lunch. Why is the system not taking this free lunch? Okay, it's meant to be free with a bottle. Oh yeah, okay, so, okay. So you can, this is just meant to be a demonstration. Yeah, you can worry about the fact that the curvature is changing in space, but even if you do it with a ping-pong ball or something with a constant curvature, you see the same thing. So it's not a feature of changing in the curvature. Okay, so what I need to do is I need to understand a little bit about this wrinkling instability, right? And the first thing I need to do is I need to understand whether or why it's happening. So the sort of simple model we have is that you've got a beach ball, so you have a sphere with some pressure inside. It's got some radius. It's got some thickness, and then you poke. And there's some calculations you can do to sort of work out what is the stress within the shell as you poke. So before you do any poking, okay, the thing is stressed because of Laplace's law. You wanna have, you've got a pressure difference between the inside and the outside. You need to have some tension in the sphere. And then as you poke, the stress near where you're poking is very, very large. And then far away from where you're poking, it goes back to the value it had originally. But the point is that as you, so this is how much I've poked. So one 10, 15 in some dimension of this unit is just how much I've poked. So for small poking, it kind of stress decreases, it dips, and then it goes up again. And the reason for that is that as you poke, you pull material in, okay? And you're doing that in a circular geometry, so you're taking a circle that was here and bringing it closer in, so it becomes relatively compressed. If you poke enough, then what you find is that that compression becomes absolute, not just relative, and that gives you, that's basically the same as taking this piece of paper and squashing it, okay? So if it's thin enough, it will just buckle and that's what causes the wrinkling. Okay, so as you increase the indentation depth, you pull material in, and then the key thing is that this hoop stress becomes compressive, okay? And the thing that you can get from this calculation is that near the indenter, the stress is still tensile, still positive, so there's no wrinkling. Then there's a little annulus where the stress is negative, and so I expect that to be wrinkling, and then outside, it's all positive again, and so I don't expect to have wrinkling. And if you look at this picture, sorry, the small r is, yeah, okay, thank you. So the small r is the distance from the poker, so the poker is at r equals zero. But as you go, if you look at this picture, as you go from the indenter, there's no wrinkling, then there's wrinkling, and then there's no wrinkling again. Okay, so what? Okay, so this is the fourth line of your abstract for those people who write abstracts with me. So okay, I've told you about wrinkling, but you know, who cares? I mean, it might just be that it's a small perturbation of mirror buckling. So actually if you look, kind of you get yourself a transparent beach ball or also in these experiments on polymerosomes, if you look through the side, then what you see is it's got exactly the wrong curvature. I told you it should look like a sort of spherical cap turned inside out, and it's got exactly the opposite curvature there. Okay, and so we see the same thing in our simulations. Here's what the black curve here is, mirror buckling would suggest you should have, and our simulations have exactly the wrong curvature. So again, I'm sorry, and then because we're really interested in indentation experiments, there's a sort of quantitative thing that this force you get from mirror buckling is too large by about 50%. So you need to understand why the wrinkled solution has a smaller wrink, yep. Okay, so it is mad, but the way that it shows it's mad is by making the stress negative. So if you think about Laplace's law to still have a positive jump from the inside to the outside, you've got to get this compression. You've got to turn the sign of your tension into a compression, and that's why it's doing that. So I'm just saying, well, no, but it's just, I'm just saying that if you change the sign of the curvature, which is what you're complaining about, if you change the sign of the curvature, then you've got to change the sign of the tension as well, right? That's all I'm saying. To get the same pressure difference, the pressure difference is definitely positive, right? So if you, and you know that the tension times the curvature equals the pressure difference. So if you change the sign of the curvature, which is what I do by doing this, then you've got to change the sign of the tension to keep the same product. So, no, I only had to cut my tennis ball in half to show you that. That was, so here, sorry, in this case, the pressure is inside. Yeah. There's not much pressure in a tennis ball. If I push a tennis ball, you'll start to see, well, it's easier with a ping-pong ball, but if you take a ping-pong ball, you know when you hit a ping-pong ball too hard, you get a square, right? That's not because of, that's just because of this breaking of the axi-symmetry. I'm not sure I've answered that. But Lorette, are you happier? So. No. Okay. Okay. This is coming in different colors. Okay, so sorry, I haven't explained it. So different colors here are increasing, changing the indentation depth, and I've rescaled everything so that it should collapse. And the pressure? So here, so okay, so everything I'm doing is at constant pressure inside, and you might worry if I have a finite volume or number of moles of gas, then the pressure should increase, but you can still, depending on exactly how big your shell is and so on, you can still be in this limit that the pressure is effectively constant. Yeah. Okay, so. Yeah. Wait, I'm gonna. Which one? This one? This one? This one. Well, this is meant to be the same as this, or that's what I'm trying to show you is that it looks similar. Wow, but hang on, it had curvature. Well, okay, I'm gonna talk a little bit about, it's not, okay, you need to calculate, it's not just, is it able to stretch a little bit? So you need to calculate the elastic energy carefully, and I will talk about that in a second, okay. What am I doing? Okay, so what should we do? Say, when you assume that, sorry, when you sort of solve things in elasticity normally, and fluid mechanics and everything, you sort of assume that the stresses in a different direction are kind of in the same order of magnitude, so when you solve the equilibrium equation, so div sigma equals zero, you're assuming that both stresses kind of play a role, or in general, you do that. So for a sort of axisymmetric geometry, with some tensions applied, it turns out that if you do that, you get what's called the lame solutions, you get that the stress is some constant, plus or minus, depending on whether you're RR or theta theta, radial or hoop, divided by R squared. But the point is that wrinkling is really caused by compression, right? So if I take this piece of paper, and I, okay, it has a high stress, and then once it buckles, it has a much smaller compressive force, effectively. So what's happening there is that the wrinkling is able to sort of relax the compressive stress, so instead of saying that these two things are the same order of magnitude, you have to say sigma theta theta is basically zero, and then when you go back to div sigma equals zero, instead of getting this solution, you find that sigma RR is one over R. So, okay, maybe that's a bit technical. What I'm really trying to say is that instead of thinking about a membrane, a wrinkled object is a bit more like a spider's web, okay, so in a spider's web, you have these radial threads, okay, and there, you know, you kind of push on the spider's web, those radial threads get stretched, but the ones in the hoop direction, the kind of circumferential threads can just buckle out of the way. They don't do anything. It's not that there's a big compressive stress in the hoop direction, okay? So this actually changes the behavior of the system qualitatively, and so now I can actually calculate the shape that my pressurized shell should have when I get to very, very large indentation depths, and that gives me this dashed black curve here. Again, I can tell you what these prefactors are in terms of the square root of five, but what really matters is that you've got a kind of universal deformation. So I've rescaled the indentation depth, but the shape, by how far I've pushed, and then I've rescaled the radial distance by this geometrical length scale, the square root of delta times R, and this shape then becomes universal, and you can see that the numerics are sort of converging onto that shape as the indentation depth increases. So the shape has got a geometrical character, just like mirror buckling had. Mirror buckling was just taking a spherical cap, turning it inside out. I found something else that has a similar geometrical characteristic, okay? But we don't understand why this, or in what sense, this wrinkly shape is better than mirror buckling, if it is, okay? So normally, as I said, better means lower energy. So what I need to do is I need to understand the energy of this system, and obviously the energy is gonna depend a little bit on how many wrinkles there are. So what I need to do is I need to understand how the system chooses the number of wrinkles. So to do this, it makes sense to sort of take a step back and to think about the sort of simplest problem of wrinkling, which is essentially to have my piece of paper now floating on the surface of water, okay? And then I do the same thing. So in the absence of water, we already said that it's just gonna do a single buckle like this, right? But if I do that when I'm floating on water, then I have to lift up a lot of water, and that's gonna cost me a lot of gravitational potential energy, okay? So instead, what I could do is I could do lots of really tiny wrinkles, and I could accommodate the same compression with very small wrinkles, but that will cost me a lot of bending energy, okay? So somewhere in between those two is kind of an optimum, okay? So it has sort of conditioned because I've compressed, which tells me what the amplitude of the wrinkle compared to its wavelength has to be in terms of this squashing distance. And then I can calculate the energy. So there's an energy because of the curvature of the sheet and there's an energy because of the gravitational energy of the liquid that I displace, okay? And what I find is that, as I expect, so bending would like me to have as large a wavelength as possible, while gravity would like me to have as small a wavelength as possible, okay? So somewhere I've gotta kind of go between these two things, and it turns out that if you sort of minimize this energy by changing the wavelength, then you find that you wanna have this particular wavelength at the fourth route of the bending stiffness compared to gravity. Okay, so this is one particular kind of wrinkle length, okay? This maybe what happens if you have a substrate stiffness, but there are other options for how these well-developed wrinkle patterns can be determined, okay? So one is if you have a lot of tension along the wrinkle length, okay? So this is, again, sort of in the early 2000s, thought about what happens when you pull on a sheet and there you get an effective stiffness because of the tension that you're pulling with. And then more recently, we've thought about what happens if you have a curvature, okay? So you have, it turns out that you have a stiffness because of the radius of curvature along the wrinkles. So again, I'm not going into much detail here, but the important thing is that you can sort of add these different stiffnesses up, and you can generalize this wavelength here. Okay, so in my problem, I've got a curvature, I've got this sort of universal shape that I know, and I can, and it turns out that it's this curvature that really matters. So you can write down what is sort of a relatively simple law for what the wrinkle number is as a function of the distance from the indenta and the indentation depth and the thickness and all of these things. But there are a few interesting features of this law, okay? One, I think is that, you know, we've got a lot of things in this problem we've got the Young's modulus, we've got the pressure, the thickness, indentation depth, all of these things. But in this law, what we see is not the Young's modulus or the pressure, rather the just the thickness, the indentation depth and the radius of the shell. And then the other thing that I think is interesting is that there's this kind of curious r to the three halves, so radial position to the three halves law. So you can sort of, well, I can, these shows you some simulations at different indentation depths and you may or may not be convinced that as I increase the indentation depth I'm getting different numbers of wrinkles. But also if you look very close to the center there's a small number of wrinkles and at the edge there's a much larger number of wrinkles. Okay, and it's not just that the system likes to keep a constant wavelength because this, that would need you to be your wrinkle number to be linear in r. Okay, so there's something a little bit complicated or non-trivial that we're able to find out here. But I still haven't told you why it's doing this. Okay, so I've calculated the number of wrinkles and now I can calculate the energy and it's again a little bit technical but if I, I've got to allow the thickness to go to zero but to really understand or for my approach to be sort of valid I also need to let the pressure go to zero and I have to let the pressure and the thickness go to zero in the right sort of way. Okay, so I need the pressure divided by thickness to go to zero but pressure over thickness squared to go to infinity. So it's maybe a little bit of a mathematicians thing but the important thing is that now it's not really an isometry in a sense of Gauss's theorem, okay? Because what we have, but if I calculate the elastic energy it's essentially not the important energy in the problem anymore. What instead I'm finding is a very, so maybe a very boring answer which is that when I look at my new shape the dashed black curve here I'm displacing less gas than the mirror buckled solution. Okay, so the mirror buckled solution would take all of this gas, push it out of the way and that's Loretz complaint. Okay, this wrinkled solution displaces less and it has a kind of a very small elastic energy. Okay, so well that's basically the answer to the problem is this is allowing you to displace less gas and so your PDV is much smaller than it would have been in the mirror buckled case. Yep. Okay, so here. Yeah. It's a pressure resistant test. Okay, you're displacing less, but why doesn't the number of wrinkles does not depend on the pressure? Where does that get the most? How do you do that? Well, it's essentially because the wrinkling allows you to get to this universal shape and that universal shape is just some geometry, it's a non-trivial geometry, but it's in geometry. So that doesn't depend on the pressure and then it's this thing about the curvature along the wrinkles is what matters. So you sort of lose the information. I mean, this is probably not of much practical interest in the sense that you have to go to very large, you have to poke a very long way to get to this regime. Okay. Okay, so I want to just give you another example of the same thing. So if we take a very thin polystyrene membrane, I'm not sure you can see it, but it's maybe a couple of centimeters in diameter and then a hundred nanometers or so in thickness and it's floating on water. So because it's floating on water, the surface tension at the edge is pulling on it. So it's again tense. That's a little bit like the Laplace tension or the tension from Laplace's law in the shell case. And then I'm gonna poke on it. And we can sort of do a comparison between the pressurized shell and the floating membrane case. So again, we had a tension because of Laplace pressure. Here we've got a surface tension. In the case of the pressurized shell, we had mirror buckling for the floating membrane. The only thing I can think of is this decone. Okay, it might not be relevant, but that's the only thing I can think of. So these are some experiments. So on the top, you're gonna see a sort of top view of this poking. And in the bottom, you can just see a sort of cross section through there. So what we're doing is just increasing the indentation depth. We're pushing from underneath. And what you see is that wrinkles very quickly reach the edge of the film, okay? So that turns out to be what's important in this problem is that it's a little bit technical, but you just need to sort of solve the problem and you find that basically as soon as you touch the film, the wrinkles are everywhere because the film is finite. Okay, so the other thing to take away is that I don't see any sign of a decone here, okay? It's just more or less looks axi-symmetric. Yes, there are some very fine wrinkles all the way around. So we can do experiments of different indentation depth and with different thicknesses and different radius sheets. And then if we do the calculation based on what's called tension field theory, this idea that the hoop stress is compressed or is relaxed by wrinkling, then what we find is a sort of universal shape, this black dashed curve, which I get very excited about because it's just an airy function, okay? And I can calculate it analytically. But the important thing, again, I'm not going into the details, but the important thing here is that the elastic energy doesn't really enter into the problem. So the force that I'm using to do this indentation is more or less independent of the elasticity of the sheet. What it does depend on is the density of the liquid, the surface tension that I'm applying and the radius of the film. So again, it's just a sort of geometric thing. So I wanted to sort of bring this back and explain a little bit better than I did in the snapshot on Monday morning or what the purpose of the hands-on session is. So the key thing in both of the problems I've just shown you, the shell and the floating sheet is that there's a small tension at the edge, right? So if there's a pressurized shell, that's the tension from Laplace's law. In the floating sheet, that's just a surface tension, okay? And so the idea here was to think about whether you can sort of do the same thing for a decone, okay? So this is an experiment with G pointing upwards, okay? So what I've done is I've sort of, so this is my, an experiment we took, photo we took yesterday with a green sheet and we've got the pen pointing up this way, the sheet is here and then we bring the cylinder down. But the image I showed you of the decone earlier had it kind of the other way around. So I just showed you that gravity is now effectively pointing upwards in this picture I've rotated it. Again, you get the nice decone, but if you do it the exactly the same cylinder size, exactly the same indenture and exactly the same sheet, but with gravity pointing downwards, then instead of the decone, you see something a little bit different. You see, you again see this big region where it's lifted up, but on the outside it kind of points down. And that's what the groups who've been working with us, I think Monday, Tuesday and Thursday did a lot of, we did lots of different sheets, lots of different cylinders, lots of different thicknesses, and we just observed whether it goes up, sorry, down like this or up like this. Last night I had to think of a snazzy name for this and the best I could do was to call this the G cone, so the gravity cone if you like, but the results from so far in the hands on school is to sort of map out this phase diagram. So we've got two parameters, we've got the diameter of the sheet, the diameter of the cylinder in which you're sort of poking the sheet, and there's a region where the cylinder is bigger than the sheet, so that's not allowed because the sheet will just fall through. But what we see is that there's a very narrow band where there's a decone and there's a very large band, roughly. We don't understand the transition between these two regimes yet, but there's a much larger regime where you see this G cone. Again, what we're trying to do now or what we'll be trying to do today is to try and understand what's going on in this regime. Okay, so I just wanna spend a few minutes talking about a different kind of elastic instability. So if you've fallen asleep, got bored of wrinkling, maybe now's a good time to wake up and, because I'm gonna talk a little bit about snap three. Okay, so I'm gonna sort of return to this mirror buckling that I told you about. Okay, and I'm gonna say that, remember I told you that I could turn, I cut the tennis ball in half, I can turn it inside out and everything is fine. Okay, but that's not quite true. If you cut too little of your tennis ball off and you try to turn it inside out, then it won't do it at all. It will just straight away return to its flat geometry. Okay, or its normal geometry. Okay, so actually if you kind of calculate what the, if you sort of look at the equations, it turns out that the important parameter is the thickness of this cap that you've cut off compared, sorry, the thickness of the shell compared to how much you've cut off. Okay, so if the shell becomes too thick compared to the height, or that you cut off too little as in this case, then you lose this second turned out solution and you just have the natural shoot solution over here. Okay, so this is an example of what's called a snap through transition yet. You go from a region where there are two stable solutions to another region where there's just one. Okay, and I forgot to bring it with me but there's this famous toy called the hopper popper, okay? Where you turn the sphere inside out and then you put it on the table and you let go and it jumps into the air. People seen this, okay? So this toy is basically just beyond this transition, okay? So you might be interested in, how long does it take to do this snap? And the way to estimate it is to say, okay, I've got some bending stiffness because that's what's trying to turn it inside out. There's also the inertia of the thing because it happens very fast and so on. And what you find is that actually, it should snap through in about 10 milliseconds. But when you do it, you have time to put it on the table. I mean, 10 milliseconds is how fast it, you know, this just doesn't resist at all, whereas a toy takes a little bit longer, okay? So people have said, okay, well, if it's slower, then that means that it must be damped. It must not be an elastic object. There must be some viscous dissipation in the system. Seems like a sensible conclusion, but there's a lot of examples of this in other systems. So for example, in the Venus flytrap, again, if you estimate how long it should take, the sort of theory says it should take between 1, 1,000th and 100th of a second and actually it takes something more like a 10th of a second. And again, people say, okay, well, it's slow. So, you know, there must be some dissipation. But again, it's generic. So there's some work on what's called the pull-in instability in MEMS. And the thing that they see there is they measure how long it takes to do this snap-through as a function of the control parameter. So if you're out here, okay, far away from the transition, you see that it's very quick, which is just what you expect. But as you decrease the control parameter, which in this case in the pull-in transition is the applied voltage, okay? As you decrease the applied voltage, the time taken to transition goes up and up and up and up and it gradually diverges as you get closer to the transition. So again, this is people who know about the dynamics close to saddle node bifurcations. This is very well-known. If you look in the book of Steve Strogatz, he says that there's what's called a bottleneck caused by the ghost of the saddle node bifurcations. So if you don't know about saddle node bifurcations, just think about this ODE, okay? So dx by dt equals some control parameter plus x squared. If lambda is negative, I have two solutions, one stable, one's unstable. And on this side, because I've got an equilibrium, I can do linear stability analysis, everything is fine. If lambda becomes positive, then there's no equilibrium anymore and I can't do a linear stability analysis. But this system is so simple that I can just solve it, right? So I can find that if I start off at x equals zero, then eventually I'm gonna get kicked off to infinity and the time it takes me to get to infinity is gonna go like one over the square root of this control parameter, okay? So as I get closer and closer and closer to this transition, I get that the time diverges. This is the bottleneck that Strogatz talks about. So the question we asked was, well, could such a bottleneck be what's causing this slow snap-through in all of these different systems? And the question is not really whether it's in that, in the hopper-popper or whatever, because that probably does have some rheology, but just is it true that if you see slow dynamics, then there must be dissipation? So we wanted to find a problem that doesn't have any dissipation and that we can do the mathematics for. And the sort of problem that we came up with is not quite this, but it's easier to show you if I do this. So you take a piece of paper and you say, okay, we know about this situation, okay? We know about this case, but if I've got my hands close together, then I can also have this one, okay? So this is by stable. It can be in either this case or this case, okay? But this inverted case is a bit like the mirror-buckled solution as you turned inside out case. This one only exists if my hands are very close together. If I start to pull my hands apart, then it's not gonna do, it doesn't do it very excitingly, but it snaps up. Yeah, so automatically it just goes to the other case. Yeah? So that's more or less the system that we studied. We had to fiddle a little bit with the boundary conditions to make sure that it's exactly a saddle node bifurcation. But here the key parameter is really how much you've kind of squashed your hands together. And it turns out that there's a sort of geometrical parameter. There's only one dimensionless parameter, essentially governing this problem, which is the angle that you have the ends at compared to how much you've squashed them together. So if I squash them together less, then this mu becomes large and I snap through. Or if I have a smaller alpha, then again this mu is small and it's gonna be, sorry, yeah, I said it the wrong way around. The important thing is that there's a bifurcation at some critical value of mu and you can sort of calculate the diagram. So if mu is large, then there's only the case out here. And if mu is small enough, then there can be two equilibria. So it turns out that the key parameter here is how far beyond the transition are you? So what we do is we basically, we take our piece of paper or actually PET in this case and we kind of push it into the regime where there's only one scenario, one equilibrium and we hold it in the sort of wrong equilibrium. And this is a movie taken at, I think a thousand frames a second, I think it's playing. Okay, so you just watch how fast it snaps through and you watch it kind of oscillate like this. And then you try to quantify this and you measure, okay, this is a time trace of the center as a function of time. So you see it does nothing, nothing, nothing and then it suddenly accelerates and then it oscillates. Okay, so we can plot. We need to work out where we actually lost contact in here but we can plot that as a function of time. And what you see is that first of all, you see two things that I think are interesting. First of all, you see that the growth on a log, the log plot at small times is a straight line. So there's power law growth. It's not an exponential as you'd expect from linear stability analysis. And then also as I change the value of this parameter mu, then you slow down the dynamics. So we can actually sort of quantify this if we measure the time taken to snap. So the time from here to the peak there and we plot this as a function of where how much end compression I had. Again, we do this for different material properties. So again, we see that the snapping as I get closer and closer to the fold or to the bifurcation, the snapping takes longer and longer. Okay, so to choose a zero you have to do, yeah, so what you do is you have to, so basically you have to sort of, you know where the zero of w is, that's okay. And then you, I've forgotten exactly what you do actually. If you know what the answer is gonna be, then it's very easy. You kind of do w squared as a function of time, sorry, the square root of w is a function of time. But you can do that in a more rational way, just by looking at how things, by allowing different power law exponents is, so I mean if you do it in the supplementary information, we show you what it looks like on a semi log plot and it definitely is not correct. There's definitely not a straight line. And then the question is if it's not an exponential, what power law is it? And you can sort of work out what power law it is just by assuming it's some power law and then differentiating one over the log or something like that. I can show you, but there is a, yeah there's a slightly delicate thing to work out exactly where t zero is. Okay, so we can, I'm not gonna go into the details, but basically we can look at the equations for the dynamic motion of this arch and we can do an asymptotic analysis which allows us to take a PDE and make an ODE. Again, we can solve that ODE up to some quadrature analytically and that gives us this black solid curve and it also suggests how we should rescale our experiments and they more or less collapse, okay? And we can also look at how long it takes to snap as a function of this parameter delta mu and again our experimental data more or less collapse and then crucially we're able to get without any fitting parameters this black straight line which surprisingly as of our analysis is only really valid for this delta mu close to very small but it works out to order one delta mu as well. Okay, so maybe you guys know all about this because this is really just an example of critical slowing down near a phase transition it's maybe not so surprising, but I think in the sort of community on elastic instability people weren't really aware about the possibility of studying this sort of quantitatively. So that's all I wanted to say really. I've hopefully convinced you that there's several interesting physical phenomena that involve elastic instability. I started off by telling you that isometries are useful to understand the deformation of elastic objects but I also maybe told you that you need to be a bit careful about that because wrinkling may let you see something different or we uncover a new kind of isometry which we call wrinkly isometries and then also the structure that you get with these isometric solutions can be interesting and the way they disappear can give you this critical slowing down and then in the spirit of the school I think I hope I've convinced you that there's a lot of potential here for some relatively simple experiments. It needs a little bit of care to think of the right question but you can actually learn something new with some relatively simple ingredients. So with that I should thank the various people I've worked with on the three problems I've talked about and even though I'm British I'm gonna thank the EU for the money and then take any questions that you have.