Welcome back. We'll continue with our study of the constitutive relations of linearized elasticity. What we've seen so far is the fact that our elasticity tensor C is symmetric, right? It has major and minor symmetries. We saw what these mean and why they hold. Continuing. Right. In the context of linearized elasticity, there is another very important property that C holds, okay? So C is positive definite. Okay? And here's what we mean by saying it is positive definite. For all tensiles, I'm thinking of what I should use as a general tensor up here. Let me use theta. Okay, I hope that doesn't disturb any of you too much if you got to use the same theta as temperature or angle or something. Theta is a tensor for me. So theta is a general tensor, so it doesn't need to be symmetric, so I'm going to say it belongs to that space. That's not a notation I've used before. GL(3) simply means the group of General Linear, I guess I should capitalize them, right? So okay. General Linear Transformations. In R3. Okay, that's what GLT is. It's just a fancy way of saying matrices in 3D if you choose at every tensors being a squared matrix. Okay? Anyway. So supposing we have this, right? Then, what we're saying is that for all theta belonging to this space, we have the following result: theta contracted with C contracted with theta, again. All right, so we form this quadratic product of C with theta. This is greater than or equal to 0, the scalar, for all theta. In coordinate notation, this is what it means: theta ij C ijkl theta kl is greater than or equal to 0. Okay, that what we're saying. It's useful to write this in coordinate notation just to be sure that this quadratic product is indeed scalar, okay? So it's greater than the real number 0. Okay. This will rule for all theta belonging to GL3. In fact, what we can also show is that actually, the equality, right? So theta contracted with C contracted with theta equal to 0, right? Which is one of the cases of the inequality on the previous line. The special case, the equality holds if and only if, I have to mean if and only if, theta itself is the zero tensor, okay? All right, this is the condition that holds, okay? Now this is very important in the context of linearized elasticity. It essentially translates to the condition that linearized elasticity talks about materials that are fundamentally, not subject to material instabilities, okay? And this is important. If you can imagine, as you may imagine, if you're going to compute with this model, right? We're going to put stresses and strains, we're going to put loads on it. It's important to know that material instabilities within the context of linearized elasticity are okay? So what this means is that linearized. Elasticity theory.. Has no.. Material Instabilities. Okay, if you're wondering what a material instability could possibly be, fracture is a material instability. That's not covered by linearized elasticity, okay? You need to do the special things beyond linearized elasticity to admit fracture. Likewise, slip bands or sheer bands, which form in the context of extreme plastic deformation are not covered by linearized elasticity. Well, you have to include plasticity theory for that, okay? So that's all this is saying. It's important, however. It's actually very important and we're going to see another result which brings out that importance. Okay, what I'm going to do next is actually give you a very special form of elasticity tensor, but one that is used very widely. Okay? So we are going to write out C for materials that are isotropic. Isotropic materials. Okay, of course, this is for isotropic materials, long as we want to create them with linearized elasticity. And of course, we talk on mechanical isotropy, right? Right, let me first write it in coordinate notation. Okay? In the context of linearized elasticity the requirement of material isotropy also translates to the fact and this can be proven that the material can be entirely specified elastically with just two constants. All right, and those two constants are what go into C So with that being stated, here is what C looks like for an isotropic material, okay? One of those constants is lambda, I'm going to denote it is lambda. So Cijkl = lambda times delta ij, delta kl. Can you guess what the delta ij refers to?. Kronecker deltas. Right, and of course this one too. Okay so we have that, plus, 2 mu times one-half (delta ik delta jl + delta il delta jk), all right? If you wonder why I put it in a 2 mu and then reintroduced the half, it is because what I have here is also often recognized as a fourth order tensor, okay? This is often denoted as sort of a bold faced I, with indices i, j, k, l, okay? And this is what we call the fourth order Symmetric identity tensor, okay? What it does is the following. If you take I as defined there, okay? And Feed to it any second order density, right? So let's use theta which we used in the previous slide, right? I remember theta just any old second order density. It doesn't need to be symmetric or anything, right? It turns out that I acting on theta spits back for us theta but the symmetric part of theta, all right. Which is since I'm using coordinate notation that is properly written as one-half theta ij + theta ji, okay. So it sort of gives us back theta but the symmetric part of it, right. So if theta were already symmetric, well it would just give us back the same tensor. However in general when we feed a non-symmetric tensor, a non-symmetric tensor, this identity gives us back just the symmetric part of that tensor, okay. All right, right, now in writing direct notations, so I'm going to use this I in writing the direct notation version of this relation. In writing the direct version notation of this relation delta ij the Kronecker delta I like to write as what I call a bold-phase one in this ij, okay? If you come to the world of matrices that's just your three by three identity matrix. Okay, all right, so direct notation from the same relation is the following C = lambda 1 dyadic 1, okay that's a dyadic product, okay. And to know what it is we'll just look at the direct notation, that's what it means, okay? I often call this the tensor product. + 2 mu times the 4th order identity, symmetric identity, okay? Note that because each of these tensors, each of these 1s is a second order tensor, they're dyadic, or the tensor product gives us a fourth order tensor. Okay, so this is direct notation. All right, what about these constants? Lambda and mu are what are called Lamé constants Okay, and if these seem unfamiliar to you, they will become familiar in just a few seconds. And here's how they're related, right. If now you use something that you're maybe more familiar with if E is the Young's modulus. And if nu is the Poisson ratio, Okay, then the relations are the following. The Lamé constants can be related to these guys. So lambda is E nu over 1 + nu times 1- 2 nu, okay? And mu is E divided by 2 of 1 + nu, okay. Chances are, the Young's modulus and Poisson ratio are something you encountered almost certainly in your second year of your study of mechanics or engineering, or something like that. Do you recognize what this is? Do you recognize what mu as defined as that also gives you? That should be a little more familiar if you recall these sort of relations. It is the sheer modulus, right? All of course for isotropic materials. Okay, lambda is not something that's related to shear rating, it's just often just called the Lamé parameter. Yet another modulus that you may be familiar with is kappa, which is the bulk modulus. Right, so kappa is related again to E and nu as E divided by 3 times 1- 2 nu, okay? All right, [COUGH] so there are limits on these constants and I'll talk about them just a bit. It emerges that nu can lie between -1, And one-half. All right, for the linearized theory of elasticity, okay? Do you recall what this limit Corresponds to. Well, this corresponds to elastic incompressibility. Okay, and as you may imagine from here, if you approach the limit mu = -1, you get a material that is sure unstable, all right? So actually well this theory properly I guess I, Just do that, okay? All right, otherwise linearized elasticity tends to break down in those limits. Let's just leave those actual limits out okay, right? Also from here you may be able to see why when you equals half is the limit of inconvisibility because it makes the bulk modules get unbounded. Okay so we have materials very, very stiff to complexion right like that okay. Fine, so let's get back to talking about, Positive definiteness, all right? So, having put down this form of the elasticity tensile for an isotropic material, The condition of positive definiteness, Okay? It translates to a requirement on the elastic constants that we were working with on the previous slide. Okay, in particular one can demonstrate the following, positive definiteness implies the following conditions. It implies that lambda + 2 mu has to be greater than 0, okay? Also implies that mu itself has to be greater than 0 okay? If you come from the world of structural dynamics or acoustics in elastic materials, these conditions may mean something to you, okay? Okay. And you recall it no? Okay. Well, let me see or maybe some of you do know. Well, the reason we said these are important conditions is the following. In elastic materials, waves propagate in at least two ways in the bulk of the material as longitudinal or shear waves. It emerges that the longitudinal wave speed is lambda + sorry that lambda looks bad. The longitudinal wave speed is lambda + 2 mu divided by the mass density under square root. Okay, so, when we see the lambda + 2 mu is than 0, what we are saying is that we have real wave speeds. It allows longitudinal waves to actually propagate through the material okay. And the shear wave speed is square root of mu divided by rho. Okay, so these conditions on lamba + 2mu and mu essentially allow us to have propagating longitudinal and shear waves. In the elastic material that we're describing with this linearized theory of elasticity. Okay. Let me see now. Now it turns that in most materials of interest the longitudinal wave speed is greater than the short wave speed. Okay, and that's a fact that is sometimes used in geophysics when people are dealing with earthquakes anyway. All right, that is, Probably the basic material that we need to know about our constitutive relations for linearized elasticity. And there is more, of course, that can be said but we don't really need to get into it, I think. Okay, so we're going to end the segment here. When we return we will work with the weak form.
