We also need, some something called Kinematics, all right? We need the kinematic relation Again, with coordinate notation. This kinematic relation for the stream. Epsilon kl is one half derivative of another vector, u with respect to x plus the symmeterizing term. I should mention that, I should make it clear that that is l, not i. Okay? Right. So we have this kinematic relation. In direct notation this would be epsilon equals the symmetric part of the gradient of u, okay? All right. So we have these components of our problem, okay? So let me just go back here for a second. So we're given data in terms of these vectors ug, t bar, and f. We're given a constitutive relation, relating the stress to the strain through the elasticity tensor. Okay? And we're given this kinematic relation. Okay, so given all this, what we are faced with is finding u. Okay? Which is, I'm sorry. We need to stick with coordinate notation here. Find ui, okay. And again, ui is a vector, right? It's a component of a vector, so this really is saying that u belongs to R 3. Okay again, parenthetically, we have the direct notation there. Okay? Find ui, right? Which, s, so ui is our solution field and you probably know if you, you know some elasticity, you know that this is the displacement field. Okay, so, find ui such that. The following hold, okay? Let's first write out the PDE itself. The PDE is the following. Sigma i j, sorry, sigma i j, j plus f i equals zero in omicron. Okay? And, if you all ready know it that simply means that the stress, the stress divergence, the divergence of the stress plus f equals zero, okay. This is sometimes called the stress equilibrium equation. Okay, so. Sometimes called the quasi-static stress equilibrium relation. All right? Okay. So that's the PDE. Boundary conditions. U i equals u given sub i on the Dirichlet boundary. But here comes the rub We have a vector problem, right? So if you look at the equation that we put down the PDE, this PDE. What we need to recall is that in all of this, we need to recall that we need to recall that i, j equals 1, 2, and 3, right? So when we look at the PDE, we actually have three PDEs there, right? For each value of phi. Not a, not a big surprise, because we are talking of finding a field which is itself a vector, right? Which has three components. So we're really solving three PDEs. Okay? So vector PDE. What this means is that we need boundary conditions for each component of our field, of our solution field, all right? But then each component of our solution field could have a Dirichlet boundary condition defined on a different Dirichlet boundary. Okay, so on this Dirichlet boundary itself we can have a subscript. What this means, when I go back, oh, well, l, let me draw it right here. What this means is that we have our basis We have our domain. It means that for each component, right, I could have a certain tertiary boundary. Okay. Right? Let me actually give it, give this a specific component. Then we suppose that this is the Dirichlet boundary for the component one. All right. However, when I come to component two and three, I am free to take on, to, to give them different Dirichlet boundaries, right? So it, and these could overlap or not. All right, they, they can be completely arbitrary all right? No implied relation with each other. This simply comes about from the fact that we're solving a vector problem. All right, and of course, it's elliptic so we do need boundary conditions everywhere. Okay? So that's for the Dirichlet boundary. Well, the same sort of thing happens for the Neumann boundary. Because each of those Dirichlet boundaries has as its compliment a Neumann boundary. And, I think I'm going to denote this as omega. Well, it's obvious, right? Omega, partial of omega t bar sub 1. We have partial of omega t bar sub 2 and Partial of omega t bar sub 3, okay? Of course, in each case, each set of Dirichlet and Neumann boundary conditions are complements of each other, okay? So, here we have, okay. So where does the Neumann boundary condition come in? It comes in to say that sigma ij, nj equals t bar i on partial omega t bar sub i, okay? What we have is that the boundary, okay? Is always equal to, partial omega u sub i, union partial omega t bar sub i. And that each Dirichlet boundary for a given component of the solution is, disjoint from its Neumann boundary. Okay? This holds for i equals 1, 2, 3. Okay? Here's what this means. [NOISE] Three dimensions. This is our continuum football now, all right? When it comes to boundary conditions, what we're seeing is that for each component of space, right? For each compo, for each, f, for each, direction as defined by our basis vectors, we have, a Dirichlet condition for the corresponding component, so u 1! So maybe u 1 is specified on the maze bar of this football And the corresponding traction component, right? Which would be obtained through this relation. All right? The corresponding value t bar 1. Right? When you're talking of this. Okay. So if you want to specify on the maize part t1 is specified on the blue part, all right. And then again if you too maybe specified on the part of the football that you can see, right? And t2 is specified on the opposite part. Okay. You u 3 maybe specified maybe on some I don't know, some combination of the maize and blue. Maybe that maize part, this blue part. On it's compliment is where u3 is specified. Okay, what this corresponds to, is if you've studied problems of elasticity or mechanics is the fact that on a given boundary we may often not be controlling all components of the displacement. Okay? So it's common in, problems of elasticity to do, so called union axial tension problems, right? So supposing this were my, this were a bar and at this end, I decide to hold things fixed. So there are different ways to do this. I may choose to specify that on, that on this surface, on this edge of the, this sort of LEGO pieces, I control all displacement components, right? U1, u2, u3, I may choose to say all of them are zero. All right. And then I may choose to load this thing up. Right? Alternately, there is another version of ten, of tension. That it, what, what I've just described is not true in the axillary tension. It turns out it develops stresses at this end in, in, in, in directions other than along the longitudinal direction. Instead, I may choose that at on this boundary I specify only this component, let's call it the e 1 component, the u 1 component, to be zero. Okay? And the other two components are allowed to move around, to relax, so that now, when I pull this bar. Because of something called the Poisson ratio, in linearized elasticity, this bar will tend to shrink in size. It will tend to contract in, in, in, in the other two dimension I pull it one way. Then I'm allowing points on this surface to also move in the, the vertical direction and the direction towards you. Okay? So elasticity gives us the freedom to do that on a single boundary. There are many ways in which we can combine the Dirichlet boundary conditions on, on the, any given surface and indeed for the traction conditions too. All right, so in the problems I just described here, in this uni axial tension, often the lateral boundaries are free of traction. Okay? Which means there are traction on those surfaces is zero. [COUGH] Sorry, zero. Now when I pull on this, on this end of the bar I may choose that but also specifying the Dirichlet condition, right? I may specify that I'm, I'm, I'm going to pull this end. There we go. I'm going to pull this end to some distance, right? That's a Dirichlet condition. Or I may choose to say that on this surface I'm specifying only that the traction along this direction, this longitude, this, this longitudinal direction is controlled, only that component of the vector. Okay? I'm a, I'm free to actually to specify the displacement components in the other two directions. That's they're all kinds of ways in which we can combine them. All right? And, and these are actually highly relevant for physical situations and lead, they're, they're completely different boundary value problems. They lead to solutions that look That look, that are different in, in important ways, okay? But the important condition to always be satisfied, is that the union of the Dirichlet and Neumann condition, boundaries for each component, always essentially, give a union that gives us the entire boundary. And that each Dirichlet boundary and Neumann boundary for a given component direction are disjoint. Okay so this is just a beginning step an early step towards the strong form of elasticity. I'm going to end the segment here, but when we return, we are going to say a little more about this. Because it's important to understand what elasticity's trying to do here. I will also write up the direct notation for this, problem. And say much more, actually about, many of the, the ingredients of, this problem. And it's important to understand that, as we are, setting up numerical methods for it. The comp, the complexity comes mainly, entirely from the fact that it's a vector problem. All right, we stop here.
